These are my live-TeXed notes for the course Math 268x: Pure Motives and Rigid Local Systems taught by Stefan Patrikis at Harvard, Spring 2014.

Any mistakes are the fault of the notetaker. Let me know if you notice any mistakes or have any comments!



Let $k$ be a field with algebraic closure $\bar k\subseteq \mathbb{C}$ (so $\Char(k)=0$). Consider smooth projective varieties $X$ over $k$ (either dropping the word smooth or projective will force us to enter the world of mixed, rather than pure, motives). There are several nice cohomology theory.

  • the Betti realization $H_B^*(X)=H_\mathrm{sing}^*(X_\mathbb{C}^\mathrm{an},\mathbb{Q})$ (a $\mathbb{Q}$-vector space), the singular cohomology of the topological space $X_\mathbb{C}^\mathrm{an}$.
  • the de Rham realization $H_\mathrm{dR}^*(X)=\mathbb{H}^*(X,\Omega_{X/k}^\cdot)$ (a $k$-vector space with a Hodge filtration), the algebraic de Rham cohomology = the hypercohomology of the sheaf $\Omega_{X/k}^\cdot$ of algebraic differential forms on $X$.
  • the $\ell$-adic realization $H_\ell^*(X)=H_\mathrm{et}^*(X_{\bar k}, \mathbb{Q}_\ell)$ (a $\mathbb{Q}_\ell$-vector space with $\Gal(\bar k/k)$-action), the $\ell$-adic etale cohomology.

It is not even clear a priori that these $\mathbb{Q}$-vector space, $k$-vector space and $\mathbb{Q}_\ell$-vector have the same dimension. But miraculously there are comparison isomorphisms between them. For example,

Theorem 1 (Comparison for B-dR) There are isomorphisms $$\alpha_{B,dR,X}: H_B^*(X) \otimes_\mathbb{Q} \mathbb{C} \cong H_{dR}^*(X/k) \otimes_k \mathbb{C}.$$

These isomorphisms are functorial and satisfy other nice properties (indeed an isomorphism of Weil cohomology, more on this later). This suggests that there is an underlying abelian category (of pure motives) that provides the comparison between different cohomology theory.

Slogan "sufficient geometric" pieces of cohomology have comparable meaning in all cohomology theory

We will spend a great amount of time on the foundation of all these different cohomology theory. But notice the comparison isomorphisms already suggest the various standard conjectures, for examples,

  • Standard conjecture D: numerical equivalence = cohomological equivalence
  • Standard conjecture C: Kunneth (the category of pure motives is graded and has a theory of weights).
  • Standard conjecture B: Lefschetz (the primitive cohomology should be "sufficiently geometric")
Question Why should one care about the existence of such a category?

One motivation is that one gets powerful heuristic for transferring the intuition between different cohomology theory.

Example 1 By the early 60's, one knew that if $X/\mathbb{C}$ is a smooth projective variety, then it follows from Hodge theory that $H_B^k(X)$ naturally carries a pure Hodge structure of weight $k$, i.e. a $\mathbb{Q}$-vector space $V$ with a bi-grading $V_\mathbb{C}=\bigoplus_{p+q=k}V^{p,q}$ such that $\overline{V^{p,q}}=V^{q,p}$, where $v\mapsto \bar v$ is the complex conjugation with respect to $V_\mathbb{R}$. On the other hand, Weil had conjectured that for a smooth projective variety $X/\mathbb{F}_q$. The $\Gal(\overline{\mathbb{F}_q}/\mathbb{F}_q)$-representation $H_\ell^k(X)$ is pure of weight $k$, in the sense that the eigenvalues $\alpha$ of the geometric Frobenius $\Frob_q\in \Gal(\overline{\mathbb{F}_q}/\mathbb{F}_q)$ are algebraic numbers and for each embedding $\iota$ of $\alpha$ into the complex numbers, $|\iota(\alpha)|=q^{k/2}$.

When $X/\mathbb{F}_q$ is smooth but not projective, people played with examples and found that $H_\ell^k(X)$ can be filtered (the weight filtration) such that the Frobenius eigenvalue is pure on each graded piece.

Example 2 Let $X$ be a smooth projective curve, $S\subseteq X$ be a finite set of points and $U=X\backslash S$. Then we have an exact sequence $$0\rightarrow H_\ell^1(X)\rightarrow H_\ell^1(U)\rightarrow H^2_S(X)\rightarrow H^2_\ell(X)\rightarrow H^2_\ell(U)=0.$$ Here $H_\ell^1(X)$ is pure of weight 1 and $H^2(X)=\mathbb{Q}_\ell(-1)$ ($=H_\ell^2(\mathbb{P}^1)$, the $\ell$-adic cyclotomic character) is pure of weight 2 and $H_S^2(X)=H^0(S)(-1)=\mathbb{Q}_\ell(-1)^{\#S}$ is also pure of weight 2. Therefore one obtains an increasing weight filtration on $H^1_\ell(U)$: $$W_0=0\subseteq W_1=H^1_\ell(X)\subseteq W_2=H^1_\ell(U).$$

The above mentioned $\ell$-adic intuition (generalized to higher dimension) lead Deligne to mixed Hodge theory. To give $H_B^*(X)$ a mixed Hodge structure for $X$ not smooth projective, the key point is to find a spectral sequence $E\Rightarrow H_B^*(X)$ such that its $E_2^{p,q}$ term is (conjecturally) pure of weight $p+2q$.

In Hodge II, Deligne treated the case of smooth but no longer projective varieties $U/k$. The ($\ell$-adic analogue of the) spectral sequence is the Leary spectral sequence for $j: U\hookrightarrow X$, where $X$ a smooth compactification of $U$ with $X\backslash U=\bigcup_{i\in I}D_i$ is a union of smooth divisors with normal crossings, $$E_2^{p,q}=H^p(X_{\bar k}, R^qj_* \mathbb{Q}_\ell)\Rightarrow H^{p+q}(U_{\bar k}, \mathbb{Q}_\ell).$$ One can explicitly compute the sheaf $$R^qj_*\mathbb{Q}_\ell=\bigoplus_{Q\subseteq I, |Q|=q} \mathbb{Q}_\ell(-q)_{D_Q},$$ where $D_Q=\bigcap_{i\in Q}D_i$ is smooth. Therefore $E_2^{p,q}=\bigoplus_{|Q|=q} H^p(D_Q,\mathbb{Q}_\ell)(-q)$ is pure of weight $p+2q$.

Let us look at the differential $E_2^{p,q}\xrightarrow{d_2} E_2^{p+2,q-1}$: notice both the target and the source are pure of weight $p+2q$ (all $E_r^{p,q}$ are pure of weight $p+2q$), nothing is weired. But on the $E_3$-page, $E_3^{p,q}\xrightarrow{d_3}E_3^{p+3,q-2}$, where the source has weight $p+2q$ and the target has weight $p+2q-1$ respectively. The mismatching of the weight of the Frobenius eigenvalues implies that $d_r^{p,q}=0$ for $r\ge3$. Therefore the Leray spectral sequence degenerates at $E_3$-page. One can compute that $$E_3^{p,q}=\ker\left(\bigoplus_{|Q|=q}H^p(D_Q,\mathbb{Q}_\ell)(-q)\rightarrow \bigoplus_{|Q'|=q-1}H^{p+2}(D_{Q'}, \mathbb{Q}_\ell)(1-q)\right)\Big/\im.$$ The Betti analogue (of maps of pure Hodge structure) is provided by the reinterpretation that $D_Q\hookrightarrow D_{Q'}$ and the differentials $d_2^{p,q}$'s are simply Gysin maps ( = Poincare dual to pullbacks), which are also maps of pure Hodge structures.

The upshot is that the $\ell$-adic Leray spectral sequence gives the weight filtration (= the Leary filtration up to shift), and the graded piece $$\Gr^W_nH^k(U)=E_3^{2k-n,n-k}$$ is pure of weight $n$. The Betti Leray sequence also gives a weight (defined to be) filtration on $H_B^k(U)$ such that we already know that $E_\infty^{p,q}$ are naturally pure Hodge structures.

Another motivation for considering the category of pure motives is toward a motivic Galois formalism.

Example 3 Let $k$ be a field. The classical Galois theory establishes an equivalence between finite etale $k$-schemes with finite sets with $\Gamma_k=\Gal(\bar k/k)$-actions. Linearizing a finite set $S$ with $\Gamma_k$-action gives finite dimensional $\mathbb{Q}$-vector spaces $\mathbb{Q}^{\#S}$ with the continuous $\Gamma_k$-action $gf(s)=f(g^{-1}s)$. The linearization of finite etale $k$-schemes are the Artin motives (motives built out of zero dimension motives). The equivalence between the two linearized categories is then given by $H^0_\ell(-)$.

Generalizing to higher dimension: the category of finite etale $k$-schemes is extended to the category of pure homological motives. The Standard conjectures then predict that it is equivalent to the category of representations of a certain group $G_k$, which is a extension of classical Galois theory $$1\rightarrow G_{\bar k}\rightarrow G_k\rightarrow \Gamma_k\rightarrow1.$$ These are still conjectural. But one can replace the category of pure homological motives by something closely related and obtain unconditional results. In this course we will talk about one application of Katz's theory of rigid local systems (these are topological gadgets but surprisingly produce motivic examples): to construct the exceptional $G_2$ as a quotient of $G_\mathbb{Q}$ (the recent work of Dettweiler-Reiter and Yun).


TopWeil cohomology

We now formulate the notion of Weil cohomology, in the frame work of motives.

Definition 1 Let $k\subseteq \mathbb{C}$ be a field. Let $X/k$ be smooth projective (not assumed to be connected) variety over $k$. Let $\mathcal{P}(k)$ be the category of such varieties. Then $\mathcal{P}(k)$ is a symmetric monoidal via the fiber product $X\times_{\Spec k} Y$ with the obvious associative and commutative constraints and the unit $\Spec k$.
Definition 2 Let $E$ be a field. Let $\Gr_E^{\ge0}$ be the category of finite dimensional graded $E$-vector spaces in degrees $\ge0$ with the usual tensor operation $$(V \otimes W)^n=\bigoplus_{i+j=k}V^i \otimes W^j.$$ It is endowed with a graded commutative constraint via $$c_{V,W}: V\otimes W\cong W \otimes V,\quad v \otimes w\mapsto (-1)^{\deg v\deg w} w \otimes v.$$
Definition 3 A Weil cohomology over $E$ (a field of characteristic 0) on $\mathcal{P}(k)$ is a tensor functor $H^*:\mathcal{P}(k)^\mathrm{op}\rightarrow \Gr_E^{\ge0}$, namely, $H^*$ comes with a functorial (Kunneth) isomorphisms $$\mathcal{K}_{X,Y}: H^*(X) \otimes H^*(Y)\cong H^*(X\times Y)$$ respecting the symmetric monoidal structure. Notice the monoidal structure induces a cup product $$\cup: H^*(X) \otimes H^*(X)\xrightarrow{\mathcal{K}_{X,X}}H^*(X\times X)\xrightarrow{\Delta^*}H^*(X)$$ making $H^*(X)$ a graded commutative $E$-algebra. We require it to satisfy the following axioms.
  1. (normalization) $\dim_E(H^2(\mathbb{P}^1))=1$. In particular, $H^2(\mathbb{P}^1)$ is invertible in $\Gr_E$. We define the Tate twists $V(r):= V \otimes (H^2(\mathbb{P}^1))^{\otimes -r}$ (this is well motivated by $\ell$-adic cohomology).
  2. (trace axiom) For any $X$ of (equi-)dimension $d$, there is a trace map $$\tr_X: H^{2d}(X)(d)\rightarrow E$$ satisfying
    1. Under $\mathcal{K}_{X,Y}$, one has $\tr_{X\times Y}=\tr_X\cdot \tr_Y$.
    2. $\tr_X$ and the cup product induces a perfect duality (Poincare duality) $$H^i(X)\times H^{2d-i}(X)(d)\rightarrow H^{2d}(d)\xrightarrow{\tr_X} E.$$
  3. (cycle class maps) Let $\mathcal{Z}^r(X)$ be the $\mathbb{Q}$-vector spaces with a basis consisting of integral closed schemes $Z\hookrightarrow X$ of codimension $r$. Then there are cycle class maps $$\gamma_X^r: \mathcal{Z}^r(X)\rightarrow H^{2r}(X)(r)$$ satisfying
    1. $\gamma_X^r$ factors through the Chow group $\CH^r(X)$ (modulo the rational equivalence).
    2. $\gamma_X^r$ is contravariant in $X$, i.e., for a morphism $X\xrightarrow {f}Y$ and a cycle $Z\hookrightarrow X$ of codimension $r$, we have $$f^*\gamma_Y^r(Z)=\gamma_X^r([f^{-1}Z])$$ whenever this makes sense. This will always make sense after passing to the Chow group. In general, one cannot always define $f^{-1}Z$ on $Z^r(Y)$. But if $f$ is flat , then one can: in fact, by flatness $f^{-1}Z$ has all its components of codimension $r$ in $X$ (but $f^{-1}Z$ is not necessarily integral). Let $W_i$ be the (reduced structure) of the irreducible components. One then associates a cycle $$[f^{-1}Z]:=\sum n_i W_i,$$ where $n_i=\Lg\mathcal{O}_{f^{-1}Z,W_i}$, the length of the local ring at $W_i$. We also require it to be compatible with pushforward (defined in Definition 10) that $$f_*\gamma_X=\gamma_Y f_*.$$
    3. For $\alpha\in \mathcal{Z}^r(X)$, $\beta\in \mathcal{Z}^s(Y)$. $$\gamma_X^r(\alpha) \otimes \gamma_Y^s(\beta)=\gamma_{X\times Y}^{r+s}(\alpha \times \beta).$$ Notice $\alpha\times\beta$ is not necessarily a combination of integral closed subscheme (e.g., $\mathbb{F}_p(\sqrt[p]{t}) \otimes_{\mathbb{F}_p} \mathbb{F}_p(t)\cong\mathbb{F}_p[X]/X^p$ is nonreduced), the cycle $\alpha\times\beta$ should be understood as the reduced structure with multiplicity.
    4. (pinning down the trace) the composite $$Z^d(X)\xrightarrow{\gamma_X^d}H^{2d}(X)(d)\xrightarrow{\tr_X}E.$$ sends $\sum n_i P_i$ to $\sum n_i [k(P_i):k]$, where $P_i$ are closed points.
Remark 1 Sometimes the Lefschetz axiom is also thrown in the definition of Weil cohomology. We will talk about this later.
Remark 2 We set $H_B^2(\mathbb{P}^1)=\frac{1}{2\pi i} \mathbb{Q}$. This means via the comparison isomorphism, $$H^2_B(\mathbb{P}^1) \otimes_\mathbb{Q} \mathbb{C} \cong H^2_{dR}(\mathbb{P}^1/\mathbb{Q}) \otimes_\mathbb{Q} \mathbb{C},$$ the image of $H^2_B(\mathbb{P}^1)$ is $\frac{1}{2\pi i} H^2_{dR}(\mathbb{P}^1/\mathbb{Q})$. If we take granted that the comparison isomorphisms are compatible with Mayer-Vietoris. Applying Mayer-Vietoris to $\mathbb{P}^1=(\mathbb{P}^1-0)\cup (\mathbb{P}^1-\infty)$, then we are reduced to the calculation on $\mathbb{P}^1-\{0,\infty\}$ ($H^1(\mathbb{A}^1)=0$). The isomorphism $$H^1_{dR}(\mathbb{P}^1-\{0,\infty\})\rightarrow H^1_B(\mathbb{P}^1-\{0,\infty\})$$ is given by $$\omega\mapsto (\sigma\mapsto\int_\sigma \omega),$$ here $\sigma$ is a smooth 1-chain. A good $\mathbb{Q}$-basis of $H^1_{dR}(\mathbb{P}^1-\{0,\infty\})$ is given by the differential $dz/z$. Choosing a simple loop $\sigma$ around the origin, then one obtains $\int_{\sigma}dz/z=2\pi i$.
Example 4 (Trace in Betti cohomology) Let $X/\mathbb{C}$ be smooth projective of dimension $d$. Define $\tr_X$ to be the composite $$\scriptstyle H^{2d}_B(X)(d)\xrightarrow{\frac{1}{(2\pi i)^d}} H^{2d}_B(X)\xrightarrow[\alpha]{\sim} H^{2d}(X, \underline{\mathbb{Q}})\rightarrow H^{2d}(X,\underline{\mathbb{Q}}) \otimes \mathbb{R} \xrightarrow[\beta]{\sim} H^{2d}(X,\mathcal{A}_X^\cdot)\xrightarrow{\int_X}\mathbb{R},$$ where $\mathcal{A}_X^\cdot$ is the smooth de Rham complex with $\mathbb{R}$-coefficients. Notice $\alpha$ is an isomorphism because the sheafy singular cochain complex is a flasque resolution of $\underline{\mathbb{Q}}$ and $\beta$ is an isomorphism because $\mathcal{A}_X^\cdot$ is a fine resolution of $\underline{\mathbb{R}}$. The choice $i=\sqrt{-1}$ will chancel out the choice of the orientation we made on complex manifold when we do integration and one can check that $H^{2d}_B(X)(d)\rightarrow \mathbb{R}$ lands in $\mathbb{Q}$.

TopAlgebraic de Rham cohomology

Suppose $k$ is a field of characteristic 0 and $X/k$ smooth (not necessarily projective).

Definition 4 We define the algebraic de Rham cohomology $H^*_{dR}(X/k)=\mathbb{H}^*(X,\Omega_{X/k}^\cdot)$ to be the hypercohomology of the sheaf $\Omega_{X/k}^\cdot$ of algebraic differential forms on $X$.
Theorem 2 $H^*_{dR}: \mathcal{P}(k)^\mathrm{op}\rightarrow \Gr_k^{\ge0}$ is a Weil cohomology.

We now selectively check a few of the axioms.

Lemma 1 $H^*_{dR}$ is a tensor functor.
Proof Given a morphism $f:X\rightarrow Y$. Let $I^\cdot$ be an injective resolution of $\Omega_Y^\cdot$ and $J^\cdot$ be an injective resolution of $\Omega_X^\cdot$. Then $f^{-1}I^\cdot$ is quasi-isomorphic to $f^{-1}\Omega_Y^\cdot$. The map $\Gamma(Y, I^\cdot)\rightarrow \Gamma(X,f^{-1}I^\cdot)\rightarrow\Gamma(X,J^\cdot)$ induces the pull-back on $H^*_{dR}$ $$f^*: H^*(Y, \Omega_Y^\cdot)\rightarrow H^*(X, f^{-1}\Omega_Y^\cdot)\rightarrow H^*(X,\Omega_X^\cdot).$$

The Kunneth isomorphism $\mathcal{K}_{X,Y}$ is explicitly given by $(\alpha, \beta)\mapsto p^*\alpha\cdot q^*\beta$, where $p: X\times Y\rightarrow X$ and $q: X\times Y\rightarrow Y$ are the natural projections.

Remark 3 To check that $H^*_{dR}(X/k)$ is finite dimensional for $X/k$ smooth projective, one can use the Hodge to de Rham spectral sequence $$E_1^{p,q}=H^q(X,\Omega^p)\Rightarrow H^{p+q}_{dR}(X/k)$$ and the fact that each $H^q(X,\Omega^p)$ is finite dimensional (this may fail when $X$ is not projective) and $(p,q)$ lives in a bounded region. For not necessarily projective or smooth varieties, $H^*_{dR}$ is still finite dimensional (by comparison, and by the resolution of singularities in characteristic 0). $H^*_{dR}(X/k)$ may fail to be finite dimensional in characteristic $p$ for nonprojective varieties (see the next example).
Example 5 When $U$ is affine, we have $H^*_{dR}(U)=H^*(\Gamma(U,\Omega_U^\cdot))$ (this follows from the vanishing of $H^i(U,\mathcal{F})$ for $U$ affine and $\mathcal{F}$ quasi-coherent; in particular, ), which makes the computation feasible. For example, the de Rham complex for $\mathbb{A}^1$ is simply $k[T]\xrightarrow{d}k[T]dT$. So taking cohomology gives $H^0_{dR}(\mathbb{A}^1)=k$ and $H^1_{dR}(\mathbb{A}^1)=0$. This also gives an example in characteristic $p$ that $H^0_{dR}(\mathbb{A}^1)$ is infinite dimensional because $d(T^p)=0$, so one don't really want to work with the algebraic de Rham cohomology in characteristic $p$!


In general, one covers $X$ by open affines $\mathcal{U}=(U_i)_{i\in I}$. For any quasi-coherent sheaf $\mathcal{F}$ on $X$, one then obtains the Cech complex $\mathcal{C}^.\langle\mathcal{U},\mathcal{F})$, a resolution of $\mathcal{F}$ by acyclic sheaves, defined by $$\mathcal{C}^q(\mathcal{U},\mathcal{F})(U)=\bigoplus_{|J|=q+1}\Gamma(U_J\cap U,\mathcal{F}).$$ Now we have a double complex $$\xymatrix{\mathcal{C}^\cdot(\mathcal{U},\mathcal{O}_X) \ar[r] & \mathcal{C}^\cdot(\mathcal{U},\Omega_X^1) \ar[r] & \mathcal{C}^\cdot(\mathcal{U},\Omega_X^2) \ar[r]  &\cdots\\ \mathcal{O}_X \ar[r] \ar[u]^{\sim} & \Omega_X^1 \ar[r] \ar[u]^{\sim}& \Omega_X^2 \ar[r] \ar[u]^{\sim}& \cdots}$$ whose columns are acyclic resolutions of $\Omega_X^p$. The general formalism implies that $$\mathbb{H}^*(X,\Omega_X^\cdot)=H^*(\Gamma(X,\mathrm{Tot}^\cdot)),$$ the cohomology of the global sections of the total complex. Recall the total complex is defined by $$\mathrm{Tot}^n=\bigoplus_{p+q=n}D^{p,q},$$ where $D^{p,q}=\mathcal{C}^q(\mathcal{U},\Omega_X^p)$.

Example 6 (normalization) Let $U_0=\mathbb{P}^1-\infty$ and $U_1=\mathbb{P}^1-0$ be a covering of $\mathbb{P}^1$. The Cech double complex looks like $$\xymatrix{ \mathcal{O}(U_0\cap U_1)\ar[r] & \Omega^1(U_0\cap U_1)\\ \mathcal{O}(U_0)\oplus\mathcal{O}(U_1) \ar[r] \ar[u] & \Omega^1(U_0)\oplus\Omega^1(U_1) \ar[u]}$$ The total complex is thus $$\mathcal{O}(U_0)\oplus \mathcal{O}(U_1)\rightarrow \Omega^1(U_0)\oplus\Omega^1(U_1)\oplus\mathcal{O}(U_0\cap U_1)\rightarrow \Omega^1(U_0\cap U_1)$$ where the two differentials are given by $$(a_0,a_1)\rightarrow(da_0,da_1,-a_0+a_1)$$ and $$(\phi,\psi,\alpha)\mapsto d\alpha-\phi+\psi.$$ One can easily compute $H^0_{dR}=k$, $H^1_{dR}=0$ and $H^2_{dR}$ is 1-dimensional generated by $dt/t\in \Omega^1(\mathbb{P}^1-\{0,\infty\})$. Indeed one sees the computation really shows $$H^0_{dR}(\mathbb{P}^1)=H^0(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1}),\quad H^1_{dR}(\mathbb{P}^1)=H^1(\mathbb{P}^1,\Omega_{\mathbb{P}^1}^1).$$ This is an instance of the Hodge to de Rham spectral sequence.
Remark 4 The Hodge de Rham spectral sequence comes from filtering the total complex of the double complex $D^{p,q}$. Take the filtration (cut out by right half planes) $$F^p\mathrm{Tot}^\cdot=\bigoplus_{a\ge p} D^{a,\cdot-a}.$$ So $\Gr^p\mathrm{Tot}^\cdot=D^{p,\cdot-p}$ itself forms a complex. The general machinery implies that the spectral sequence associated to this filtered complex is $$E_1^{p,q}=H^{p+q}(\Gr^p\mathrm{Tot}^\cdot)\Rightarrow E_\infty^{p,q}=\Gr^p(H^{p+q}\mathrm{Tot}^\cdot)$$ Here $$\im(H^n(F^p\mathrm{Tot}^\cdot)\rightarrow H^n(\mathrm{Tot}^\cdot))$$ defines the filtration on $H^n(\mathrm{Tot}^\cdot)$ (so the grading on the right hand side makes sense). Notice in our case, $H^n(F^p\mathrm{Tot}^\cdot)$ is simply $\mathbb{H}^n(X,\Omega_X^{\cdot\ge p})$ and $E_1^{p,q}=H^{p+q}(D^{p,\cdot-p})$ is nothing but $H^q(X,\Omega^p)$.
Remark 5 What happens when one filter the total complex in the other way (by upper half planes)? Namely $$F'^p\Tot^\cdot=\bigoplus_{a\ge p}D^{\cdot-a,a}.$$ Then we see that $E'^{p,q}_1=\bigoplus_{|J|=q+1} H_{dR}(U_J)$. The corresponding sequence is the Mayer-Vietoris spectral sequence. When the covering $\mathcal{U}=\{U_0,U_1\}$ consists of two affines, it recovers the usual Mayer-Vietoris long exact sequence.
Theorem 3 If $\Char(k)=0$ and $X/k$ is projective, then the Hodge to de Rham spectral sequence degenerates at the $E_1$-page.
Remark 6 It is not clear how to obtain the splitting without Hodge theory.

To define the trace for the algebraic de Rham cohomology, we proceed in two steps. We first show that $H^{2d}_{dR}(X)(d)$ is abstractly the right thing, i.e., $H^{2d}_{dR}(X)(d)\cong H^0(X,\mathcal{O}_X)$ and is $k$ if $X$ is geometrically connected. Then we pin down that actual map $\Tr_X: H^{2d}_{dR}(X)(d)\rightarrow k$ after defining the de Rham cycle class map.

The first step uses the Serre duality. By the Hodge to de Rham spectral sequence, we have a map $$H^d(X,\Omega_X^d)\rightarrow H^{2d}_{dR}(X).$$ By Serre duality ($\Omega_X^d$ is the dualizing sheaf), one has the trace map $$H^d(X,\Omega_X^d)\rightarrow k.$$ So we want to say that the map $H^d(X,\Omega_X^d)\rightarrow H^{2d}_{dR}(X)$ is an isomorphism. By the Hodge de Rham spectral sequence, it is enough to show that $d_1^{d-1,d}=0$ (or $H^{2d}_{dR}(X)\ne0$ because $H^d(X,\Omega_X^d)$ is a free $H^0(X,\mathcal{O}_X)$-module of rank 1). This can be checked bare-handed by reduction to $X=\mathbb{P}^d$: choose a finite flat map $f: X\rightarrow \mathbb{P}^d$ to get the trace map $\tr_f: H_{dR}^*(X)\rightarrow H_{dR}^*(\mathbb{P}^d)$ (so $\tr_f\circ f^*=\deg f$). It follows that $f^*: H_{dR}^{2d}(\mathbb{P}^d)\rightarrow H^{2d}_{dR}(X)$ is injective. It suffices to prove that $H^{2d}_{dR}(\mathbb{P}^d)\ne0$ which boils down to the direct computation that $H^d(\mathbb{P}^d,\Omega^d)=k$ using the Hodge to de Rham spectral sequence.

To pin down $\tr_X$, we need to choose carefully a generator $u_X$ of $H^{2d}_{dR}(X)$ and set $$\tr_X(c\cdot u_X)=\frac{\tr_{H^0(X,\mathcal{O}_X)/k}(c)}{[H^0(X,\mathcal{O}_X):k]},$$ i.e., $\tr_X(u_X)=1$.

TopDe Rham cycle class maps (via Chern characters)

We seek a cycle class map such that $\tr_X\gamma_X^d(P)=[k(P):k]$ as follows.

  1. Define the Chern class $c_1$ of line bundles.
  2. Define the Chern class $c^{dR}$ of vector bundles.
  3. Define the Chern character $\ch^{dR}$ of vector bundles on $X$.
  4. One knows that $\ch^{dR}:\mathrm{Vect}_X\rightarrow\bigoplus H^{2i}(X)$ factors through $K^0(X)$, Grothendieck group of vector bundles on $X$. Using the fact that $X$ is smooth, the latter can be identified with $K_0(X)$ , the Grothendieck group of coherent sheaves on $X$.
  5. For $Z\hookrightarrow X$ a codimension $p$ cycle, $\ch^{dR}(\mathcal{O}_Z)$ makes sense and we define the cycle class map $$\gamma^p_X(Z)=\ch^{dR}_p(\mathcal{O}_Z).$$

Now we describe each step in details.

Step a We want a group homomorphism $c_1:\Pic(X)\rightarrow H^2_{dR}(X)(1)$. Identify $\Pic(X)=H^1(X,\mathcal{O}_X^\times)$. The map $$d\log : \mathcal{O}_X^\times\rightarrow \Omega_X^1,\quad f\mapsto df/f$$ induces a map $\mathcal{O}_X^\times\rightarrow \Omega_X^\cdot[1 ]$ and hence induces the desired map $H^1(X,\mathcal{O}_X^\times)\rightarrow H_{dR}^2(X)$.

Step b Let $E$ be a vector bundle of rank $r$ on $X$. Denote the projective bundle $f: \mathbb{P}(E)\rightarrow X$. Notice $f^*E$ on $\mathbb{P}(E)$ has a tautological line subbundle $L_E^\vee\subseteq f^*E$. Let $c=c_1(L_E)\in H^2_{dR}(\mathbb{P}(E))$. The fact (the Leray-Hirsch theorem, a special case of the Leray spectral sequence) is that $H_{dR}^*(\mathbb{P}(E))$ is a free module over $H_{dR}^*(X)$ with basis $1,c,c^2,\cdots c^{r-1}$. We then define $c_j^{dR}(E)\in H^{2j}_{dR}(X)$ by $$c^r+\sum_{j=1}^r c_j^{dR}(E)\cdot c^{r-j}=0.$$ Notice this agrees with the previous definition of $c_1$ of line bundles and is functorial in $X$.


Define the total Chern class $$c^{dR}(E)=\sum_{j=1}^r c_j^{dR}(E).$$ The key is the following multiplicative property.

Proposition 1 For any sort exact sequence of vector bundles $$0\rightarrow E_1\rightarrow E\rightarrow E_2\rightarrow 0,$$ we have $c(E)=c(E_1)\cdot c(E_2)$.
Proof To show this, one first show that if $E=\oplus L_i$ is a direct sum of line bundles, then $$c(E)=\prod(1+c_1(L_i)).$$ Then reduce the general to the first case by showing that there exists a map $f: Y\rightarrow X$ such that $f^*(0\rightarrow E_1\rightarrow E\rightarrow E_2\rightarrow0)$ splits as direct sum of line bundles and $f^*$ is injective on $H^*_{dR}$ (the splitting principle).

For the first case, since the statement is invariant under twist, one can assume each $L_i$ is very ample of the form $f^*_i(\mathcal{O}_{\mathbb{P}^{d_i}}(1))$ and reduce to the case to the case of $X$ being a product of projective spaces. Notice that each $L_i$ gives a section $s_i: X=\mathbb{P}(L_i)\subseteq \mathbb{P}(E)$ and by definition $s_i^* L_E^\vee=L_i$. Write $x_i=c_1(L_i)$. Pullback the defining relation for $c_j(E)$ along each $s_i$, we obtain the relation $$(-x_i)^r+\sum c_j(E)(-x_i)^{r-j}=0.$$ So the polynomial $$(-t)^r+\sum c_j(E) (-t)^{j}=0$$ in $H^*(X)[t]$ has roots $x_1,\ldots, x_r$. But $H^*(X)[t]$ has the advantage of being like a polynomial ring, $$H^*\left(\prod_{i=1}^r \mathbb{P}^{d_i}\right)\cong k[x_1,\ldots,x_r]/(x_1^{d_i+1},\ldots, x_r^{d_r+1}).$$ Assume $d_i\gg0$, then the defining relation must be $\prod_{i=1}^r(x_i-t)$, which shows that $c_j(E)$ is the $j$-th symmetric polynomial of $x_i $ as desired.

To reduce the general case to the first case, arrange $g:Y\rightarrow X$ so that $g^*E$ has a full flag of subbundles by iterating the projective bundle construction, then split the extension by further pullback: if one has a surjection $E\rightarrow E'$ of vector bundles, then the sections $E'\rightarrow E$ form an affine bundle over $X$; pulling back along this affine bundle splits $E\rightarrow E'$ and induces an isomorphism on cohomology.

Step c Using the multiplicative property, we can define formally the Chern roots $x_i $ of $E$ so that $c(E)=\prod_{i=1}^r(1+x_i)$. Here the Chern roots don't not make sense but their the symmetric polynomials do make sense in cohomology. Define the Chern character $$\ch(E)=\sum_{i=1}^re^{x_i}.$$ This makes sense in cohomology. Now we have the additivity $$\ch(E)=\ch(E_1)+\ch(E_2)$$ in exact sequences. Moreover $\ch(E_1 \otimes E_2)=\ch(E_1)\ch(E_2)$. Therefore we obtain a ring homomorphism $$\ch: K^0(X)\rightarrow H^{2*}(X)({*}).$$

When $X$ is smooth, one can form finite locally free resolutions of any coherent sheaves on $X$, and taking the alternating sum of the terms in the resolutions induces the inverse of natural map $K^0(X)\rightarrow K_0(X)$. Thus $K^0(X)\cong K_0(X)$ (see Hartshorne, Ex III.6.8).

Step d For $Z\hookrightarrow X$ a codimension $p$ cycle, $\ch^{dR}(\mathcal{O}_Z)$ makes sense and we define the cycle class map $$\gamma^p_X(Z)=\ch^{dR}_p(\mathcal{O}_Z)\in H_{dR}^{2p}(X)(p).$$ In particular, our choice of the basis $u_X$ for $H_{dR}^{2d}(X)$ is given by for any closed point $P$ of $X$, $$u_X=\frac{\gamma_X^d(P)}{[k(P):k]}=\frac{\ch_d(k(P))}{[k(P):k]}.$$ This is the choice we made to normalize the trace map. We need to check that $u_X$ is independent on the choice of $P$ (this follows from connecting two points by a curve in $X$ and the invariance of $\ch$ in a flat family). We also need to check that $u_X\ne0$. This reduce to the case of projective spaces. Let $P\in \mathbb{P}^d$ be a closed point. One can put $P$ in a chain $$P\subseteq \mathbb{P}^1\subseteq \mathbb{P}^2\subseteq \cdots \subseteq\mathbb{P}^d.$$ Using the short exact sequences of the form (given a choice of a section of $\mathcal{O}_{\mathbb{P}^n}(1)$), $$0\rightarrow \mathcal{O}_{\mathbb{P}^n}(-1)\rightarrow \mathcal{O}_{\mathbb{P}^n}\rightarrow\mathcal{O}_{\mathbb{P}^{n-1}}\rightarrow 0,$$ for each $n$, it follows that in $K_0(X)$, we have $$k(P)=\sum_{i=0}^d(-1)^i{d \choose i}\mathcal{O}_{\mathbb{P}^d}(-i).$$ Applying the Chern character we obtain that for $x=c_1(\mathcal{O}_\mathbb{P}^1(1))$, $$\ch(k(P))=\sum (-1)^i{d\choose i}e^{-ix}=(1-e^{-x})^d=x^d\in H^{2d}(\mathbb{P}^{d}),$$ which is nonzero.

TopFormalism of cohomological correspondences

Let $H^*: \mathcal{P}(k)^\mathrm{op}\rightarrow \Gr_E^{\ge0}$ be a Weil cohomology.

Definition 5 Given a morphism $f:X\rightarrow Y$, we define the Gysin map $$f_*:H^*(X)\rightarrow H^{*-2(d_X-d_Y)}(Y)(-d_X+d_Y)$$ to be the transpose of $f^*$ under the Poincare duality. At the level of cycles, $f_*(Z)$ is basically $f(Z)$ when $\dim f(Z)=\dim Z$ and zero otherwise (this matches the degree shift in $f_*$).
Proposition 2 (Projection formula) Let $\alpha\in H^*(X)$, $\beta\in H^*(Y)$, then $f_*(f^*\alpha\cdot \beta)=\alpha\cdot f_*\beta$.
Proof The property $\tr_Y(f_*\alpha\cdot\beta)=\tr_X(\alpha\cdot f^*\beta)$ characterizes $f_*$.
Remark 7 Using Gysin maps, one has an alternative construction of cycle class maps. For a smooth cycle $i: Z\hookrightarrow X$ of codimension $p$, we define $$\gamma_X^p(Z)=i_*[Z],$$ where $[Z]$ is "1" in $H^0(Z)$. This can be extended to non-smooth cycles by a resolution $\tau: \tilde Z\rightarrow Z\hookrightarrow X$ and defining $\gamma_X^p(Z)=\tau_*[\tilde Z]$.
Definition 6 A cohomological correspondence from $X$ to $Y$ is an element $u\in H^*(X\times Y)$ interpreted (using the Poincare duality and the Kunneth formula) as a linear map $H^*(X)\xrightarrow{u_*} H^*(Y)$. Explicitly, if $u=a \otimes b\in H^*(X) \otimes H^*(Y)$, then $u_*(c)=\tr_X(c\cdot a)b$ (extended to be zero away from top degree). Let $p:X\times Y\rightarrow X$, $q:X\times Y\rightarrow Y$ be the natural projections. Then another way of writing $u_*$ is $$u_*(c)=q_*(p^*(c)\cdot u).$$ Namely, pullback $c$, intersect with $u$, then pushforward to $Y$. One can check that $u_*(c)=q_*(p^*c \cdot p^*a\cdot q^*b)=\tr_X(c\cdot a)b$ by the projection formula.
Definition 7 Define $u^*: H^*(Y)\rightarrow H^*(X)$, $b\mapsto p_*(u\cdot q^*b)$.
Definition 8 The transpose $^tu$ of $u$ is defined to be the image of $u$ in $H^*(Y\times X)$ under $\text{swap}^*$. One can check that $u^*=(^{t}u)_*$.
Definition 9 (Composition of correspondences) For $u\in H^*(X\times Y)$, $v\in H^*(Y\times Z)$, we define $v\circ u=p_{XZ,*}(p_{XY}^*(u)\cdot p_{YZ}^*(v))$.
Lemma 2 $(v\circ u)_*=v_*\circ u_*: H^*(X)\rightarrow H^*(Z)$.
Proof Notice that $u_*(a)=u\circ a$. The claim then follows from the associativity of composition of correspondences. For details, see Fulton, Intersection theory, Chapter 16.
Lemma 3 Let $\Gamma_f$ be the graph of the morphism $f: X\rightarrow Y$. Then $(\Gamma_f)^*=f^*$, $(\Gamma_f)_*=f_*$, $(^t\Gamma_f)^*=f_*$.
Remark 8 Everything makes sense in Chow groups too. We will not repeat it later.

Our next goal is to deduce the Weil conjecture (except the Riemann hypothesis) from a Weil cohomology (hence the name). We will later see that the Riemann hypothesis follows from the standard conjectures.


TopFormal consequences of a Weil cohomology

Let $H^*: \mathcal{P}(k)^\mathrm{op}\rightarrow \Gr_E^{\ge0}$ be a Weil cohomology.

Proposition 3 (Lefschetz fixed point) Suppose $k$ is algebraically closed. Let $X, Y\in \mathcal{P}(k)$ be connected. If $v\in H^*(X\times Y)$, $w\in H^*(Y\times X)$ are of degree $+r$ and $-r$ respectively (namely, $v\in H^{2d_X+r}(X\times Y)$ and induces $v_*: H^k(X)\rightarrow H^{k+r}(Y)$; similar for $w$). Then $$\tr_{X\times Y}(v\cdot ^tw)=\sum_{i=0}^{2d_X}(-1)^i\tr((w\circ v)_*| H^i(X)).$$
Proof We compute by each Kunneth component so let $v\in H^{2d_X-i}(X)\otimes H^j(Y)$, $w\in H^{2d_Y-j}(Y)\otimes H^i(X)$. Let $\{a_l\}$ be a basis of $H^i(X)$ and let $\{a_l'\}$ be a dual basis of $H^{2d_X-i}(X)$ such that $\tr_{X}(a_l' a_m)=\delta_{l,m}$. So we can write $$v=\sum_l a_l' \otimes b_l,\quad w=\sum_k c_k \otimes a_k.$$ Here $b_l,c_l\in H^j(Y). $So the left hand side is equal to 
  &\tr_{X\times Y}(\sum a_l' \otimes b_l\cdot \sum a_k \otimes c_k (-1)^{\deg a\deg c}) \\ &=\tr_{X\times Y}(\sum p^*a_l'\cdot q^*b_l\cdot p^* a_k\cdot q^*c_k (-1)^{\deg a\deg c}).
Switching $q^* b_l$ and $p^* a_k$ introduces another sign $(-1)^{\deg b\deg a}$ which cancels out the sign $(-1)^{\deg a\deg c}$ since $\deg b+\deg c=2d_Y$. So the left hand side is equal to $$\tr_{X^\times Y} (\sum p^*(a_l'a_k)\cdot q^*(b_lc_k))=\sum_l \tr_Y(b_l c_l).$$ To compute the trace on the right hand side, we notice that $$w_*v_*(a_l)=w_*(\sum_k \tr_X(a_l a_k') b_k)=(-1)^i\sum_k \tr_Y(b_kc_k)a_l.$$ Since we care only about the $k=l$-term when taking the trace, this matches the left hand side.

Let $\Delta$ be a cohomological correspondence so that $\Delta_{X,*}=\Id$ on $H^*(X)$. Write $$\Delta_{X,*}=\sum_{i=0}^{2d_X}\Pi_X^i,$$ where $\Pi_X^i$ is the cohomological correspondence $H^*(X)\twoheadrightarrow H^i(X)\hookrightarrow H^*(X)$. So $\Pi_X^i\in H^{2d_X-i}(X) \otimes H^i(X)$.

Corollary 1 Let $u\in H^{2d_X}(X\times X)$ (so $u$ is of degree zero). Then $$\tr_{X\times X}(u\cdot \Delta)=\sum_{i=0}^{2d_X}(-1)^i\tr(u_*|H^i(X)).$$
Remark 9 When $u$ is the graph of $f:X\rightarrow X$ then $u\cdot\Delta$ is the fixed points of $f$ counted with suitable weights.

Taking $w=\Pi_X^i$ and using $^t\Pi_X^i=\Pi_X^{2d_X-i}$, we obtain the following refinement.

Corollary 2 $$\tr_{X\times X}(u\cdot \Pi_X^{2d_X-i})=(-1)^i\tr(u_*|H^i(X)).$$
Remark 10 The existence of a cycle giving rise to $\Pi_X^i$ is still conjectural.

Now let $X\in\mathcal{P}(k=\mathbb{F}_q)$ and $F: X_{\bar k}\rightarrow X_{\bar k}$ be the (absolute) Frobenius morphism. Then $X(\mathbb{F}_{q^m})$ is the fixed point of $F^m$ for any $m\ge1$.

Theorem 4 (Grothendieck and others) There exists a Weil cohomology on $\mathcal{P}(\mathbb{F}_q)$, $X\mapsto H^*_{et}(X_{\bar k},\mathbb{Q}_\ell)$.
Corollary 3 $$\tr_{X\times X}(\Gamma_{F^m}\cdot\Delta)=\sum(-1)^i\tr(F^m|H^i(X)).$$

To interpret the left hand side as the fixed points of $F^m$, we need the following lemma.

Lemma 4 $\Gamma_F$ and $\Delta_X$ intersect properly: every irreducible component of $\Gamma_F\cap \Delta_X$ is of codimension $2d_X$ (i.e., the codimensions add). So $\Gamma_F\cdot\Delta$ can be computed as a sum of local terms, one for each point in $\Gamma_F\cap\Delta_X$. Moreover, the local terms are multiplicity-free (by computing the tangent space intersection $T_{P,P}\Gamma_F\cap T_{P,P}\Delta=0$ at an intersection point $(P,P)$).

Therefore we conclude that $$X(\mathbb{F}_{q^m})=\sum (-1)^i\tr(F^m|H^i(X)).$$

The Weil conjecture (expect the Riemann hypothesis) the follows.

Corollary 4 The zeta function $$Z(X,t)=\exp \sum_{m\ge1}|X(\mathbb{F}_{q^m})| \frac{t^m}{m}$$ can be computed as $$Z(X,t)=\prod_{i=0}^{2d_X}\det (1-F t|H^i(X))^{(-1)^{i+1}}.$$
Proof The previous corollary of the Lefschetz fixed point theorem and the easy linear algebra identity $$\sum_{m\ge1}\tr(F^m|H^i(X) )\frac{t^m}{m}=\log\frac{1}{\det(1-F t|H^i(m))}$$ proves the claim.

Combining this cohomological expression of $Z(X,t)$ with the Poincare duality, we also obtain the functional equation of $Z(X,t)$ (part of the Weil conjecture).

Corollary 5 $$Z(X,\frac{1}{q^{d_X}t})=\pm q^{d_X\cdot\chi(X)/2}\cdot t^{\chi(X)}Z(X,t).$$ Here $\chi(X)=\sum(-1)^i\dim H^i(X)$ is the Euler characteristic of $X$.

TopIntersecting cycles

Definition 10 Let $X$ be a smooth quasi-projective variety. For any $f:X\rightarrow Y$ proper (this is not serious since we will be working in $\mathcal{P}(k)$), we define pushforward cycles by $f_*(Z)=[k(Z):k(f(Z))]f(Z)$ when $\dim f(Z)=\dim Z$ and 0 otherwise.

On the other hand, we defined pullback of cycles along a flat morphism $f:X\rightarrow Y$ (Definition 3 c)).

We would like to make sense of pullback for more general classes of morphisms. Moreover, such pullback should be compatible with the pullback on cohomology under the cycle class maps. This can be done if there is a cup product (intersection pairing) on the group of cycles, $$\mathcal{Z}^k(X)\times \mathcal{Z}^l(X)\rightarrow \mathcal{Z}^{k+l}(X),$$ by intersecting with the graph of $f$. This is not naively true since the two cycles may not intersect properly (the codimension is wrong). So first we restrict to properly intersecting cycles $Z_1,Z_2$ whose intersection $Z_1\cap Z_2$ has all components of the right codimension. Then $Z_1\cdot Z_2$ should be a sum of irreducible components of $Z_1\cap Z_2$ with multiplicities $$\sum(-1)^i\Lg\Tor_i^A(A/I(Z_1),A/I(Z_2)),$$ here $A$ is the local ring of $X$ at the an irreducible component of the intersection $Z_1\cap Z_2$. This formula of intersection multiplicities (due to Serre) defines an intersection product for properly intersecting cycles.

To deal the general case, the classical approach is to jiggle $Z_1$ to make the intersection properly meanwhile staying in the same rational equivalence class (moving lemma).

Definition 11 We say two cycles $Z, Z'$ of dimension $m$ are rationally equivalence if $Z-Z'$ is generated by terms of the following form. Let $W\hookrightarrow X$ be a $(m+1)$-dimensional closed subvariety and take its normalization $\tau: \tilde W\rightarrow X$; these generators are the proper pushfowards $\tau_*\div(f)$ for $f\in k(W)^\times$.

An alternative approach is to consider $W\hookrightarrow X\times \mathbb{P}^1$ a $m+1$ dimension closed subvariety. Then the rationally equivalent to zero cycles are generated by $[W(0)]-[W(\infty)]$, here $W(x)$ is the fiber of $x$ in $W$.

These two definitions are equivalent. One can check that being rational equivalent is a equivalence relation. We denote it by $\sim_\mathrm{rat}$.

Definition 12 The Chow group (with $\mathbb{Q}$-coefficient) $\CH^k(X):=\mathcal{Z}^k(X)/\sim_\mathrm{rat}$.

Chow's Moving Lemma then gives a well defined intersection pairing on the Chow groups $$\CH^k(X)\times\CH^l(X)\rightarrow\CH^{k+l}(X).$$ This makes $\CH^*(X)$ a graded and commutative unital ring. The proper pushforward $f_*$ descends to the level of Chow groups.

Definition 13 We define the pullback $f^*$ on Chow groups for $Y$ proper by $f^*(Z)=p_{X,*}(\Gamma_f\cdot p_Y^{-1}(Z))$.


TopAdequate equivalences on algebraic cycles

Definition 14 An adequate equivalence is an equivalence relation on $\sim$ on $\mathcal{Z}^*(X)$ for any $X\in \mathcal{P}(k)$ such that
  1. it respects the linear structure;
  2. $\mathcal{Z}^*(X)/\sim$ becomes a ring under intersection product (the intersection product is defined by demanding the analogue of Chow's moving lemma for $\sim$).
  3. For any $f:X\rightarrow Y$ (since $f$ is proper, $f_*$ makes sense at the level of cycles), if $\alpha\sim0$, then $f_*\alpha\sim0$. So $f_*$ descends to $\mathcal{Z}^*(X)/\sim\rightarrow \mathcal{Z}^*(Y)/\sim$.
  4. Similarly, the pullback $f^*$ descends to $\mathcal{Z}^*(X)/\sim$.
  5. $f_*$ and $f^*$ are related by the projection formula $f_*(\alpha\cdot f^*\beta)=f_*(\alpha)\cdot\beta$.
Example 7 We showed last time that $\sim_\mathrm{rat}$ on $\mathcal{Z}^*(X)$ is an adequate relation.
Example 8 For any Weil cohomology $H^*$, the cohomological equivalence $\sim_{H^*}$ is an adequate relation. Here $\alpha\sim_{H^*}\beta$ if $\gamma_X(\alpha)=\gamma_X(\beta)$ in $H^*(X)$. Notice that a priori these cohomological equivalences may not be independent of the choice of $H^*$. If two such Weil cohomology theories are related by comparison, e.g., $H_B^*$ and $H_{dR}^*$, then the corresponding cohomological equivalences are the same.
Example 9 We say $\alpha\in \mathcal{Z}^k(X)$ is numerically equivalent 0 if for all $\beta\in \mathcal{Z}^{d_X-k}(X)$, $\deg(\alpha\cdot\beta)=0$, here the degree map $\deg: \mathcal{Z}^d(X)\rightarrow \mathbb{Z}$, $\sum n_iP_i\mapsto\sum n_i[k(P_i):k]$ (one can think of it as $\pi_*:\mathcal{Z}^d(X)\rightarrow \mathcal{Z}^0(\Spec k)\cong \mathbb{Z}$, for the structure map $\pi: X\rightarrow\Spec k$). Then $\sim_\mathrm{num}$ is an adequate relation.
Lemma 5
  1. $\sim_\mathrm{rat}$ is the finest adequate equivalence relation.
  2. $\sim_\mathrm{num}$ is the coarsest adequate equivalence relation.
  1. Let $\sim$ be an adequate relation. We want to show that if $\alpha\sim_\mathrm{rat}=0$, then $\alpha\sim0$. By definition, $\alpha$ is linear combination $\sum_i [W_i(0)]-[W_i(\infty)]$. Let $p:X\times \mathbb{P}^1\rightarrow\mathbb{P}^1$ and $q: X\times \mathbb{P}^1\rightarrow X$ be the projections. Then $$\sum_i q_*(p^*([0]-[\infty])\cdot W_i)=\alpha.$$ Suppose we knew that $[0]\sim[\infty]$. Then by the definition of adequate relation $\sim$, we know $\alpha\sim0$. So we reduced to show that $[0]\sim[\infty]$ on $\mathbb{P}^1$. Let $x,y\in \mathbb{P}^1(k)$ (assume $k=\bar k$ for simplicity). Since $\sim$ is adequate, we can find $z\sim x$ intersecting properly with $x $, i.e., $z=\sum z_i$ with $z_i\ne x$. We can certainly write down a map $f:\mathbb{P}^1\rightarrow\mathbb{P}^1$ such that $f_*(x)=y$ and $f_*(z_i)=z_i$. Explicitly, $$f(T)=T+(y-x)\prod\frac{T-z_i}{x-z_i}.$$ Therefore we have a chain of equivalences $y=f_*(x)\sim f_*(z)=z\sim x$ as desired.
  2. The second part is basically a tautology.
Definition 15 Let $\sim$ be an adequate equivalence relation on $\mathcal{Z}^*(X)$. Let $E$ be field of characteristic 0 (e.g., $E=\mathbb{Q}_\ell$). We define $A^*(X)=\mathcal{Z}^*(X) \otimes E/\sim$, the ring of cycles on $X$ modulo $\sim$.
Remark 11 The composition law we gave for cohomological correspondences works as well for $\sim$ correspondences (Definition 9). Namely, the composition $$A^{d_Y+s}(Y\times Z)\times A^{d_X+r}(X\times Y)\rightarrow A^{d_Z+r+s}(X\times Z)$$ is given by $$(\beta,\alpha)\mapsto p_{XZ,*}(p_{XY}^*(\alpha)\cdot p_{YZ}^*(\beta)).$$ In particular, $A^{d_X}(X\times X)$ becomes a ring, which will end up being endomorphisms of $X$ as a motive modulo $\sim$.
Definition 16 Let $\mathrm{Corr}_\sim(k)$ be the category with objects $X\in \mathcal{P}(k)$ (usually write it as $h(X)$ thought as a cohomological object), and $$\Hom(h(X), h(Y))=A^{d_X}(X\times Y).$$ (Think: graphs of homomorphisms $Y\rightarrow X$.) This is an $E$-linear category, with $$h(X)\oplus h(Y)=h(X\sqcup Y).$$ There is a functor $$\mathcal{P}(k)^\mathrm{op}\rightarrow \mathrm{Corr}_\sim(k),\quad, X\mapsto h(X), f\mapsto ^t\Gamma_f.$$

We want to enlarge $\mathrm{Corr}$ to include images of projectors. There is a universal way of doing this by taking the pseudo abelian envelope. We also want duals to exist in our theory (this amounts adding Tate twists). Combining these two steps into one,

Definition 17 We define the category $\mathcal{M}_K^\sim$ (the coefficient field $E$ is implicit) of pure motives over $k$ modulo $\sim$. Its object is of the form $(X,p,n)$, here $p$ is an idempotent in $A^{d_X}(X\times X)$ and $n$ is an integer (Think: $pH^*(X)(n)$). The morphisms are given by $$\Hom((X,p,n), (Y,q,m))= qA^{d_X+m-n}(X\times Y)p$$ Here the existence of Tate twists allows one to shift dimensions (e.g, a map $H^0(\mathbb{P}^1)\rightarrow H^2(\mathbb{P}^1)(1)$.
Proposition 4
  • $\mathcal{M}_k^\sim$ is pseudo abelian ( = preadditive and every idempotent has a kernel).
  • $\mathcal{M}_k^\sim$ is $E$-linear. The addition is given by (if $n\ge m$) $$(X,p,n)\oplus (Y,q,m)=((X,p,0)\oplus (Y,q,m-n))(n)=(X\sqcup Y\times (\mathbb{P}^1)^{n-m},p\oplus q', n).$$ Here we think of $(\Spec k, \Id, -1))$ as the summand $H^2(\mathbb{P}^1)$ of $h(\mathbb{P}^1)$ and identify $$(Y\times (\mathbb{P}^1)^{n-m}, q', 0){}=(Y, q, m-n).$$
  • (next time) There is a $\otimes$-structure $$h(X)\otimes h(Y)=h(X\times Y).$$

Grothendieck conjectured (Standard Conjecture D) that for $X\in \mathcal{P}(k)$, $\sim_{H^*}=\sim_\mathrm{num}$ for any Weil cohomology $H^*$. He also conjectured that $\mathcal{M}_k^\sim$ is abelian. Hence under Conjecture D, $\mathcal{M}_k^{H^*}$ is abelian. Conjecture D is still widely open, but in the early 90s, Jannsen proved the following startling theorem.

Theorem 5 (Jannsen) The followings are equivalent:
  1. $\mathcal{M}_k^\sim$ is semisimple abelian.
  2. $\sim=\sim_{num}$.
  3. For any $X\in \mathcal{P}(k)$, $A^{d_X}(X\times X)$ is a finite dimensional semisimple $E$-algebra.

That means that the numerical equivalence is arguably the "unique" right choice for the theory of motives.


TopTannakian theory

Let $(\mathcal{C}, \otimes)$ be an additive tensor ( = symmetric monoidal) category. One can check for the unit object $\mathbf{1}$, then endomorphisms $\End(\mathbf{1})=E$ is a commutative ring and $\mathcal{C}$ becomes an $E$-linear category.

Definition 18 We say a category $\mathcal{C}$ is rigid if for any $X\in \mathcal{C}$ there exists $X^\vee\in\mathcal{C}$ ("dual") and morphisms $$\mathrm{ev}_X: X^\vee\otimes X\rightarrow\mathbf{1},\quad \mathrm{coev}_X:\mathbf{1}\rightarrow X \otimes X^\vee$$ such that the composite map $$X\xrightarrow{\mathrm{coev_X}\times \Id_X} X \otimes X^\vee \otimes X\xrightarrow{\Id_X \otimes \mathrm{ev}_X} X,$$ is $\Id_X$ and the composite map $$X^\vee\xrightarrow{\Id_{X^\vee}\times \mathrm{coev}_X} X^\vee \otimes X \otimes X^\vee\xrightarrow{\mathrm{ev}_X \otimes \Id_{X^\vee}} X^\vee,$$ is $\Id_{X^\vee}$.
Remark 12 The rigidity condition gives internal homs: the functor $T\mapsto \Hom(T \otimes X,Y)$ is represented by an object $\underline{\Hom}(X,Y)=Y \otimes X^\vee$. Indeed, given a morphism $T\mapsto Y \otimes X^\vee$, one obtains a morphism $T \otimes X\rightarrow Y \otimes X^\vee \otimes X\xrightarrow{\mathrm{ev}_X}Y$ (and vice versa).
Definition 19 Let $E$ be a field. A neutral Tannakian category $\mathcal{C}/E$ is a rigid abelian tensor category with $\End(\mathbf{1})=E$ and for which there exists a fiber functor $w:\mathcal{C}\rightarrow\mathrm{Vec}_E$. By a fiber functor, we mean a faithful, exact, $E$-linear tensor functor. It is neutralized by a choice of such a fiber functor. (Think: the category of locally constant sheaves of finite dimensional $E$-vector spaces on a topological space $X$; a fiber functor is given by taking the fiber over $x\in X$).
Definition 20 Define a functor $\underline{\Aut}^\otimes(w): E\mathbf{-Alg}\rightarrow\mathbf{Sets}$ sending $R$ to the collection of $(g_X)_{X\in\mathcal{C}}$ such that for any $f:X\rightarrow Y$,
  1. the diagram $$\xymatrix{w(X) \otimes R \ar[r]^{g_X}\ar[d]  & w(X)\ar[d]  \otimes R\\ w(Y) \otimes R \ar[r]^{g_Y} &w(Y) \otimes R}$$ commutes.
  2. the diagram $$\xymatrix{(w(X) \otimes R) \otimes_R (w(Y) \otimes R) \ar[d]^{g_X \otimes g_Y} \ar[r] & (w(X) \otimes w(Y)) \otimes R \ar[d]^{g_{X \otimes Y}}\\ (w(X) \otimes R) \otimes_R (w(Y) \otimes R) \ar[r] & (w(X) \otimes w(Y)) \otimes R}$$ commutes.
  3. $g_{\mathbf{1}}$ is the identity on $w(\mathbf{1}) \otimes R$.

We have a natural functor $\mathcal{C}\rightarrow\mathbf{Rep}_E(\underline{\Aut}^\otimes(w))$ sending $X$ to the representation which on $R$-points is given by $g_X: w(X) \otimes R \rightarrow w(X) \otimes R$ for $(g_X)_{X\in\mathcal{C}}\in \underline{\Aut}^\otimes(w)(R)$.

The main theorem of Tannakian theory is the following.

Theorem 6 Let $\mathcal{C}$ be a neutral Tannakian category over $E$ and let $w:\mathcal{C}\rightarrow\mathrm{Vec}_E$ be a fiber functor. Then the functor on $E$-algebras $\underline{\Aut}^\otimes(w)$ is represented by an affine group scheme over $E$ and $\mathcal{C}\rightarrow\mathbf{Rep}_E(\underline{\Aut}^\otimes(w))$ is an equivalence of categories.
Remark 13 Even if we assume the Standard Conjecture D that $\sim_{H^*}=\sim_{\mathrm{num}}$, $\mathcal{M}_k^{\sim_{H^*}}$ is not Tannakian. This is because any rigid tensor category has an intrinsic notion of rank: for $X\in\mathcal{C}$, the composite $$\mathbf{1}\xrightarrow{\mathrm{coev}_X}X \otimes X^\vee\xrightarrow{c_{X,X^\vee}} X^\vee\times X\xrightarrow{\mathrm{ev}_X}\mathbf{1}$$ in $\End(\mathbf{1})$ is called the rank of $X$. For example, in $\mathrm{Vec}_E$, the rank of $V$ is simply the dimension of $V$; in $\mathrm{Gr}_E$, because $c_{V^i,V^{i\vee}}$ introduces a sign $(-1)^i$, the rank of $V$ is the alternating sum $\sum_i(-1)^i\dim V^i$, where $V^i$ is the $i $-th graded piece of $V$. But any tensor functor preserves the rank, so the tensor functor $H^*: \mathcal{M}_k^{\sim_{H^*}}\rightarrow\mathrm{Gr}_E$ tells us that $\mathcal{M}_k^{\sim_{H^*}}$ has objects of negative rank, hence $\mathcal{M}_k^{\sim_{H^*}}$ (using the usual commutativity constraint) does not admit any fiber functors.


TopThe Kunneth Standard Conjecture (Conjecture C)

Lemma 6 Suppose $\Char(k)=0$ . Assume $\sim_{H^*}=\sim_\mathrm{num}$ (Conjecture D) and that all Kunneth projectors $\Pi_X^i\in H^*(X\times X)$ are all algebraic cycles (Kunneth). Then $\mathcal{M}_k^{\sim_{H^*}}$ (with $E$-coefficients) is an a neutral Tannakian category over $E$.
Proof By Jannsen's theorem, $\mathcal{M}_k$ is abelian. We saw last time that $\mathcal{M}_k$ with its given naive commutative constraint could not be Tannakian. So we will keep the same tensor structure but modify the commutativity constraint using Kunneth. Kunneth tells us that $\mathcal{M}_k$ is $\mathbb{Z}$-graded via the projectors $\Pi_X^i$, i.e., for any $M\in \mathcal{M}_k$, we get a weight decomposition $$M=\bigoplus_{i\in \mathbb{Z}}\Pi^i(M).$$ Now for any $M,N$, we define the modified commutativity constraint $$c_{M,N}: M \otimes N\cong N \otimes M,\quad\bigoplus_{i,j} \Pi^iM \otimes \Pi^jN\rightarrow\bigoplus_{i,j} \Pi^iN \otimes \Pi^jM$$ given by $$c_{M,N}'{}=\bigoplus_{i,j} (-1)^{ij}c_{M,N}^{ij}.$$ Now $H^*: \mathcal{M}_k\rightarrow\mathrm{Vec}_E$ is a fiber functor.
Example 10
  1. For any $k$, and $X/k$ an abelian variety, $C(X)$ is true.
  2. For $k=\mathbb{F}_q$ a finite field, then $C(X)$ is true for any $X/k$ (with respect to any Weil cohomology satisfying weak Lefschetz). This is a theorem of Katz-Messing. Deligne's purity theorem on $H^i_\ell(X_{\bar k})$ allows one to distinguish different degrees. Katz-Messing shows that for any Weil cohomology with weak Lefschetz, the characteristic polynomial $P^i(t)$ of the Frobenius $F $ on $H^i$ agrees with that on the $\ell$-adic cohomology. Choose a polynomial $\Pi^i(t)\in \mathbb{Q}[t]$ such that $P^j(t)\mid \Pi^i(t)$ (for $j\ne i$) and $\Pi^i(t)\equiv1\bmod P^i(t)$, then $\Pi^i(F)$ is algebraic (as the combinations of the graphs of $F^m$) and is the projection onto $H^i(X)$.
Remark 14 Here is a consequence of Kunneth: For any $u\in \mathcal{Z}^d(X\times X)$, $\tr(u_*|H^i(X))$ lies in $\mathbb{Q}\subseteq E$ by Corollary 3. In particular, the minimal polynomial of $u_*$ on $H^i(X)$ has $\mathbb{Q}$-coefficients. If $u_*$ is further is an isomorphism on $H^i(X)$, then $u^{-1}$ is also algebraic as $u^{-1}\in \mathbb{Q}[u]$.

TopThe Lefschetz Standard Conjecture (Conjecture B)

Definition 21 Let $H^*$ be a Weil cohomology. We say $H^*$ satisfies the hard Lefschetz theorem if for any $X\in\mathcal{P}(k)$, any ample line bundle $\eta$ and any $i\le d$, $$L^{d-i}: H^i(X)\rightarrow H^{2d-i}(X)(d-i)$$ is an isomorphism. Here $L:=\cup c_1(\eta):H^*(X)\rightarrow H^{*+2}(X)(1)$.
Example 11 When $k=\mathbb{C}$ and $H^*=H_B^*$, this is part of Hodge theory. For any $k$ and $H^*=H_\ell^*$, this is proved by Deligne in Weil II.

The hard Lefschetz gives the primitive decomposition of $H^*(X)$.

Definition 22 For all $i\le d$, define $\mathrm{Prim}_\eta^i(X)=\ker(L^{d-i+1})$ (this depends on the choice of $\eta$). Then $$H^*(X)=\bigoplus_{i=0}^d \bigoplus_{j=0}^{d-i} L^j\mathrm{Prim}_\eta^i(X).$$
Remark 15 $H^0$ and $H^1$ are always primitive.

One should think of $L$ as a nilpotent operator on $H^*(X)$, then the Jacobson-Morozov theorem implies that this action can be extended to a representation of $\mathfrak{sl}_2$. The primitive parts are exactly the lowest weight spaces for this $\mathfrak{sl}_2$-action.

Theorem 7 (Jacobson-Morozov) Let $\mathfrak{g}$ be a semisimple Lie algebra over a field $E$ of characteristic 0. Let $X\in \mathfrak{g}$ be nonzero nilpotent element. Then
  1. There exists a $\mathfrak{sl}_2$ in $\mathfrak{g}$ extending $x $.
  2. Given $x $, for any semisimple $h$ such that $[h,x]=2x$, there exists a unique $\mathfrak{sl}_2$-triple $\{x,y,h\}\subseteq \mathfrak{g}$.

Let $\Pi=\sum_{i=0}^{2d}(i-d)\Pi_X^i$. Then $\Pi$ is semisimple and $\Pi\circ L-L\circ\Pi$ sends $v\in H^i(X)$ to $(i+2-d)Lv-(i-d)L_v=2L_v$. So applying Jacobson-Morozov gives a unique $\mathfrak{sl}_2$-triple $\{\Pi,L,{}^c\Lambda\}$ (the name ${}^c\Lambda$ comes from Hodge theory). Moreover, it follows that $\mathrm{Prim}_\eta^i(X)=\ker({}^c\Lambda|H^i(X))$.

Remark 16 Explicitly, we can write $x\in H^j(X)$ as $$x=\sum_{k\ge \max(0,j-d)} L^k x_{j-2k}.$$ Here $x_{j-2k}\in \mathrm{Prim}^{j-2k}(X)$. Then $$^c\Lambda(x)=\sum_{k\ge \max(0,j-d)} k(d-j+k+1)L^{k-1}x_{j-2k}.$$


Definition 23 A more convenient operator, the Hodge star $*_H$, can be extracted as follows. The $\mathfrak{sl}_2$-action on $H^*(X)$ gives rise to a representation of $SL_2$ on $H^*(X)$. Suppose $V$ is the weight $i$ eigenspace for $\Pi$. Then $\left(\begin{smallmatrix} 0 & -1\\1 &0\end{smallmatrix}\right) v$ is in the $(-i)$-eigenspace. But $\left(\begin{smallmatrix} 0 & -1\\1 &0\end{smallmatrix}\right)^2=-1$ is not quite an involution. So we renormalize and define $*_H=(-1)^{j(j+1)/2} \left(\begin{smallmatrix} 0 & -1\\1 &0\end{smallmatrix}\right)$ on $H^j(X)$ and then $*_H^2=1$.
Definition 24 Another variant is the Lefschetz involution $*_L(x)=\sum_k L^{d-j+k}x_{j-2k}$ for $x=\sum_k L^k x_{j-2k}$. Then $*_L^2=1$ as well. It differs from $*_H$ from certain rational coefficients on each primitive component.

Now we have the following cohomological correspondences:

  1. $L$, $*_L$, $*_H$, $^c\Lambda$, $\Lambda=*_L L{*}_L$ ($\Lambda$ is the inverse to $L$ on the image of $L$),
  2. Kunneth projectors $\Pi^0,\ldots,\Pi^{2d}$,
  3. Primitive projectors $p^0,p^1,\ldots,p^{2d}$:
    1. For $0\le j\le d$, $p^j(x)=x_j$ for $x\in H^j(X)$ and 0 on $H^{i\ne j}$;
    2. For $d<j \le 2d$, $p^j(x)=x_{2d-j}$ for $x=L^{j-d}x_{2d-j}$ (so it satisfies $p^j=p^{2d-j}\Lambda^{j-d}$).

The following lemma is immediate.

Lemma 7 $\Lambda$, $^c\Lambda$, $*_H$, $*_L$, $\Pi^0,\ldots, \Pi^{2d}$, $p^0,\ldots, p^{d-1}$ are all given by universal (noncommutative) polynomials in $L$ and $p^{d},\ldots, p^{2d}$.
Corollary 6 The following $\mathbb{Q}$-subalgebra of $\End(H^*(X))$ are equal:
  1. $\mathbb{Q}[L,\Lambda]$,
  2. $\mathbb{Q}[L,{}^c\Lambda]$,
  3. $\mathbb{Q}[L,*_L]$,
  4. $\mathbb{Q}[L,*_H]$,
  5. $\mathbb{Q}[L,p^d,\ldots, p^{2d}]$.

All of them contain $p^0,\ldots,p^{d-1}$ and $\Pi^0,\ldots,\Pi^{2d}$.

Proof One can show that $p^d,\ldots, p^{2d}\in \mathbb{Q}[L,\Lambda]$.

Now we can state various versions of the Lefschetz Standard conjecture.

Conjecture 1 (Weak form $A(X,L)$) For $2p\le d$, $A^p(X)\subseteq H^{2p}(X)\xrightarrow{L^{d-2p}} A^{d-p}(X)\subseteq H^{2d-2p}$ is an isomorphism (i.e., it is surjective).
Conjecture 2 (Strong form $B(X) $) The operator $\Lambda: H^*(X)\rightarrow H^{*-2}(X)(-1)$ is algebraic. Namely, it equals to the cohomology class a cycle in $A^{d_X-1}(X\times X)$.
Proposition 5 The followings are equivalent:
  1. $A(X,L)$,
  2. $A^*(X)$ is stable under $p^d,\ldots, p^{2d}$,
  3. $A^*(X)$ is stable under $*_H$ (or $*_L$),
  4. $A^*(X)$ is stable under $\Lambda$ (or $^c\Lambda$).

In particular, $B(X)\Longrightarrow A(X,L)$.

Proposition 6 The followings are equivalent:
  1. $B(X) $,
  2. $p^d,\ldots, p^{2d}$ are algebraic,
  3. $*_H$ (or $*_L$) is algebraic,
  4. $^c\Lambda$ is algebraic,
  5. For all $i\le d$, the inverse of $L^{d-i}: H^{i}(X)\cong H^{2d-i}$ is algebraic.
Remark 17 It follows from previous discussion that a)-d) are equivalent. e) $\Longrightarrow$ a) uses something we haven't written down (but not harder).

Because $\Pi^i\in \mathbb{Q}[L,\Lambda]$, we know that

Corollary 7 $B(X)\Longrightarrow C(X)$.
Corollary 8 Under $B(X) $ and $B(Y)$, if $u\in A^{d_X+(j-i)/2}(X\times Y)$ is algebraic and induces an isomorphism $H^i(X)\rightarrow H^j(Y)$. Then $u^{-1}$ is also algebraic (see Remark 14).
Proof Notice $v=*_{L,X}^t\cdot u\cdot {*}_{L,Y}\in A^{d_Y+d_x-d_Y+(j-i)/2}(Y\times X)$ gives a map $H^j(Y)\rightarrow H^i(X)$. Under $B(X) $ and $B(Y)$, this map is algebraic and an isomorphism. Hence $v\circ u: H^i(X)\rightarrow H^i(X)$ is an algebraic and an isomorphism. Therefore $(v\circ u)^{-1}$ is algebraic by Remark 14, so $(v\circ u)^{-1}\circ v=u^{-1}$ is also algebraic.
Corollary 9 $B(X) $ is independent of the choice of the ample line bundle $\eta$ giving rise to $L$.
Proof Suppose $L'$ is given by another ample line bundle $\eta'$. Then the hard Lefschetz tells us that $L'^{d-i}: H^i(X)\rightarrow H^{2d-i}(X)$ is an algebraic isomorphism (notice the correspondence $\cup a$ is equal to $\Delta_*(a)$, hence $\cup a$ is algebraic when $a$ is algebraic). Hence its inverse is also algebraic by the previous corollary. Now use e) of the previous proposition.

TopThe Hodge Standard Conjecture (Conjecture I)

The standard conjectures B and C both follow from the Hodge conjecture. The only standard conjecture does not follow from Hodge conjecture is the Hodge Standard conjecture. It concerns a basic positivity property of motives.

Take $k=\mathbb{C}$. For any $i\ge 0$, $H^i_B(X)$ carries a pure $\mathbb{Q}$-Hodge structure of weight $k$. More fundamental in algebraic geometry is the polarizable $\mathbb{Q}$-Hodge structure.

Definition 25 For $X\in\mathcal{P}(\mathbb{C})$ and $L$ an ample line bundle, we have $$c_1(\mathcal{L})\in H^2(X,\mathbb{Q})(1),\quad\omega=c_1(\mathcal{L})/2\pi i\in H^2(X,\mathbb{Q}).$$ The class $\omega$ can be thought of as the Kahler form in $H^2(X,\mathbb{R})$ (valid for general Kahler manifolds). Define $$Q_\omega: H^k(X,\mathbb{R})\times H^k(X,\mathbb{R})\rightarrow \mathbb{R},\quad (\alpha,\beta)\mapsto \int_X \alpha\cup\beta\cup\omega^{d-k}.$$ Extending $\mathbb{C}$-linearly we define the sesquilinear pairing $$H_\omega: H^k(X,\mathbb{C})\times H^k(X,\mathbb{C})\rightarrow \mathbb{C} ,\quad (\alpha,\beta)\mapsto i^k Q_w(\alpha,\bar\beta).$$
Remark 18 Notice that $Q_\omega(H^{p,q}, H^{p',q'})=0$ unless $p+p'{}=q+q'$ (i.e., $(p,q)=(q',p')$), hence $H_\omega(H^{p,q},H^{p',q'})=0$ unless $(p,q)=(p',q')$.
Remark 19 Notice also that different pieces of the primitive decomposition are orthogonal. Namely, if $\alpha=L_\omega^r\alpha_0$, $\beta=L_\omega^s\beta_0$, where $\alpha_0,\beta_0$ are primitive, then $H_\omega(\alpha,\beta)=0$ unless $r=s$. In fact, we may assume that $r<s$, then $$Q_\omega(\alpha,\beta)=\int_X L_\omega^{d-k+r+s}\alpha_0\cup\beta_0.$$ The claim follows because $r+s\ge 2r+1$ and $\alpha_0$ is primitive.

We would like to study the positivity properties of $H_\omega$ by reducing to particular pieces of the bigrading and the primitive decomposition.

Theorem 8 (Hodge index theorem) On $L_\omega^r\mathrm{Prim}^{p,q}(X)\subseteq H^k(X,\mathbb{C})$, $H_\omega$ is definite of sign $(-1)^{k(k-1)/2}\cdot i^{p-q-k}$.
Example 12 On a curve $X$, $H_\omega$ has sign $-1$ on $H^{0,1}$ and $+1$ on $H^{1,0}$. Suppose $\alpha\in H^{1,0}=H^0(X,\Omega^1)$. Then $$H_\omega(\alpha,\alpha)=iQ(\alpha,\bar\alpha)=i\int_X\alpha\wedge \alpha=i\int|f|^2 (-2i)dx\wedge dy=2\int|f|^2 dx\wedge dy>0.$$
Example 13 On a surface $X$, $H_\omega$ on $\mathrm{Prim}^{1,1}$ has sign $+1$ , and sign $-1$ on $H^{2,0}\oplus H^{0,2}$. So $Q_\omega$ is negative definite on $\mathrm{Prim}^{1,1}_\mathbb{R}$, positive definite on $(H^{2,0}\oplus H^{0,2})_\mathbb{R}$ and positive definite on $\mathbb{R}\omega\subseteq H^{1,1}$. For example, if $X$ is a K3 surface, then $Q_\omega$ has signature $(2,19)$ on $\mathrm{Prim}^{2}$ and has signature $(3,19)$ on $H^2(X,\mathbb{R})$.

This theorem is the source of polarization in Hodge theory.

Definition 26 A weight $k$ $\mathbb{Q}$-Hodge structure $(V,h)$ ($V$ is a $\mathbb{Q}$-vector space, $h: \Res_{\mathbb{C}/\mathbb{R}}\mathbb{G}_m\rightarrow GL(V_\mathbb{R})$) is polarizable if there exists a morphism of Hodge structures $$Q: V\otimes V\rightarrow \mathbb{Q}(-k),$$ such that $(2\pi i)^k Q(v, h(i)w): V_\mathbb{R}\times V_\mathbb{R} \rightarrow \mathbb{R}$ is positive definite.

So the Hodge index theorem has the following corollary.

Corollary 10 For any $X\in\mathcal{P}(\mathbb{C})$, $\mathrm{Prim}^k(X,\mathbb{Q})$ is a polarizable $\mathbb{Q}$-Hodge structure. A polarization is given by $(-1)^{k(k+1)/2} Q$, where $$Q(\alpha,\beta)=\tr_X(L^{d-k}\alpha\cdot\beta)=\frac{1}{(2\pi  i)^k}\int_X L^{d-k}\alpha\cup\beta.$$


Proof Let $\omega=c_1(\eta)/2\pi i$. We need to show that $$Q_\omega(\alpha,\beta)=(2\pi i)^k Q(\alpha,\beta)=\int_X\omega^{d-k}\cup\alpha\cup\beta$$ satisfies $Q_\omega(\cdot, h(i)\cdot)$ is positive on $\mathrm{Prim}^k(X,\mathbb{R})=V_\mathbb{R}$. Let $v=v^{p,q}+v^{q,p}$ Then 
Q_\omega(v,h(i)v)&=i^{-k}H_\omega(v,\overline{h(i)v})=i^{-k}H_\omega(v^{p,q}+v^{q,p},\overline{i^{p-q}v^{p,q}+i^{q-p}v^{q,p}})\\ &=i^{-k}(i^{q-p}H_\omega(v^{p,q},v^{p,q})+i^{p-q}H_\omega(v^{q,p},v^{q,p})).
Now using the Hodge index theorem we see the sign cancels out and takes value in $(-1)^{k(k+1)/2}\mathbb{R}_{>0}$.
Remark 20 For general Kahler manifolds, the Kahler form only gives rise to polarization of the real Hodge structure.
Remark 21 One can further polarize $H^*(X,\mathbb{Q})$ by variants of $Q$ with sign changes for non-primitive pieces.

Now we would like a (weak) version of this that makes sense for any field $k$ and any Weil cohomology $H^*$ satisfying hard Lefschetz (so the primitive cohomology $\mathrm{Prim}^*(X)$ still makes sense). Inside $H^{2p}(X)$ there is $\mathbb{Q}$-vector subspace $A^p(X)$.

Conjecture 3 (Hodge Standard Conjecture $I(X) $) For any $2p\le d$, the pairing on $A^p(X)\cap \mathrm{Prim}^{2p}(2X)(p)$ given by $(x,y)\mapsto (-1)^p\tr_X(L^{d-2p}x\cdot y)\in \mathbb{Q}$ is positive definite.

By Corollary 10(take $k=2p$),

Corollary 11 For $k=\mathbb{C}$ and $H^*=H_B^*$, $I(X) $ holds unconditionally.

We now explain that for $k=\mathbb{F}_q$, the Hodge Standard conjecture implies the Riemann hypothesis. A more convenient reformulation of $I(X) $ is that the pairing $$A^*(X)\times A^*(X)\rightarrow \mathbb{Q},\quad (x,y)\mapsto \langle x,* _Hy\rangle:=\tr_X(x\cdot * _Hy).$$ is positive definite. It follows that there is a positive involution on $A^{d_X}(X\times X)$ (acting on $A^*(X)$) given by $$\alpha\mapsto\alpha',\quad\langle\alpha v,*_H w\rangle=\langle v,*_H\alpha'w\rangle.$$ Explicitly, $\alpha'{}=* _ H{}^t\alpha * _ H$ (which is algebraic under Lefschetz).

So we want that the eigenvalues of the Frobenius $F $ on $H^i(X)$ are pure of weight $i $. We renormalize the Frobenius (acting on $H^*(X)[q^{1/2}]$) as $$\Phi=\sum_{i=0}^{2d}q^{-i/2}\Pi^i_X F^*.$$ Under Lefschetz, $\Phi\in A^{d_X}(X\times X) \otimes \mathbb{Q}[q^{1/2}]$. We want all eigenvalues of $\Phi$ has absolute value 1 for all complex embeddings. This can be obtained by realizing $\Phi$ as a unitary operator on the inner product space ($\langle\cdot,*\cdot\rangle$). We notice that $*$ commutes with $\Pi_X^i$ and $F^*$, so $$\Phi'{}= * {}^t\Phi * = {}^t\Phi * ^2= {}^t\Phi.$$ We claim that $\Phi'{}=\Phi^{-1}$, so that $\langle\cdot,*\cdot\rangle$ is $\Phi$-invariant. This follows from the following more general lemma. One can check that $F^*\eta=q\eta$ ($\eta$ is the chosen ample line bundle), so the following lemma applies to $\Phi$.

Lemma 8 If $g\in\End(H^*(X))$ such that
  1. $\deg g=0$,
  2. $g(a\cdot b)=g(a)g(b)$,
  3. $g(\eta)=\eta$.

Then $g$ is invertible and ${}^tg=g^{-1}$.

Proof b), c) implies that $g|_{H^{2d}}=\Id$. Therefore $g$ is invertible. In fact, for $a\in H^i$ nonzero, find $b\in H^{2d-i}$ such that $\tr(a\cdot b)\ne0$, then $g(a)g(b)=g(a\cdot b)=a\cdot b$ has nonzero trace, so $g(a)\ne0$. Now $$\langle g^{-1}(a),b\rangle=\langle g(g^{-1}a),g(b)\rangle=\langle a, g(b)\rangle.$$ So $g^{-1}={}^tg$.

It follows that $\mathbb{Q}[q^{1/2}][\Phi]\subseteq A^{d_X}(X\times X)[q^{1/2}]$ is unitary with respect to the inner product $(\alpha,\beta)=\tr(\alpha'\cdot\beta)$ (the positivity follows from $I(X\times X)$ and the fact that $\tr(\alpha'\alpha)=(-1)^**_{X\times X}(\alpha)$). In particular, the eigenvalues $\Phi$ acting on $\mathbb{Q}[q^{1/2}][\Phi]$ have all absolute values 1. Hence by Cayley-Hamilton, the roots of characteristic polynomials of $\Phi$ on $H^*(X)[q^{1/2}]$ have all absolute values 1, as desired.

Remark 22
  1. $B(X)+I(X)\Longrightarrow D(X)$ (so in characteristic 0, the hard Lefschetz implies everything). In fact, if the intersection pairing $A^p(X)\times A^{d-p}(X)\rightarrow \mathbb{Q}$ is non-degenerate, then $D(X) $ follows. But we know from $I(X) $ that $$A^p(X)\times A^p(X)\rightarrow \mathbb{Q}$$ is non-degenerate, so $B(X) $ implies that the first intersection pairing is also non-degenerate.
  2. $D(X\times X)\Longrightarrow B(X)$: Jannsen's theorem implies the algebra $A^{d_X}(X\times X)$ is semisimple, then Smirnov's algebraic result on semisimple algebra's with raising operators implies $B(X) $.

TopAbsolute Hodge cycles

Our next goal is to construct a modified category $\mathcal{M}_k$ of pure motives such that

  1. Under the standard conjectures, $\mathcal{M}_k\cong \mathcal{M}_k^{\sim\Hom}$.
  2. $\mathcal{M}_k$ has all categorical properties we want: (say $\Char(k)=0$) it is $\mathbb{Q}$-linear, semisimple, neutral Tannakian (this gives unconditional motivic Galois formalism).
  3. $\mathcal{M}_k$ lets you prove some unconditional results and formulate interesting but hopefully more tractable than the standard conjecture problems.

The basic strategy is to redefine correspondences using one of these larger classes of cycles:

algebraic cycles $\subseteq$ motivated cycles (Andre) $\subseteq$ absolute Hodge cycles (Deligne) $\subseteq$ Hodge cycles

Remark 23 These inclusions are unconditionally true. The Hodge conjecture says that the Hodge cycles are algebraic cycles, so all inclusions are conjectured to be equal. One can try to prove the last two inclusions are equal, which would already be a big step further towards the Hodge conjecture.
Definition 27 An absolute Hodge cycle on $X/k$ (suppose $k=\bar k$ has characteristic 0 and finite transcendence degree) is a class $t $ in $$H^{2p}_\mathrm{dR}(X)(p)\times H^{2p}_\mathbb{A}(X)(p),$$ where $H_\mathbb{A}^n(X):=H^n_\mathrm{et}(X,\hat{\mathbb{Z}})\otimes \mathbb{Q}$, such that for all $k\hookrightarrow \mathbb{C}$, the pullback class $$\sigma^*t\in H^{2p}_\mathrm{dR}(\sigma X)(p)\times H^{2p}_\mathbb{A}(\sigma X)(p)$$ comes from a Hodge cycle in $H_B^{2p}(\sigma X)(p)$ (a $\mathbb{Q}$-vector space) via the comparison isomorphisms.
Remark 24 Due to the transcendental nature of $\sigma$, the last imposed condition is rather deep.
Theorem 9 (Deligne) Any Hodge cycle on an abelian variety ($\Char(k)=0$) is absolutely Hodge.

One should think of this as a weakening of the Hodge conjecture for abelian varieties.

We will define Andre's notion of motivated cycles next time. Along this line,

Theorem 10 (Andre) Any Hodge cycle on an abelian variety ($\Char(k)=0$) is motivated.
Example 14 One classical application of absolute Hodge cycles is the algebraicity of (products of) special values of the $\Gamma$-function like $$(2\pi i)^*\prod \Gamma(a_i/d)\in \overline{\mathbb{Q}}$$ (with refinements giving the Galois action). The origin of this comes the periods (i.e. coefficients of the matrices in the B-dR comparison theorem) of the Fermat hypersurface $$X: X_0^d+\ldots+ X_{n+1}^d=0.$$ For an algebraic cycle $Z\hookrightarrow X_\mathrm{\mathbb{C}}$ (defined over $k$) and a differential form $\omega$ such that $[\omega]\in H^*_\mathrm{dR}(X/k)$, then one obtains a period $$\int_Z\omega\in (2\pi i)k.$$ The same principle applies for $Z$ an absolute Hodge cycle. A good supply of absolute Hodge cycles for Fermat hypersurfaces are the Hodge cycles by Deligne's theorem for abelian varieties (the motive of Fermat hypersurfaces lie in the Tannakian subcategory generated by Artin motives and CM abelian varieties).


More generally, let $k$ be a number field and $X/k$ a smooth projective variety. Let $k(\Omega_X)$ be the field generated by the coefficients of the period matrix. The relations between periods are predicted by the existence of algebraic cycles. The transcendence degree of $k(\Omega_X)$ is equal to the dimension of the motivic Galois group (when one makes sense of it). For the motive $H^1(A)$ (defined by absolute Hodge cycles), we have $\tr\deg_k(k(\Omega_{H^1(A)}))\le\dim(G^{H^1(A)})$. Deligne's theorem implies that the later is equal $\dim \mathrm{MT}(H^1(A))$, the dimension of the Mumford-Tate group (the Hodge theoretic analogy of the motivic Galois group).

Definition 28 Let $(V,h)$ be a $\mathbb{Q}$-Hodge structure. The Mumford-Tate group $\mathrm{MT}(V)\subseteq GL(V)$ is the $\mathbb{Q}$-Zariski closure of the image of $h$ (i.e., the smallest $\mathbb{Q}$-subgroup of $GL(V)$ whose $\mathbb{R}$-points containing $h(\mathbb{S})$.
Example 15 $\mathrm{MT}(\mathbb{Q})=\{1\}$; $k(\Omega_{\mathbb{Q}})=\mathbb{Q}$.
Example 16 $\mathrm{MT}(H^2(\mathbb{P}^1))=\mathbb{G}_m$; $k(\Omega_{\mathbb{P}^1})=\mathbb{Q}(2\pi i)$.

Notice a priori, one only knows the inequality $\dim(G^{H^1(A)})\ge \dim\mathrm{MT}(H^1(A))$ (since absolute Hodge cycles $\subseteq $ Hodge cycles).

Example 17 Here is another application due to Andre. Suppose $F/\mathbb{Q} $ is finitely generated. Let $X$, $X'$ are K3 surfaces over $F $ with polarizations (the important fact is that $h^{2,0}(X)=1$ for K3 surfaces). Then any isomorphism of $\Gal(\bar F/F)$-modules $$\mathrm{Prim}^2(X,\mathbb{Q}_\ell)\cong\mathrm{Prim}^2(X', \mathbb{Q}_\ell)$$ arises from a $\mathbb{Q}_\ell$-linear combination of motivated cycles. Also the Mumford-Tate conjecture is true for $\mathrm{Prim}^2(X)$: namely, $\mathrm{MT}(\mathrm{Prim}^2(X))\otimes \mathbb{Q}_\ell$ is equal to the connected component of the Zariski closure of the image of $\Gal(\bar F/F)$ on $\mathrm{Prim}^2(X,\mathbb{Q}_\ell)$. This is not known even for abelian varieties: there are a lot of possibilities of Mumford-Tate groups for abelian varieties, but for K3 surfaces they are quite restricted. Let $T_\mathbb{Q}\subseteq \mathrm{Prim}^2(X)$ be the orthogonal complement of Hodge cycles (the transcendence lattice which is 21 dimensional generically). Then $\End_{\mathbb{Q}-\mathrm{HS}}(T_\mathbb{Q})$ is a field because $h^{2,0}(X)=1$, and is either totally real or CM due to the polarization. A theorem of Zarhin shows that in the totally real case the Mumford-Tate is a special orthogonal group over $F $ and in the CM case a unitary group over $F $, with the pairing coming from the polarization.

TopMotivated cycles

Let $H^*$ be a Weil cohomology with hard Lefschetz. Fix a subfield $E_0\subseteq E$ (e.g. $E_0=\mathbb{Q}$).

Definition 29 (Motivated cycles) $A_\mathrm{mot}^*(X)_{E_0}$ is defined to be the subset of elements of $H^*(X)({*}/2)$ of the form $p^{XY}_{X,*}(\alpha\cup *\beta)$ for any $Y$ and any $\alpha,\beta$ algebraic cycles. Here $*$ is the Lefschetz involution associated to a product polarization $\eta_X \otimes [Y] + [X] \otimes \eta_Y $ on $X\times Y$. The idea is that we don't know Lefschetz and so we manually to add all classes produced by the Lefschetz operators to algebraic cycles to get motivated cycles.
Remark 25 One can relate $*_{X\times Y}$ to $*_X\otimes*_Y$ (this is is cleaner in terms of the Hodge involution $*_H$). Under Kunneth, $L_X \otimes 1+ 1 \otimes L_Y$ maps to the raising operator $L_{X\times Y}$ and the semisimple element $\sum (i-d_X)\Pi_X^i \otimes \sum(j-d_Y)\Pi_Y^j$ maps to $\sum (i-d_X-d_Y)\Pi_{X\times Y}^i$. Since Kunneth is an isomorphism of $SL_2$-representations, we know that $*_H(-1)^i \otimes*_H(-1)^j$ is equal to $(-1)^{ij}*_{H,X}\otimes*_{H,Y}=*_{H,X\times Y}$.

The basic calculation (with the above remark) shows the following.

Lemma 9
  1. $A_\mathrm{mot}(X)_{E_0}$ is an $E_0$-subalgebra of $H^*(X)$ (with respect to the cup product).
  2. $p^{XZ}_{X,*}(A_\mathrm{mot}(X\times Z)_{E_0})\subseteq A_\mathrm{mot}(X)_{E_0}$.
  3. $p^{XZ,*}_X(A_\mathrm{mot}(X)_{E_0})\subseteq A_\mathrm{mot}(X\times Z)_{E_0}$.

As for algebraic cycles, we define the motivated correspondences similarly.

Definition 30 Define $C_\mathrm{mot}^r(X,Y)=A_\mathrm{mot}^{d+r}(X\times Y)$ with the similar composition law (the target is correct by the previous lemma). Then $C_\mathrm{mot}^0(X,X)$ is a graded $E_0$-algebra. We also have a formalism of $f_*, f^*$ and projection formulas for $C_\mathrm{mot}$.
Lemma 10 $*_L,*_H,\Pi_X^i\in C_\mathrm{mot}^*(X,X)$
Remark 26 For comparable Weil cohomology theories, one obtains a canonical identifications of corresponding spaces of motivated cycles.
Remark 27 One can restrict the auxiliary varieties $Y$ to some full subcategory of $\mathcal{P}(k)$ stable under product, disjoint union, passing to connected components and containing $\mathbb{P}^1$. The following definition works with $\mathcal{P}(k)$ replaced by these $Y$'s.
Definition 31 The category $\mathcal{M}_k^\mathrm{mot}$ of motivated motives is defined as
  1. An object is a triple $(X,p,m)$, $X\in\mathcal{P}(k)$, $p$ an idempotent in $C_\mathrm{mot}^0(X,X)$, $m\in \mathbb{Z}$.
  2. Morphism: $\Hom((X,p,m),(Y,q,n))=q C_\mathrm{mot}^{n-m}(X,Y)p$.

We will write $\mathcal{M}_k=\mathcal{M}_k^\mathrm{mot}$ for short.

Remark 28
  1. If $B(X) $ is true for all $X$, then $\mathcal{M}_k=\mathcal{M}_k^{\sim_H}$.
  2. As before, $\mathcal{M}_k$ is $\mathbb{Q}$-linear and pseudo-abelian.
Theorem 11 (analogue of Jannsen's theorem) For any $X$, $C^0_\mathrm{mot}(X,X)$ is a finite dimensional semisimple $\mathbb{Q}$-algebra, hence $\mathcal{M}_k$ is semisimple abelian.
Proof We define an analogue of numerical equivalence: $x\in A_\mathrm{mot}(X)$ is called to be motivated numerically equivalent to 0 if for any $y\in A_\mathrm{mot}(X)$, $\int_X x\cdot y=0$. Then Jannsen's argument shows that $C^0_\mathrm{mot}(X,X)/\sim_\mathrm{mot-num}$ is semisimple. But since $B(X) $ and $I(X) $ holds for motivated cycles by construction, the motivated equivalence is the same as the motivated numerical equivalence (Remark 22). Therefore $C^0_\mathrm{mot}(X,X)$ is semisimple.
Remark 29 $\mathcal{M}_k$ is also a rigid tensor category. Because we always have Kunneth projectors, we can modify the commutativity constraint to obtain a neutral Tannakian category over $\mathbb{Q}$ with fiber functor given by $H_B^*:\mathcal{M}_k\rightarrow \mathrm{Vec}_\mathbb{Q}$. This gives an unconditional Tannakian formalism.
Remark 30 Restricting $\mathcal{P}(k)$ to some family of varieties $\mathcal{W}\subseteq \mathcal{P}(k)$, we also obtain a neutral Tannakian category $\mathcal{M}_k^\mathcal{W}$, the smallest full Tannakian subcategory containing $\mathcal{W}$ (the objects are subquotients of direct sums of $M^{\otimes n} \otimes(M^\vee)^{\otimes  m}$). For example, one can take $\mathcal{W}$ to be a singleton. Then one can define the motivic Galois group $G_k=\underline{\Aut}^\otimes(H_B^*)$, or $G_k^\mathcal{W}=\underline{\Aut}^\otimes(H_B^*|_{\mathcal{M}_k^\mathcal{W}})$. This allows us to talk about the motivic Galois group $G^M$ of a particular object $M\in\mathcal{M}_k$. This is the motivic analogue of $\mathrm{MT}(H_B^*(M))$.


Remark 31 For any field $E\supseteq \mathbb{Q}$, we define $\mathcal{M}_{k,E}$ to be the category of motives with $E$-coefficients (the objects are $E$-modules in $\mathcal{M}_k$, Tannakian over $E$). Define $G_{k,E}$ to be the motivic Galois group.
Proposition 7 (Properties of $G_k$)
  1. $G_k$ is pro-algebraic, even pro-reductive over $\mathbb{Q}$.
  2. $G_k$ splits over the maximal CM extension of $\mathbb{Q}$ (i.e., for any $E$ and $M\in \mathcal{M}_{k,E}$, $M $ is isomorphic to $N \otimes_{E_\mathrm{cm}}E$ for some $N\in \mathcal{M}_{k,E_\mathrm{cm}}$, where $E_\mathrm{cm}\subseteq E$ is the maximal CM subfield).
Remark 32 The second part follows from the existence of polarization (the Hodge index theorem). Other manifestations of the principle "arithmetic objects have CM coefficients": Frobenius eigenvalues of pure $\ell$-adic Galois representations are Weil numbers, hence lies in CM fields; the finite component of algebraic automorphic representations should be defined over CM fields (automorphic representations are unitary).

TopThe Motivated variational Hodge conjecture

Source of motivated cycles: the motivated analogue of variational Hodge conjecture.

Conjecture 4 (Variational Hodge conjecture) Over $\mathbb{C}$, let $f:X\rightarrow S$ be a smooth projective morphism, let $\xi\in H^0(S, R^{2p}f_*\mathbb{Q}(p))$. If $\xi_s\in H^{2p}_B(X_s)(p)$ is algebraic for some $s\in S$, then for any $t\in S(\mathbb{C})$ , $\xi_t\in H^{2p}_B(X_t)(p)$ is also algebraic.
Theorem 12 (Andre) The variational Hodge conjecture holds with "motivated" in place of "algebraic".
Remark 33 The Key arguments:
  1. $\mathcal{M}_\mathbb{C}$ is abelian;
  2. The theorem of the fixed part (from Hodge II).

Let us review the theorem of the fixed part and the necessary background in mixed Hodge theory.

Theorem 13 Suppose $f:X\rightarrow S$ is smooth projective and $S$ is smooth. Let $j: X\hookrightarrow \bar X$ be a smooth compactification. Then $$H^n(\bar X,\mathbb{Q})\rightarrow H^0(S, R^nf_*\mathbb{Q})$$ is surjective. In other words, the image $H^0(S, R^nf_*\mathbb{Q})\hookrightarrow H^n(X_s,\mathbb{Q})$ is the fixed part under the monodromy, i.e., $H^n(X_s,\mathbb{Q})^{\pi_1(S,s)}$.
Proof The above maps are given by $$H^n(\bar X,\mathbb{Q})\xrightarrow{a} H^n(X,\mathbb{Q})\xrightarrow{b} H^0(S, R^nf_*\mathbb{Q})\hookrightarrow^{c} H^n(X_s,\mathbb{Q}).$$ Here
  1. $b$ is the edge map in the Leary spectral sequence associated to $X\rightarrow S\rightarrow \Spec \mathbb{C}$. By a theorem of Deligne, when $f$ is smooth projective, the Leary spectral sequence degenerates at $E_2$. So $b$ is surjective.
  2. $c\circ b$ and $c\circ b\circ a$ have the same image. Since $c$ is injective, it follows that $b$ and $b\circ a$ have the same image. Hence $b\circ a$ is surjective.

    More generally, if $Y$ is smooth projective (applied to $Y=X_s$), $X$ is smooth, Then the image of the composite map $$H^n(\bar X,\mathbb{Q})\rightarrow H^n(X,\mathbb{Q})\rightarrow H^n(Y,\mathbb{Q})$$ is the same as the image of the latter map. The reason is that each of these cohomology groups has a weight filtration $W_i\subseteq W_{i+1}\cdots$ such that $\Gr_i^W$ is a pure Hodge structure of weight $i $. Since $\bar X$, $Y$ are smooth, their weight filtration looks like $$W_{n-1}=0\subseteq W_n=H^n.$$ Since $X$ is smooth but not projective, its weight filtration looks like $$W_{n-1}=0\subseteq W_n\subseteq\cdots\subseteq W_{2n}=H^n.$$ The general important fact is that the morphisms of mixed Hodge structure are strict for the weight filtration (one consequence: mixed Hodge structures form an abelian category), i.e., for $f: V_1\rightarrow V_2$ of mixed Hodge structure, then for any $i$, $$f(W_i V_1)=W_iV_2\cap f(V_1).$$ Now the strictness implies that it suffices to check the images on each $\Gr_i^W$ are the same. The results then follows from $\im(j^*)=W_nH^n(X)$. To see this, it essentially follows from the definition of the weight filtration as the shift of the Leray filtration associated to $X\hookrightarrow \bar X\rightarrow \Spec \mathbb{C}$: $$W_nH^n(X,\mathbb{Q})=E_3^{n,0}=H^n(\bar X,\mathbb{Q})/\im(d_2=\mathrm{Gysin})$$ and by definition $\im(j^*)$ is the whole thing.

Proof (Theorem 12) To prove the motivated variational Hodge conjecture. We may assume that $S$ is connected, smooth and affine. Given $\xi_s$ motivated, we want to show that all $\xi_t$ are motivated.

By the theorem of the fixed part, we have $$j_{s'}^*: H_B^{2p}(\bar X)(p)\twoheadrightarrow H^0(S, R^{2p}f_*\mathbb{Q}(p))\hookrightarrow H^{2p}_B(X_{s'}(p)).$$ Notice that $j_{s'}^*$ has kernel independent of the choice of $s'$. In the abelian category $\mathcal{M}_\mathbb{C}$, $$j^*_{s'}:h^{2p}(\bar X)(p)\rightarrow h^{2p}(X_{s'})(p)$$ then also has kernel independent of $s'$ (since the fiber functor $H_B^*$ is exact and faithful). So $\im(j_s^*)\cong\im (j_t^*)$ in $\mathcal{M}_k$. Applying $H_B^*$, we obtain that $H^{2p}(X_s)^{\pi_1(S,s)}\cong H^{2p}(X_t)^{\pi_1(S,t)}$ carries motivated cycles to motivated cycles.

Remark 34 The argument shows that the standard conjecture for $\mathcal{M}_\mathbb{C}$ implies the variational Hodge conjecture.
Corollary 12 Let $\alpha\in H(X_s)^{\otimes m} \otimes (H(X_s)^\vee)^{\otimes n}$ be a motivated cycle such that a finite index subgroup of $\pi_1(S,s)$ acts trivially on $\alpha$, then all parallel transport of $\alpha$ are still motivated.
Proof Apply the previous theorem after a finite base change.
Example 18 Let $A/\mathbb{C}$ be an abelian variety. Then the Hodge cycles on $A$ are known to be motivated, due to Deligne-Andre. The idea of the proof is to put $A$ in a family with the same generic Mumford-Tate group, prove for Hodge cycles special abelian variety in the family and then use the variational Hodge conjecture. More precisely, any Hodge cycle $\xi $ on $A$ has the form $\xi=p_{A,*}^{A\times Y}(\alpha\cup*\beta)$, where we can take $Y$ to be the product of an abelian variety and abelian schemes over smooth projective curves. So the Hodge conjecture for abelian varieties (not known) reduces to the Lefschetz standard conjectures for abelian schemes over smooth projective curves.
Corollary 13 For any abelian variety $A/\mathbb{C}$, $G^{H^1(A)}=\mathrm{MT}(H_B^1(A))$.
Proof This follows from Hodge cycles on abelian varieties are motivated and that the product of abelian varieties are still abelian varieties: $H^1(A)^{\otimes m }\otimes (H^1(A)^\vee)^{\otimes n}\subseteq H^{m+n(2d-1)}(A^{m+n})({*})$.

TopMumford-Tate groups

Lemma 11 For $\nu=\{(a_i,b_i)\}_{i}$, define $T^\nu=\bigoplus_i V^{\otimes a_i} \otimes (V^\vee)^{\otimes b_i}$.
  1. A $\mathbb{Q}$-subspace $W\subseteq T^\nu$ is a sub Hodge structure if and only if $W$ is stabilized by $\mathrm{MT}(V)$.
  2. $t\in T^\nu$ is a Hodge class if and only if $t $ is fixed by $\mathrm{MT}(V)$.
  1. Let $\mathrm{Stab}(W)\subseteq GL(V)$ be the stabilizer of $W$. Then $W$ is a sub Hodge structure if and only if $\mathbb{S}$ stabilizes $W_\mathbb{R}$ if and only if $h$ factors through $\mathrm{Stab}(W)$ if and only if $\mathrm{MT}(V)\subseteq \mathrm{Stab}(W)$.
  2. Apply the first part to the subspace $\mathbb{Q}(1,t)\subseteq \mathbb{Q}(0)\oplus T^\nu$.
Corollary 14 The natural functor from $\mathrm{Rep}(\mathrm{MT}(V))$ to the category of $\mathbb{Q}$-Hodge structures is fully faithful and realize $\MT(V)$ as the Tannakian group of $\langle V\rangle^\otimes$ (as a subcategory $\mathbb{Q}$-Hodge structure).
Lemma 12 The full subcategory of polarized $\mathbb{Q}$-Hodge structures $\mathbb{Q}-HS^\mathrm{pol}\subseteq \mathbb{Q}-HS$ is semisimple.
Corollary 15 When $V$ is polarizable, $\MT(V)$ is a (connected) reductive group.
Proof The connectedness follows from the definition. To show that $\MT(V)$ is reductive, we only need to exhibit a faithful and completely reducible representation of $\MT(V)$. The standard representation $\MT(V)\subseteq GL(V)$ works: the subrepresentations exactly corresponds to the sub Hodge structures of $V$, whose complete reducibility is ensured by the previous lemma.
Corollary 16 When $V$ is polarizable, $\mathrm{MT}(V)\subseteq GL(V)$ is exactly the subgroup that fixes all Hodge tensors.


Proof This follows from the following general results. Let $G\subseteq GL(V)$ be a reductive group and $H $ be a subgroup of $G$. Define $$H'{}=\{g\in G: g\text{ fixes all tensors fixed by }H\}.$$ If $H $ is reductive, then $H=H'$ ($H\subseteq H'$ a priori). The claim follows from taking $G=GL(V)$ and $H=\mathrm{MT}(V)$. For any $H $ (reductive or not), by the theorem of Chevalley, there exists a representation $W$ of $G$ and a line $\ell\subseteq W$ such that $H $ is the stabilizer of $\ell$. If $H $ is further reductive, there exists a $H $-complement $W=\ell\oplus W'$. Then $\ell \otimes \ell^\vee\subseteq W \otimes W^\vee$ and $H $ consists of the elements fixing any generator of this line. So $H'\subseteq H$.
Corollary 17 Let $M\in \mathcal{M}_k$, then $\langle M\rangle^{\otimes}\subseteq \mathcal{M}_k$ giving rises to $G^M$ is exactly the subgroup of $GL(H_B^{*}(M))$ fixing all motivated cycles in all tensor constructions.

Because motivated cycles $\subseteq $ Hodge cycles,

Corollary 18 Let $M\in \mathcal{M}_\mathbb{C}$, then $\mathrm{MT}(H_B^*(M))\subseteq G^M$.
Remark 35 The Hodge conjecture implies that this is indeed an equality.
Remark 36 The following much weaker conjecture is incredibly hard: $G^M$ is connected. Unknown except for abelian varieties.
Remark 37 The calculation of possible Mumford-Tate groups of abelian varieties, or more generally Mumford-Tate groups of objects of $\mathbf{AV}_\mathbb{C}=\langle AVs\rangle^{ \otimes}$ is essentially the Hodge theoretic content of Deligne's canonical models paper in Corvallis.
Remark 38 The soft general result of Zarhin gives an upper bound on possible Mumford-Tate groups and algebraic representations occurring in $H^k(X) $ for any $X$ smooth projective. In the $k=1$ case, Zarhin showed that simple factors of $\mathrm{MT}(H^1(A))$ are all classical groups; any nontrivial representations of a simple factor must be minuscule (the weights have only a single orbit under the Weyl group). In general, the degree $k$ controls how large the group and representations can occur. In particular, any exceptional group can't occur as Mumford-Tate groups of abelian varieties.

Does $G_2$ even arises as $\mathrm{MT}(V)$ for some polarized $\mathbb{Q}$-Hodge structure $V$? This is at least necessary for it to be a motivic Galois group.

Proposition 8 A semisimple adjoint group $M/\mathbb{Q}$ is a Mumford-Tate group of a polarizable $\mathbb{Q}$-Hodge structure if and only if $M_\mathbb{R} $ contains a compact maximal torus.
Remark 39 So $SL_n$ or $GL_n$ for $n>2$ can't arise as Mumford-Tate groups. Any $\mathbb{Q}$-form of $SO(3,19)$ can't arise (though it does arise as the Mumford-Tate group of non-projective K3 surfaces); for a generic projective K3 surface, the Mumford-Tate group is $SO(2,19)$ ($SO(p,q)$ has a compact maximal torus if and only if $pq$ is even).

Let explain the case when $M $ is simple with compact maximal torus over $\mathbb{R}$. Write $\mathfrak{m}=\Lie M$. Let $T_\mathbb{R}\subseteq M_\mathbb{R}$ be a compact maximal torus, fixed by some Cartan involution $\theta$ of $\mathfrak{m}_\mathbb{R}$. The Cartan involution is essential for the polarization. Namely, $\theta$ is an involution on $\mathfrak{m}$ satisfying the following positive condition: $$\kappa_\theta(X, Y)=-\kappa(X,\theta Y)$$ is positive definite. Decompose $$\mathfrak{m}_\mathbb{R}=\mathfrak{k}_\mathbb{R} + \mathfrak{p}_\mathbb{R}$$ into the $+1$ and $-1$ eigenspaces for $\theta$. Here $\mathfrak{k}_\mathbb{R}$ matches up with the Lie algebra of the maximal compact subgroup $T_\mathbb{R}$. Now any $h: \mathbb{S}\rightarrow M_\mathbb{R} \rightarrow GL(\mathfrak{m}_\mathbb{R})$ yields a polarizable $\mathbb{Q}$-Hodge structure on $\mathfrak{m}$ if and only if $\ad h(i)$ is a Cartan involution on $\mathfrak{m}_\mathbb{R}$.

Let us write down $h$. Choose a cocharacter $\lambda :\mathbb{S}^1\rightarrow T_\mathbb{R}$ such that $$\langle \lambda,\alpha\rangle\equiv0\bmod4$$ for any compact roots $\alpha$ and $$\langle \lambda,\beta\rangle\equiv 2\bmod4$$ for any noncompact roots $\beta$. Notice such cocharacters is in bijections with $X_\cdot(T_\mathbb{C})$. Extend $\lambda$ (trivial on $G_{m,\mathbb{R}}$) to obtain $$h: \mathbb{S}\rightarrow T_\mathbb{R}\subseteq M_\mathbb{R}.$$ Then $\ad h(i)$ acts on the root space $\mathfrak{m}_\gamma$ by $i^{\langle\lambda,\gamma\rangle}$, which is 1 when $\gamma$ is compact and $-1$ when $\gamma$ is noncompact. Now use $\kappa$ is negative definite on $\mathfrak{k}_\mathbb{R}$ and positive definite on $\mathfrak{p}_\mathbb{R}$. One knows that $-\kappa(\cdot,\cdot)$ gives a polarization on $(\mathfrak{m} ,\ad h)$. Using this framework, it is easy to check $G_2$ can't arise as $\mathrm{MT}(H^1(A))$.

Example 19 Consider the split form of $G_2$. $\mathfrak{k}_\mathbb{R}=\mathfrak{su}(2)\oplus \mathfrak{su}(2)$. The two compact roots are $(1,0)$, $(0,\sqrt{3})$.

After the break we will construct $G_2$ as a motivic Galois group via the theory of rigid local systems. This is originally due to Dettweiler-Reiter using Katz's theory. Zhiwei Yun gives an alternative proof (also for $E_7$ and $E_8$). We will focus on the former, since the latter needs more machinery from geometric Langlands.

TopApplications of the motivated variational Hodge conjecture

Example 20 The Kuga-Satake construction is "motivated", i.e., for $X/\mathbb{C}$ a projective K3 surface, the attached abelian variety $KS(X)$ such that $$H^2(X)\hookrightarrow H^1(KS(X))^{\otimes 2},$$ which is a priori a morphism of $\mathbb{Q}$-Hodge structure, is indeed a motivated cycle (i.e., a morphism in $\mathcal{M}_\mathbb{C}$). This implies the Mumford-Tate conjecture for K3 surfaces, etc..
Example 21 (Variation of motivic Galois group in families). Take a $\mathbb{Q}$-variational Hodge structure (e.g., $R^nf_*\mathbb{Q}$, for $f: X\rightarrow S$ smooth projective) with the holomorphically varying Hodge filtrations on the fibers $H^n(X_s,\mathbb{Q})$. (In general, a homomorphic family of Hodge structures on $S/\mathbb{C}$ is a local system $\mathbb{V}$ on $S$ with a filtration by holomorphic subbundles $\mathcal{F}^\cdot\subseteq \mathbb{V}_\mathbb{Q}\otimes \mathcal{O}_S$). How does $\mathrm{MT}(\mathbb{V}_s)$ vary for $s\in S(\mathbb{C})$?

For example, let $Y$ be a modular curve and $f: \mathcal{E}\rightarrow Y$ be the universal elliptic curve. Let $\mathbb{V}_s=R^1f_*\mathbb{Q}$. At CM points $s$, $\mathrm{MT}(\mathbb{V}_s)$ is simply a rank 2 torus over $\mathbb{Q}$. At non-CM points, $\mathrm{MT}(\mathbb{V}_s)=GL_2$. Notice that the CM points are dense in the analytic topology. Roughly speaking, there is a generic Mumford-Tate group ($GL_2$) and it drops on a countable union of closed analytic subvarieties (the CM points).

Now let us give the motivated analogue of a refinement of this assertion. So we need a notion of a family of motivated motives.

Definition 32 A family of motivated motives parametrized by $S(\mathbb{C})$ (assume $S/k$ is smooth, $k\subseteq \mathbb{C}$) is given by
  1. smooth projective $S$-schemes $X$, $Y$ of relative dimension $d_X$, $d_Y$, equipped with relatively ample line bundle $\alpha_{X/S},\alpha_{Y/S}$.
  2. $\mathbb{Q}$-linear combinations $Z_1$, $Z_2$ of closed integral $S$-subschemes of $X\times_S X\times_S Y$, flat over $S$, such that $$q_s=p_{X_s\times X_s,*}^{X_s\times X_s\times Y_s}[Z_{1,s}]\cup*[Z_{2,s}]$$ lies in $A^{d_X}(X_s\times X_s)$ and is idempotent for any $s\in S(\mathbb{C})$.
  3. $j\in \mathbb{Z}$.

We denote this family by $s\mapsto M_s(X_s,q_s,j)$.

Theorem 14
  1. Let the exceptional locus $\mathrm{Exc}=\{s\in S(\mathbb{C}): G^{M_s}$ does not contain the image of a finite index subgroup of $\pi_1(S(\mathbb{C}),s)\rightarrow GL(H_B(M_s))\}$. Then $\mathrm{Exc}$ is contained in a countable union of closed analytic subvarieties of $S(\mathbb{C})$.
  2. (Refinement) There exists a countable collection $V_i$ of algebraic subvarieties $V\subseteq S_{\bar k}$ such that $\mathrm{Exc}$ is contained in the union of $V(\mathbb{C})$. (In Hodge theory, this continues to hold for arbitrary $\mathbb{Z}$-polarized variational Hodge structure. This "algebraicity of the Hodge loci" is a strong evidence for the Hodge conjecture.)
  3. There exists a local system $(G_s)_{s\in S(\mathbb{C})}$ of algebraic subgroups of $GL(H_B(M_s))$ such that
    1. $G^{M_s}\subseteq G_s$ for any $s\in S(\mathbb{C})$.
    2. $G^{M_s}=G_s$ for all $s\not\in \mathrm{Exc}$.
    3. $G_s$ contains the image of a finite index subgroup of $\pi_1(S(\mathbb{C}),s)$ (notice the latter is a purely topological input!)


TopRigid local systems

Let $X/\mathbb{C}$ be a smooth projective connected curve. Let $S\subseteq X$ be a finite set of points. Let $U=X-S$. For the time being, we work with the associated complex analytic spaces (so $U=U^\mathrm{an}$ implicitly).

Definition 33 A local system $\mathcal{F}$ of $E$-vector spaces on $U$ is a locally constant sheaf of $E$-vector spaces.
Remark 40 For $x\in U$, $\mathcal{F}$ gives an $E$-vector space $\mathcal{F}_x$ and for any path $\gamma: [0,1]\rightarrow U$, $\mathcal{F}$ gives an isomorphism (depending only on the homotopy class of $\gamma$) $\rho(\sigma):\mathcal{F}_{\gamma(0)}\cong\mathcal{F}_{\gamma(1)}$. So choosing a base point $x\in U$ gives rises an equivalence between local systems on $U$ with $E$-representations of $\pi_1(U,x)$.
Example 22 The case $X=\mathbb{P}^1$ is most interesting for our purpose. Here $\pi_1(U,x)$ is a free group on $|S|-1$ generators.
Question Given a local system $\mathcal{F}$ on $U$, when does $\mathcal{F}$ come from geometry?

By coming from geometry, we mean there exists a smooth projective family $f: Y\rightarrow U$ such that $\mathcal{F}\subseteq R^nf_* E$ for some $n$ (notice $R^nf_*E$ itself is a local system).

One necessary condition for $\mathcal{F}$ to come from geometry is that the local monodromy at each puncture should be quasi-unipotent (some power of it is unipotent, equivalently, all its eigenvalues are roots of unity). This follows from the local monodromy theorem:

Theorem 15 Any polarizable $\mathbb{Z}$-variational Hodge structure over a punctured disc $\Delta^*$ has quasi-unipotent monodromy.
Remark 41 This is a hard theorem. See "Periods of integrals on algebraic manifolds III", Publ. Math. IHES 38 (1970) by Griffiths; "Variation of Hodge structure: the singularities of the period mapping" Invent. math. 22 (1973) by Schmid. Both the integral structure and the polarization are important for the theorem to be true.
Remark 42 Recall that a $\mathbb{Q}$-variational Hodge structure is a $\mathbb{Q}$-local system $\mathcal{F}$ on $U$ and a filtration by holomorphic subbundles of $\mathcal{E}=\mathcal{F} \otimes_\mathbb{Q} \mathcal{O}_U$ such that on each fiber one obtains a $\mathbb{Q}$-Hodge structure with the Hodge filtration on $\mathcal{F}_x \otimes \mathbb{C}$ induced from $\mathrm{Fil}^\cdot \mathcal{E}$, satisfying the Griffiths transversality: $\nabla(\mathrm{Fil}^i\mathcal{E})\subseteq \mathrm{Fil}^{i-1}\mathcal{E} \otimes \Omega_{U}^1$.
Remark 43 Though the local monodromy generators are quasi-unipotent, when $f: X\rightarrow U$ is smooth projective, the global monodromy representation of $R^nf_*E$ is semisimple (Hodge II). This is because one gets a polarizable variational Hodge structure.

The sufficient condition to come from geometry is still a total mystery. Simpson's guiding philosophy is that rigid local systems shall always come from geometry. Katz's book proves this is the case for irreducible rigid local systems on $\mathbb{P}^1-S$.

There are several notions which you may want to call rigid local systems.

Definition 34 Let $\mathcal{F}$ be a $E$-local system on $U$. $\mathcal{F}$ is physically rigid if for local system $\mathcal{G}$ such that for any $s\in S$, $\mathcal{F}|_{\Delta_s^*}\cong\mathcal{G}|_{\Delta_s^*}$, then $\mathcal{F}\cong\mathcal{G}$. In terms of the monodromy representation: if the generators are conjugate (possibly by different matrices), then they are globally conjugate.

A slight variant:

Definition 35 $\mathcal{F}$ is physically semi-rigid if there exists finitely many local systems $\mathcal{F}_1,\ldots,\mathcal{F}_r$ such that if $\mathcal{G}$ is locally isomorphic to $\mathcal{F}$ for all $s\in S$ (as in the previous definition), then $\mathcal{G}\cong\mathcal{F}_i$ for some $i $.
Remark 44 One can also define general notion of $G$-rigid local system for any reductive group $G$. For $G=GL_n$ (the notion defined above), physically semi-rigid implies physically rigid.

These two notions are very intuitive but extremely hard to check. The following definition provides a numerical condition and is easier to check.

Definition 36 $\mathcal{F}$ is cohomologically rigid if $H^1(X,j_*\End\mathcal{F})=0$. Here $j: U\hookrightarrow X$. Notice $\End\mathcal{F}$ is still a local system on $U$ (but $j_*\End\mathcal{F}$ is no longer a local system on $X$).
Remark 45 Intuitively, $\mathcal{F}$ being cohomologically rigid means that there is no infinitesimal deformation of $\mathcal{F}$ with prescribed local monodromy. We shall now explain this intuition in more detail.

Fix $\rho: \pi_1(U,x)\rightarrow GL_d(E)$. Let $\mathrm{Art}_\mathbb{C}$ be the category of local Artinian $\mathbb{C}$-algebras with residue field $\mathbb{C}$. We define the functor $\mathrm{Lift}_\rho: \mathrm{Art}_\mathbb{C}\rightarrow\mathbf{Sets}$, such that $\mathrm{Lift}_\rho(R)$ is the set of all liftings of $\rho$ to $GL_d(R)$. Then the familiar fact is that the tangent space $$\mathrm{Lift}_\rho(\mathbb{C}[\varepsilon]/\varepsilon^2)\cong Z^1(\pi_1(U,x),\ad\rho).$$ Taking the $GL_d$-equivalence into account, and assume that $\rho$ is irreducible (so ${}^{\bar\alpha}\rho=\rho$ implies that $\bar\alpha\in\mathbb{G}_m$), we are motivated to consider the deformation functor $\mathrm{Def}_\rho: \mathrm{Art}_\mathbb{C}\rightarrow\mathbf{Sets}$, such that $\mathrm{Def}_\rho(R)$ is the set of all liftings of $\rho$ up to $1+M_d(\mathfrak{m}_R)$-equivalence. Then $$\mathrm{Def}_\rho(\mathbb{C}[\varepsilon]/\varepsilon^2)\cong H^1(\pi_1(U,x), \ad\rho).$$ So when $\rho$ is irreducible, $H^1(\pi_1(U,x),\End\mathcal{F})$ measures the space of the infinitesimal deformations of $\rho$.

We further want the deformations with prescribed local monodromy. Let $\gamma_s$ be the generator of $\pi_1(\Delta_s^*)$. We now consider $\mathrm{Def}_\rho^S:\mathrm{Art}_\mathbb{C} \rightarrow\mathbf{Sets}$, sending $R$ to the set of $1+M_d(\mathfrak{m}_R)$-equivalence classes of liftings $\rho_R$ such that for $s\in S$, $\rho(\gamma_s)\sim\rho_R(\gamma_s)$ are conjugate by an element of $1+M_d(\mathfrak{m}_R)$. Then the tangent space $\mathrm{Def}_\rho^S(\mathbb{C}[\varepsilon]/\varepsilon^2)$ consists of cocycles $g\mapsto A_g$ such that $(1+\varepsilon A_{\gamma_s})\rho(\gamma_s)$ is conjugate to $\rho(\gamma_s)$ by an element of $1+M_d(\mathbb{C})$. It follows that for any $s\in S$, $A_{\gamma_s}=B_s-{}^{\rho(\gamma_s)}B_s$ for some $B_s\in M_d(\mathbb{C})$. Namely, $$\mathrm{Def}_\rho^S(\mathbb{C}[\varepsilon]/\varepsilon^2)=\ker\left(H^1(\pi_1(U),\ad\rho)\xrightarrow{\mathrm{res}}\bigoplus_{s\in S} H^1(\pi_1(\Delta_s^*),\ad\rho)\right).$$ Moreover, we claim that this restriction map can be identified with the edge map in the Leray spectral sequence for $j: U\hookrightarrow X$, $$H^1(U,\End\mathcal{F})\rightarrow H^0(X, R^1j_*\End\mathcal{F}).$$ So we can identify $$\mathrm{Def}_\rho^S(\mathbb{C}[\varepsilon]/\varepsilon^2)=H^1(X, j_*\End\mathcal{F}).$$ This motivates the definition of cohomologically rigid local systems.

We briefly indicate why the claim is true. For any local system $\mathcal{G}$ on $U$, notice $R^1j_*\mathcal{G}$ is the sheaf associated to the presheaf $V\mapsto H^1(j^{-1}(V), \mathcal{G})$ (which is $V\mapsto H^1(V,\mathcal{G})$ if $V\subseteq U$). Covering $V$ by simply-connected opens, we know that $R^1j_*\mathcal{G}|_U=0$. In a neighborhood of a puncture $s\in S$, we get $H^1(\Delta_s^*,\mathcal{G})\cong R^1j_*\mathcal{G}|_{\Delta_s}$, which glue to get $H^0(X, R^1j_*\mathcal{G})$.

The following lemma gives a very useful numerical criterion for cohomologically rigidity.

Lemma 13 Let $\mathcal{F}$ be an irreducible local system of rank $d$ on $U$. Then $\mathcal{F}$ is cohomologically rigid if and only if $\chi(X,j_*\End\mathcal{F})=2$, if and only if $$2=\chi(U)(\dim\mathcal{F})^2+\sum_{s\in S}\dim \mathrm{Cent}_{M_d}(\rho(\gamma_s)).$$


Proof Notice for any local system $\mathcal{H}$, the long exact sequence associated to $$0\rightarrow j_!\mathcal{H}\rightarrow j_*\mathcal{H}\rightarrow\mathrm{punctures}\rightarrow 0$$ gives $H^2_c(U,\mathcal{H})\cong H^2(X, j_*\mathcal{H})$. Also by Poincare duality, $H^2_c(U,\mathcal{H})\cong H^0(U,\mathcal{H}^\vee)$. The first equivalence then follows from the irreducibility of $\mathcal{H}=\End(\mathcal{F})$ (i.e., $h^0(X, j_*\mathcal{H})=h^2(X, j_*\mathcal{H})=1$).

For the second equivalence, we use the fact that for any local system $\mathcal{G}$ on $U$, we have the Euler characteristic formula $$\chi(X, j_*\mathcal{G})=\chi(U)\rank\mathcal{G}+\sum_{s\in S}\dim (j_*\mathcal{G})_s.$$ Notice $(j_*\mathcal{G})_s$ is nothing but the $\pi_1(\Delta_s^*)$-invariants of $\mathcal{G}_x$, the desired equivalence follows by applying to $\mathcal{G}=\End\mathcal{F}$ (so $\mathcal{G}_x=\End(\mathbb{C}^d)=M_d$.

It remains to prove the Euler characteristic formula. The Leary spectral sequence for $j:U\hookrightarrow X$ formally implies that $$\chi(U,\mathcal{G})=\sum_{b}(-1)^b\chi(X, R^bj_*\mathcal{G}).$$ The $b=0$ term gives $\chi(X,j_*\mathcal{G})$ and the $b=1$ term gives is $-\sum_{s\in S} h^1(\pi_1(\Delta_s^*),\mathcal{G})$ (see the previous remark). But $\pi_1(\Delta_s^*)$ is a cyclic group, $H^1(\pi_1(\Delta_s^*), \mathcal{G})$ is simply the coinvariants $\mathcal{G}_x/(\gamma_s-1)\mathcal{G}_x$, which has the same dimension as the invariants $\mathcal{G}_x^{\gamma_s=1}$. So $$\chi(U,\mathcal{G})=\chi(X, j_*\mathcal{G})-\sum_{s\in S}\dim (j_*\mathcal{G})_s.$$ But the left hand side is equal to $\chi(U)\rank\mathcal{G}$, as $\mathcal{G}$ is locally constant.

Remark 46 In the algebraic setting, the lemma is true for lisse $\overline{\mathbb{Q}_\ell}$-sheaves on $U$, as long as they are tamely ramified (i.e., $\ell$ is invertible in the base field, which is automatic in characteristic 0). When they are wildly ramified, more correction terms for the wild ramification are needed (known as the Grothendieck-Ogg-Shafarevich formula).
Example 23 We denote the Jordan block of length $k$ with the eigenvalue $\alpha$ by $\alpha\cdot U(k)$. Take $h_1=\left(\begin{smallmatrix}1 & 0 \\ 2 & 1\end{smallmatrix}\right)$ , $h_0=\left(\begin{smallmatrix} 1 & -2 \\ 0 & 1\end{smallmatrix}\right)$ and $h_\infty^{-1}= h_1h_0 =\left(\begin{smallmatrix}1 & -2 \\ 2 &-3\end{smallmatrix}\right)$. They give a local system on $\mathbb{P}^1-\{0,1,\infty\}$. They have Jordan forms $U(2)$, $U(2)$ and $-U(2)$, all are quasi-unipotent. It actually comes from geometry (classically known) as the local monodromies of the Legendre family $$f: \mathcal{E}\rightarrow \mathbb{P}^1-\{0,1,\infty\}, \quad y^2=x(x-1)(x-\lambda)\mapsto\lambda.$$ Namely, it comes from the local system $\mathcal{F}=R^1f_*\mathbb{C}$ (e..g, one can see these matrices by Picard-Lefschetz). Moreover, it is cohomologically rigid by the previous lemma: $\chi(\mathbb{P}^1-\{0,1,\infty\})=-1$ and each $\dim\mathrm{Cent}(h_i)=2$.
Example 24 More general classes of examples are provided by the hypergeometric local systems. These are given by $h_0,h_1,h_\infty\in GL_n(\mathbb{C})$ such that
  1. $h_\infty h_1 h_0=1$,
  2. $h_1$ is a pseudo reflection (i.e., $\rank(h_1-1)=1$).
  3. $\det (t-h_\infty)=\prod(t-a_i)$, $\det (t-h_0^{-1})=\prod(t-b_j)$ with $a_i\ne b_j$ for any $i,j$.

Let $H\subseteq GL_n(\mathbb{C})$ be the monodromy group generated by $h_0,h_1,h_\infty$ (hypergeometric group).

Lemma 14 Hypergeometric local systems are irreducible.
Proof If not, let $V_1 $ be a subrepresentation and $V_2$ be the corresponding quotient. Since $h_1$ is a pseudo-reflection, we know that it must acts trivially on one of $V_1,V_2$, hence $h_\infty=h^{-1}_0$ on one of them, which contradicts the assumption $a_i\ne b_j$.

Given $a_i,b_j$, one can write down the explicit matrix description for the local monodromies.

Theorem 16 (Levelt) Let $a_1,\ldots,a_n$, $b_1,\ldots, b_n\in \mathbb{C}^\times$ such that $a_i\ne b_j$. Define $A_1,\ldots, A_n$, $B_1,\ldots, B_n$ by $$\prod(t-a_i)=t^n+ A_1 t^{n-1}+\cdots + A_n$$ and $$\prod(t-b_j)=t^n+ B_1 t^{n-1}+\cdots + B_n.$$ Define the companion matrices $$A=
  0 & & & & -A_n\\
  1 &0 & & & -A_{n-1}\\
  & & \ddots & & \vdots\\
  & & 1&0 & -A_2\\
  & & &1 & -A_1
\end{bmatrix},\quad B=
  0 & & & & -B_n\\
  1 &0 & & & -B_{n-1}\\
  & & \ddots & & \vdots\\
  & & 1&0 & -B_2\\
  & & &1 & -B_1
\end{bmatrix},\quad$$ Then
  1. $h_\infty=A$, $h_0=B^{-1}$ and $h_1=A^{-1}B$ gives a hypergeometric local system with parameters $a_i,b_j$.
  2. Any hypergeometric local system with parameters $a_1,\ldots, a_n$, $b_1,\ldots, b_n$ is $GL_n(\mathbb{C})$-conjugate to the one of the above form. (This is stronger than the physical rigidity because we don't need to specify the Jordan forms).
  1. It suffices to show that $h_1$ is pseudo-reflection: $$h_1-1=A^{-1}B-1=A^{-1}(B-A)$$ indeed has rank 1.
  2. Given such an $h_\infty, h_1, h_0$, Set $A=h_\infty$, $B=h_0^{-1}$. Let $W=\ker(B-A)$. Then $W$ has dimension $n-1$ since $h_1$ is a pseudo-reflection). Hence $\cap_{j=0}^{n-2}A^{-j}W$ has dimension at least one; let $v$ be a nonzero vector of this space. Thus $v, Av, \ldots, A^{n-2}v\in\ker(B-A)$. Therefore $Bv=Av$, $B^2v=BAv=A^2v$, ..., $B^{n-1}v=A^{n-1}v$. We claim that the span $\langle v, Av,\ldots A^{n-1}v\rangle=\langle v, Bv \ldots, B^{n-1}v$ is the whole space. In fact, by Caylay-Hamilton $A,B$ stabilize on this span, so it must be the whole space by the irreducibility. In this basis, $h_\infty$, $h_1$, $h_0$ have the desired form.
Remark 47 The Jordan form of a companion matrix: when an eigenvalue $\alpha$ has multiplicity $r$, we obtain a Jordan block $\alpha\cdot U(r)$. Using this, one can check that a hypergeometric local system is cohomologically rigid. The dimension of the centralizer of $h_1$ is $(n-1)^2+1$, and the dimension of the centralizer of $h_0,h_\infty$ is $\sum r_i=n$. Now $-n^2+(n-1)^2+1+n+n$ adds to 2!
  1. What are (the Zariski closure) of the monodromy group $H $ of hypergeometric local systems? For example, does $G_2$ appear?
  2. Are those with $a_i,b_j$ roots of unity always geometric?
  3. We saw that hypergeometric local systems are both physically rigid and cohomologically rigid. What is the relationship between physical and cohomological rigidity in general?
  1. Not $G_2$. Beukers-Heckman computed all possibilities: $Sp(n,\mathbb{C})$, $O(n,\mathbb{C})$, $SL(n,\mathbb{C})$ and some specific finite groups.
  2. Yes, by Katz's theory.

Now let us come to the third question in more detail.

Proposition 9 Let $\mathcal{F}$ be an irreducible local system on $U$. Suppose $\mathcal{F}$ is cohomologically rigid, then $\mathcal{F}$ is physically rigid.
Remark 48 The same result (with the same proof) works for lisse $\ell$-adic sheaves (tamely ramified) in the algebraic setting.
Proof Let $\mathcal{G}$ be a local system with the same local monodromy as $\mathcal{F}$. Since $\End\mathcal{F}$ and $\Hom(\mathcal{\mathcal{F}},\mathcal{G})$ has the same local monodromies, the Euler characteristic formula implies that $$2=\chi(X,j_*\End\mathcal{F})=\chi(X, j_*\Hom(\mathcal{F},\mathcal{G})).$$ So $$h^0(X,j_*\Hom(\mathcal{F},\mathcal{G}))+h^2(X,j_*\Hom(\mathcal{F},\mathcal{G}))\ge0,$$ and by Poincare duality, $$h^0(U,\Hom(\mathcal{F},\mathcal{G}))+h^0(U,\Hom(\mathcal{G},\mathcal{F}))\ge2.$$ So at least one of the local systems $\Hom(\mathcal{F},\mathcal{G})$, $\Hom(\mathcal{G},\mathcal{F})$ has a global section. Since $\mathcal{F}$ is irreducible and $\rank \mathcal{F}=\rank \mathcal{G}$, this global section gives an isomorphism $\mathcal{F}\cong\mathcal{G}$.

For the other direction, we will use a transcendental argument. This direction is not known in the $\ell$-adic setting (knowing local monodromy matrices is not enough in the $\ell$-adic setting: one needs to know continuity).

Proposition 10 Suppose $X=\mathbb{P}^1$. Let $\mathcal{F}$ be an irreducible local system on $U$. Suppose $\mathcal{F}$ is physically rigid, then $\mathcal{F}$ is cohomologically rigid.
Proof We know a prior that $\chi(\mathbb{P}^1, j_*\End\mathcal{F})\le2$. We need to show it is $\ge2$. Let $\gamma_1\cdots\gamma_k=1$ be the local generators around the punctures. Suppose $\mathcal{F}$ is given by matrices $A_i$ and $\mathcal{G}$ is given by matrices $D_i$. Since $\mathcal{F}$ is physically rigid, if there exists $B_i$ such that $D_i=B_iA_iB_i^{-1}$, then there exists $C\in SL_n(\mathbb{C})$ such that $D_i=CA_iC^{-1}$. We want to show that $$(2-k)n^2+\sum\dim \mathrm{Cent}(A_i)\ge2.$$ Consider the map $$\pi: GL_n(\mathbb{C})^k\rightarrow SL_n(C),\quad (B_i)\mapsto \prod B_iA_i B_i^{-1}.$$ The fiber $\pi^{-1}(1)$ corresponds to the local systems with the same local monodromies as $\mathcal{F}$. The group $G=SL_n(\mathbb{C})\times \prod\mathrm{Cent}(A_i)$ acts on $\pi$ equivariantly, where $(C, Z_i)\in G$ acts on the domain and codomain by $$(B_i)\mapsto (CB_iZ_i^{-1}),\quad A\mapsto CAC^{-1}.$$ In particular, $G$ acts on the fiber $\pi^{-1}(1)$. Now $\mathcal{F}$ is physically rigid if and only if $G$ acts transitively on $\pi^{-1}(1)$. Therefore $\dim G\ge \dim \pi^{-1}(1)$. But $\dim\pi^{-1}(1)\ge\dim GL_n(\mathbb{C})^k-(n^2-1)$, it follows that $$\dim G=n^2-1+\sum\dim\mathrm{Cent}(A_i)\ge n^2k-(n^2-1),$$ which gives the desired inequality!
Remark 49 The implication physically rigid $\Longrightarrow$ cohomologically rigid works for $H $-local systems for general groups $H $. But the converse is not true for general $H $ (on the automorphic side: multiplicity one may fail for general groups other than $GL_n$).


TopPerverse sheaves

Theorem 17 Let $k$ be a field. For a separated finite type $k$-scheme $X$, we have a triangulated category $D_c^b(X,\overline{\mathbb{Q}_\ell})$ (the bounded derived category of constructible $\ell$-adic sheaves on $X$) equipped with a standard $t $-structure such that there is an equivalence of categories $$\mathcal{H}^0: D_c^b(X,\overline{\mathbb{Q}_\ell})^\heartsuit\rightarrow\{\overline{\mathbb{Q}_\ell}-\text{sheaves}\},$$ For $f: X\rightarrow Y$ a morphism, we have adjoint pairs $(f^*, Rf_*)$ and $(Rf_!, f^!)$. We also have adjoint pairs $( - \otimes^L K, \mathcal{RH}om(K,-))$ for $K\in D_c^b(X,\overline{\mathbb{Q}_\ell})$.
Remark 50 The category of $\overline{\mathbb{Q}_\ell}$-adic sheaves is an abelian category, whose objects are colimits of $E$-sheaves, where $E$ runs over all finite extensions of $\mathbb{Q}_\ell$. Here an $E$-sheaf is a constructible $\mathcal{O}_E$-sheaves with $\varpi_E$-inverted.
Remark 51 There is an analytification functor from $D_c^b(X, \overline{\mathbb{Q}_\ell})$ to the corresponding analytic category $D_c^b(X^\mathrm{an}, \overline{\mathbb{Q}_\ell})$. This functor is fully faithful but not essentially surjective. For example, take $X=\mathbb{G}_m$. Then $\pi_1(X^\mathrm{an})\cong \mathbb{Z}\rightarrow \overline{\mathbb{Q}_\ell}^\times$ sending the generator to $1/\ell$ does not extend to the etale fundamental group ($\pi_1(X)\cong\hat{\mathbb{Z}}$). Nevertheless, given $K\in D_c^b(X^\mathrm{an},\mathbb{C})$, for almost all $\ell$, one can choose an isomorphism $\iota: \mathbb{C}\cong \overline{\mathbb{Q}_\ell}$, such that $\iota(K)$ lies in the essential image of the analytification functor.

The triangulated category $D_c^b(X,\overline{\mathbb{Q}_\ell})$ is defined to be the colimit of $D_c^b(X,\mathcal{O}_E)_E$. The latter triangulated category $D_c^b(X,\mathcal{O})$ is hard to define (it is not defined as the derived category of $\mathcal{O}$-sheaves, which do not have enough injectives). There are 3 approaches to define $D_c^b(X,\mathcal{O})$..

  1. Use the pro-etale topology introduced by Bhatt-Scholze ($\mathcal{O}$ becomes a genuine sheaf).
  2. Taking limit is well-behaved for the stable $\infty$-category version of $D_c^b(X,\mathcal{O}/\varpi^r)$. The triangulated limit comes for free.
  3. Deligne's classical approach: replace $D_c^b(X, \mathcal{O}/{\varpi^r})$ with the full subcategory of very well-behaved complexed (these are quasi-isomorphic to bounded complexes of constructible $\mathcal{O}/\varpi^r$-flat sheaves). Call this full subcategory $D_\mathrm{ctf}^b(X, \mathcal{O}/{\varpi^r})$. Then $\mathcal{D}=\varprojlim_r D_\mathrm{ctf}^b(X, \mathcal{O}/{\varpi^r})$ is naturally triangulated: $X\rightarrow Y\rightarrow Z\rightarrow$ is a distinguished triangle if $X_r\rightarrow Y_r\rightarrow Z_r\rightarrow$ is a distinguished triangle for any $r$.
Definition 37 $K\in D_c^b(X,\overline{\mathbb{Q}_\ell})$ is semi-perverse if for any $i\in \mathbb{Z}$, $\dim \supp\mathcal{H}^{-i}K\le i$; $K $ is perverse if $K $ and $\mathbb{D}(K)$ are both semi-perverse, where $\mathbb{D}K=\mathcal{RH}om(K,\pi^{!}\overline{\mathbb{Q}_\ell})$ is the Verdier dual of $K $.
Example 25 Suppose $X$ is smooth of dimension $d$. For $\mathcal{F}$ lisse $\overline{\mathbb{Q}_\ell}$-sheaf on $X$. Then $\mathcal{F}[d]$ is perverse (since $\pi^! \overline{\mathbb{Q}_\ell}=\overline{\mathbb{Q}_\ell}[2d](d)$) but not for other shifts. In general, perverse sheaves are built out of lisse sheaves on smooth varieties.

Introducing perverse sheaves allows one to define intersection cohomology $IH^*(X)$ for singular proper varieties $X$ satisfying the Poincare duality and purity. Another major motivation for us is the following function-sheaf dictionary.

Definition 38 Take $X/\mathbb{F}_q$ and a $\overline{\mathbb{Q}_\ell}$-sheaf $\mathcal{G}$ on $X$. We define for any $m $, $$f_m^\mathcal{G}: X(\mathbb{F}_{q^m})\rightarrow \overline{\mathbb{Q}_\ell},\quad x\mapsto \tr \Frob_m| x^*\mathcal{G}.$$ For example, when $\mathcal{G}=R^ng_* \overline{\mathbb{Q}_\ell}$, $f^\mathcal{G}$ produces the trace of the Frobenii on the cohomology of the fibers of the morphism $g: X\rightarrow Y$. Generalizing this, for any $K\in D_c^b(X,\overline{\mathbb{Q}_\ell})$, we define $$f^K=\sum(-1)^if^{\mathcal{H}^i(K)}.$$

The key thing is that these functions interact nicely with the sheaf-theoretic operations. For example,

  1. When $K\rightarrow L\rightarrow M\rightarrow$ is a distinguished triangle, we have $f^K+f^M=f^L$.
  2. $f^{K \otimes L}=f^K\cdot f^L$.
  3. For $g: X\rightarrow Y$, we have $f^{g^*K}=f^K\circ g$ for $K\in D^b_c(Y,\overline{\mathbb{Q}_\ell})$.
  4. For $g: X\rightarrow Y$, we have $$f^{Rg_!K}(y)=\sum_{x\in X(\mathbb{F}_{q^m}), x\mapsto y} f_m^K(x).$$ (think: $Rf_!$ is the integration over the fibers) This is essentially the Lefschetz trace formula.

The moral is that if you have some classically understood operations on functions, you can mimic them at the level of sheaves. The key role of perverse sheaves that one can recover the perverse sheaves from their functions:

Theorem 18 Suppose $K $ and $L$ are two semisimple perverse sheaves. Then $K $ and $L$ are isomorphic if and only if $f_m^K=f_m^L$ for any $m $.
Remark 52 A basic fact hinted in this theorem: the full subcategory $\mathrm{Perv}(X)$ of perverse sheaves is an abelian category and all objects have finite length.

How do we produce more perverse sheaves from the "lisse on smooth" case (Example 25)?

Theorem 19
  1. Suppose $f:X\rightarrow Y$ is an affine morphism, the $Rf_*$ preserves semi-perversity (but not perversity).
  2. Suppose $f:X\rightarrow Y$ is a quasi-finite morphism, then $Rf_!=f_!$ preserves semi-perversity (but not perversity).
Corollary 19 If $f$ is both affine and quasi-finite (e.g., $f$ is an affine immersion), then both $f_!$ and $Rf_*$ preserve perversity.
Proof Suppose $K $ is perverse, then $Rf_*K$ is semi-perverse (by the previous theorem). Now $\mathbb{D}(Rf_*,K)=Rf_!\mathbb{D}K$ (by duality). Since $K $ is perverse, $\mathbb{D}K$ is perverse (by definition), hence $Rf_!\mathbb{D}(K)$ is also semi-perverse (by the previous theorem).

Here comes the key construction: intermediate extensions. Let $j:Y\hookrightarrow \bar Y\hookrightarrow X$ be a locally closed immersion. For simplicity, let us assume that $Y$ is affine, so $j$ is affine and quasi-finite. If $K\in \mathrm{Perv}(Y)$. Then both $j_!K$ and $Rj_*K$ lie in $\mathrm{Perv}(X)$. There is a natural map $j_!K\rightarrow Rj_*K$.

Definition 39 Define the intermediate extension (or middle extension)$j_{!*}K=\im(j_!K \rightarrow Rj_*K)$ (in the abelian category $\mathrm{Perv}(X)$).
Proposition 11
  1. $j_{!*}: \mathrm{Perv}(Y)\rightarrow\mathrm{Perv}(X)$ is fully faithful.
  2. $j_{!*}\mathbb{D}=\mathbb{D}j_{!*}$.
  3. $j_{!*}$ preserves simple objects, injections and surjections.
Theorem 20 Any simple perverse sheaf $K $ on $X$ is of the form $j_{!*}(\mathbb{F}[\dim Y])$ for some smooth affine $Y$ locally closed subvariety of $X$, for some lisse sheaf $\mathcal{F}$ on $Y$.
Proof (Sketch) Define $\bar Y$ to be the closure of $\mathrm{supp}(\oplus_i \mathcal{H}^i(K))$. Choose $Y\subseteq \bar Y$ such that the constructible sheaves $\mathcal{H}^i(K)$ become lisse when restricted to $Y$. Take $\mathcal{F}=\mathcal{H}^{-\dim Y}(K)|_Y$. This works.

Interesting things happen when extending $K $ to the boundary of $Y$.

Example 26 Let $X$ be a smooth geometrically connected curve. For $U\subseteq X $ dense open and $\mathcal{F}$ lisse on $U$, we have $j_{!*}(\mathcal{F}[ 1 ])=j_*\mathcal{F}[ 1 ]$ (here $j_*=R^0j_*$, see Katz 2.8 or 2.9). Let $i: X-U\hookrightarrow X$ and $\mathcal{G}$ be a sheaf on $X-U$ , then $\Hom(i_*\mathcal{G}, j_*\mathcal{F})=\Hom(j^*i_*\mathcal{G}=0,\mathcal{F})=0$. Namely, $j_*\mathcal{F}$ has no punctual sections. More generally, $K\in D_c^b(X, \overline{\mathbb{Q}_\ell})$ is perverse if and only if
  1. $\mathcal{H}^i(K)=0$ for any $i\ne-1,0$;
  2. $\mathcal{H}^{-1}(K)$ has no punctual sections;
  3. $\mathcal{H}^0(K)$ is punctual.

Hence the simple perverse sheaves on $X$ are either punctual or of the form $j_*\mathcal{F}[ 1 ]$ for $\mathcal{F}$ lisse on an open dense $U$.


TopThe middle convolution

Today we will introduce the key operation on perverse sheaves in Katz's classification of rigid local systems: the middle convolution.

Example 27 The rigid local system considered in Example 23 is the sheaf of the local solutions of the Gauss hypergeometric equation. The solution has an integral representation $$F(\frac{1}{2},\frac{1}{2},1;\lambda)=\int_1^\infty\frac{1}{\sqrt{x(1-x)(\lambda-x)}}dx.$$ Here the parameter $(\frac{1}{2},\frac{1}{2},1)$ determines the local monodromies. More generally, $$F(a,b,c;\lambda)=\int_1^\infty x^{a-c}(1-x)^{c-b-1}(\lambda-x)^{-a}dx$$ is the solution of $$\lambda(1-\lambda)\frac{d^2}{d\lambda^2}-(c-(a+b+1)\lambda)\frac{d}{d\lambda}-ab)f=0.$$ This integral looks like $$\int f(x) g(\lambda-x)dx,$$ namely the (additive) convolution of $g(x)=x^{-a}$ and $f(x)=x^{a-c}(1-x)^{c-b-1}$. The function $x^{-1/2}$ corresponds to the rank 1 Kummer sheaf associated to the representation $\pi_1(\mathbb{G}_m)\twoheadrightarrow \mu_2$. Similarly, the function $f(x)$ corresponds to a tensor product of (translated) Kummer sheaves. So rigid local system $\mathcal{F}\cong\mathcal{L}_{x^{-1/2}}*\mathcal{L}_{(1-x)^{-1/2}x^{-1/2}}$ can be expressed in terms of the convolution of simpler objects.

Here is the precise construction of the convolution.

Definition 40 Let $k$ be an algebraically closed field. Let $G/k$ be a connected smooth affine algebraic group. Let $\pi: G\times G\rightarrow G$ be the multiplication map. For $K,L\in D_c^b(G,\overline{\mathbb{Q}_\ell})$, we can define two kinds of convolutions $K*_!L=R\pi_!(K \boxtimes L)$ and $K*_*L=R\pi_*(K \boxtimes L)$.
Remark 53 Even if $K,L$ are perverse, these two convolution may not be perverse.
Remark 54 Since $\pi$ is affine, if $K,L$ is semi-perverse, then $K*_!L$ is also semi-perverse.
Definition 41 Suppose $K\in \mathrm{Perv}(G)$ such that for all $L\in \mathrm{Perv}(G)$, both $K*_!L$ and $K*_*L$ are perverse. We define the middle convolution $K*_\mathrm{mid}L$ to be the image of $K*_!L\rightarrow K*_*L$ in the abelian category of perverse sheaves.
Example 28 Take $G=\mathbb{A}^1$ and $K$. Let $\mathcal{L}_\chi$ be the local system on $\mathbb{G}_m$ associated to a nontrivial character $\pi_1(\mathbb{G}_m)\rightarrow \overline{\mathbb{Q}_\ell}^\times$. Let $K=j_*\mathcal{L}_\chi[ 1 ]$, where $j:\mathbb{G}_m\hookrightarrow\mathbb{A}^1$. Then the middle convolution $*_\mathrm{mid}K$ makes sense: both $*_!K$, $*_*K$ preserve perversity, by the following proposition.
Proposition 12 Suppose $\dim G=1$. Let $K\in \mathrm{Perv}(G)$ be irreducible such that its isomorphism class is not translation invariant. Then $*_!K$ and $*_*K$ both preserve perversity.
  1. The $*_*K$ statement follows from the $*_!K$ statement: because $\mathbb{D}(K)$ is also perverse and not translation invariant, so $\mathbb{D}K*_!L$ is perverse; taking dual implies that $K*_*\mathbb{D}L$ is perverse.
  2. For $K,L\in \mathrm{Perv}(G)$, then $K*_!L$ is perverse if and only if $K*_!L$ is semi-perverse: $\mathbb{D}(K*_!L)\cong\mathbb{D}K*_*\mathbb{D}L$ is semi-perverse since $\pi$ is affine.
  3. If $K $ is perverse, then the followings are equivalent:
    1. $K*_!L$ is perverse for any $L\in \mathrm{Perv}(G)$;
    2. $K*_!L$ is perverse for any irreducible $L\in \mathrm{Perv}(G)$.

    In fact, because $\mathrm{Perv}(G)$ is an abelian category with all objects having finite length, we can induct on the length of $L\in\mathrm{Perv}(G)$. A distinguished triangle $(M,L,N)$ (with $M,N$ lower lengths) gives a distinguished triangle $(K*_!M, K*_!L, K*_!N)$; the long exact sequence in cohomology then implies that $$\dim \supp \mathcal{H}^i(K*_!L)\le \max \dim \supp\{ \mathcal{H}^i(K*_!M), \mathcal{H}^i(K*_!N)\}\le -i.$$

  4. So we reduce to the case of irreducible perverse sheaves $L\in\mathrm{Perv}(G)$. We now use the assumption that $\dim G=1$. Namely, we need to check that $$\dim\supp \mathcal{H}^0(K*_!L)\le0,\quad \mathcal{H}^{i>0}(K*_!L)=0.$$ By Example 26, an irreducible perverse sheaf is either punctual or an intermediate extension $j_*\mathcal{F}[ 1 ]$. If either $K $ or $L$ is punctual, then $K*_!L$ is a translate of $L$ or $K$, hence is perverse. So we can assume that there exists $j: U\hookrightarrow G$ and lisse $\mathcal{F},\mathcal{G}$ on $U$ such that $K=j_*\mathcal{F}[ 1 ]$ and $L=j_*\mathcal{G}[ 1 ]$. The stalk of $\mathcal{H}^i(K*_!L)$ at a geometric point $g\in G$ is $$R^i\pi_!(K \boxtimes L)_g=H^{i+2}_c(\pi^{-1}(g), j_*\mathcal{F}\boxtimes j_*\mathcal{G}|_{\pi^{-1}(g)})$$ This vanishes for $i>0$ since $\dim\pi^{-1}(g)=1$. It remains to check that for $i=0$, this vanishes for at most finitely many $g\in G$. Now we need to use the assumption that $K $ is not translation invariant. Notice the fiber $\pi^{-1}(g)=\{(gx,x^{-1})\}\cong G$, so for $i=0$, we have $$H^{i+2}_c(\pi^{-1}(g), j_*\mathcal{F}\boxtimes j_*\mathcal{G}|_{\pi^{-1}(g)})=H^2_c(G, g^*(j_*\mathcal{F}) \boxtimes  \inv^*(j_*\mathcal{G})).$$ Since both $g^*j_*\mathcal{F}$ and $\inv^*j_*\mathcal{G}$ are lisse on $U_g= g^{-1}U\cap \inv^{-1}(U)$, it is equal to $$H^2_c(U_g, g^*\mathcal{F} \boxtimes \inv^*\mathcal{G})\cong H^0(U_g, g^*\mathcal{F}^\vee \boxtimes \inv^*\mathcal{G}^\vee)^\vee=\Hom_{\pi_1(U_g)}(g^*\mathcal{F},\inv^*(\mathcal{G}^\vee))^\vee.$$ Since both source and target are irreducible, this is zero unless there is an isomorphism $g^*\mathcal{F}\cong\inv^*\mathcal{G}$. Since the right hand side does not depend on $g$, either we win or there exists infinitely many $g$ such that there is such an isomorphism. Since these $g$ lie in the support of a constructible sheaf on a curve, the same would happen for $g$ in an open dense subset $V\subseteq G$. Let $g_0\in V$, then the isomorphism class of $g_0^*\mathcal{F}$ is translation invariant under $g_0^{-1}V$. Thus we obtain a subgroup containing $g_0^{-1}V$, which must be the whole group, under which $K $ is translation invariant. A contradiction!

Henceforth we take $G=\mathbb{A}^1$.

Definition 42 Let $\mathcal{T}_\ell$ be the full subcategory of constructible $\overline{\mathbb{Q}_\ell}$-sheaves $\mathcal{F}$ on $\mathbb{A}^1$ satisfying
  1. $\mathcal{F}$ is an irreducible intermediate extension, i.e., there exists $j:U\hookrightarrow \mathbb{A}^1$ open dense on which $j^*\mathcal{F}$ is lisse, irreducible and $\mathcal{F}\cong j_*j^*\mathcal{F}$.
  2. $\mathcal{F}$ is tame, i.e., the corresponding $\pi_1(U)$-representation is tamely ramified at the punctures $\mathbb{P}^1-U$.
  3. $\mathcal{F}$ has at least two singularities in $\mathbb{A}^1$. Notice if $\rank\mathcal{F}\ge2$, then a) and b) implies c). (In particular, $j_*\mathcal{L}_\chi\not\in \mathcal{T}_\ell$.

Now we can state the main results (slightly specialized) about the middle involution.

Theorem 21Fix a nontrivial tame character $\chi: \pi_1^\mathrm{tame}(\mathbb{G}_m)\rightarrow \overline{\mathbb{Q}_\ell}^\times$. Let $\mathrm{MC}_\chi(\mathcal{F})=(\mathcal{F}[ 1 ]*_\mathrm{mid} j_*\mathcal{L}_\chi[ 1 ])[ -1 ].$ Then
  1. $\mathrm{MC}_\chi$ preserves $\mathcal{T}_\ell$.
  2. We have composition laws $$\mathrm{MC}_\chi\circ\mathrm{MC}_\rho\cong\mathrm{MC}_{\chi\rho},$$ if $\chi\rho\ne\mathbf{1}$ and $$\mathrm{MC}_\chi\circ \mathrm{MC}_{\chi^{-1}}=\Id.$$
  3. For $\mathcal{F}$ lisse on $U$, define the index of rigidity $\mathrm{rig}(\mathcal{F})=\chi(\mathbb{P}^1,j_*\End\mathcal{F})$ (so $\mathcal{F}$ is rigid if and only if $\mathrm{rig}(\mathcal{F})=2$). Then $$\mathrm{rig}(\mathrm{MC}_\chi(\mathcal{F}))=\mathrm{rig}(\mathcal{F}).$$ for any $\mathcal{F}\in \mathcal{T}_\ell$.
  4. The local monodromies of $\mathrm{MC}_\chi(\mathcal{F})$ can be computed using those of $\mathcal{F}$. (See Dettweiler-Reiter.)


Let us explain some of the ideas of the proof without going into details.

Remark 55 The key step is to show that $*_\mathrm{mid}j_*\mathcal{L}_\chi[ 1 ]$ preserves the subcategory $\mathcal{C}$ consisting of irreducible perverse sheaves $K $ such that $K*_!$ and $K*_*$ preserve perversity (we will say $K $ satisfies property $\mathcal{P}$ for short). Then $\mathcal{T}_\ell[ 1 ]\subseteq\mathcal{C}$ and its complement can be described explicitly (e.g., $\overline{\mathbb{Q}_\ell}[ 1 ]$). Bare-hand calculation shows this complement is also preserved under $*_\mathrm{mid}j_*\mathcal{L}_\chi[ 1 ]$. The first key step has a proof which works in any characteristic. But in characteristic $p$, the approach of Fourier transform is more pleasant, which we shall now briefly discuss.

For any algebraic closed field $k$ and any $X/k$ separated and of finite type, we define the subcategory of middle extensions $\mathrm{ME}(X)\subseteq\mathrm{Perv}(X)$ consisting of $K\cong j_{!*}j^*K$, where $j^*K$ is lisse for some $j: U\subseteq X$. We have an operation $$\otimes_\mathrm{mid}: \mathrm{ME}(X)\times \mathrm{ME}(X)\rightarrow\mathrm{ME}(X),\quad (j_{!*}\mathcal{F}[d]), j_{!*}\mathcal{G}[d])\mapsto j_{!*}(\mathcal{F}\otimes \mathcal{G})[d].$$

If $k$ has characteristic $p$ and $X=\mathbb{A}^1$. It turns out that the category $\mathcal{C}$ on $\mathbb{A}^1$ satisfying $\mathcal{P}$ is equivalent to $\mathrm{ME}(\mathbb{A}^1)$ via the Fourier transform. The middle convolution on $\mathcal{C}$ then corresponds to $\otimes_\mathrm{mid}$ on $\mathrm{ME}(\mathbb{A}^1)$ (as in the classical Fourier theory: the Fourier transform of the convolution is the product of the Fourier transforms).

Definition 43 We now define the Fourier transform, which is a functor $D_c^b(\mathbb{A}^1)\rightarrow D_c^b(\mathbb{A}^1)$. Fix an additive character $\psi:\mathbb{F}_p\rightarrow \overline{\mathbb{Q}_\ell}^\times$ and denote the associated the Artin-Schrier sheaf on $\mathbb{A}^1$ by $\mathcal{L}_\psi$. Let $p_1,p_2$ be the two projections $\mathbb{A}^1\times\mathbb{A}^1\rightarrow\mathbb{A}^1$. Motivated by the classical Fourier transform $$f\mapsto\int_\mathbb{R}f(x)e^{ixy}dx,$$ we define $$\mathrm{FT}_{\psi,!}(\mathcal{F}):=R_{p_2,!}(p_1^*\mathcal{F}\otimes \mathcal{L}_{\psi(xy)}[ 1 ]),$$ where $\mathcal{L}_{\psi(xy)}$ is the pullback of $\mathcal{L}_\Psi$ via the multiplication map. Similarly define $\mathrm{FT}_{\psi,*}$ using $R_{p_2,*}$. It turns out that $\mathrm{FT}_{\psi,!}=\mathrm{FT}_{\psi,*}$ and we denote it by $\mathrm{FT}$ for short. It follows that $\mathrm{FT}$ preserves $\mathrm{Perv}(\mathbb{A}^1)$ since projection maps are affine and the duality switches $\mathrm{FT}_{\psi,!}$ and $\mathrm{FT}_{\psi,*}$. Moreover, $\mathrm{FT}$ is involutive: $$\mathrm{FT}_\psi\circ\mathrm{FT}_\psi\cong[-1]^*(-1).$$ In particular, $\mathrm{FT}$ is an auto-equivalence of $\mathrm{Perv}(\mathbb{A}^1)$.
Remark 56 Now we can check that $K*_\mathrm{mid}j_*\mathcal{L}_\chi[ 1 ]$ has property $\mathcal{P}$. Using the Fourier transform, it suffices to check that $$\mathrm{FT}(K) \otimes_\mathrm{mid} \mathrm{FT}(j_*\mathcal{L}_\chi[ 1 ])$$ lies in $\mathrm{ME}(\mathbb{A}^1)$. The first factor in $\mathrm{ME}(\mathbb{A}^1)$ since $K $ has property $\mathcal{P}$, and the second factor is in $\mathrm{ME}(\mathbb{A}^1)$ because $j_*\mathcal{L}_{\chi^{-1}}[ 1 ])$ also has property $\mathcal{P}$ (Example 28). Hence the $\otimes_\mathrm{mid}$ is in $\mathrm{ME}(\mathbb{A}^1)$.
Remark 57 Suppose $K $ is irreducible, then $\mathrm{FT}(K)$ is also irreducible by the exactness of the Fourier transform. It is easy to see that $\otimes_\mathrm{mid}$ with a rank one object is invertible on $\mathrm{ME}(\mathbb{A}^1)$: i.e., if $L$ is rank one with $j^*L$ lisse and $K $ with $j^*K$ lisse, one check that $K\mapsto K \otimes_\mathrm{mid}L\mapsto K \otimes_\mathrm{mid}L \otimes_\mathrm{mid}\mathbb{D}L$ is the identity map. Applying this to $j_*\mathcal{L}_{\chi^{-1}}[ 1 ]$, we know that $K*_\mathrm{mid}j_*\mathcal{L}_\chi[ 1 ]$ is again irreducible.

TopKatz's classification

Using Theorem 21, we can prove the Katz's classification theorem of tamely ramified cohomological rigid local systems. Besides $\mathrm{MC}_\chi$, we also need a simpler twisting operation: If $\mathcal{L}$ is rank 1 lisse on $U$, we define $$\mathrm{MT}_\mathcal{L}:\mathcal{T}_{\ell,\mathrm{rank}\ge2}\rightarrow\mathcal{T}_{\ell,\mathrm{rank}\ge2},\quad \mathcal{F}\mapsto j_*(j^*\mathcal{F}\otimes j^*\mathcal{L}).$$ The index of rigidity is easily seen to be preserved under $\mathrm{MT}_\mathcal{L}$.

Given an irreducible tame cohomological rigid local system on $\mathbb{P}^1-S$, our aim is to apply a series of $\mathrm{MC}_\chi$ and $\mathrm{MT}_\mathcal{L}$'s (these are all invertible operations) to obtain a rank 1 object (which is easy to understand).

Theorem 22 (Katz) Suppose $\mathcal{F}\in\mathcal{T}_{\ell,\rank\ge2}$. Assume $\mathcal{F}$ is lisse on $\mathbb{P}^1-S=\mathbb{A}^1-D$ (so $S=D\cup\{\infty\}$) and cohomologically rigid. Then there exists a generic rank 1 $\mathcal{L}$ lisse on $\mathbb{A}^1-D$ and a nontrivial character $\chi:\pi_1(\mathbb{G}_m)^\mathrm{tame}\rightarrow \overline{\mathbb{Q}_\ell}^\times$ such that $\mathcal{G}=\mathrm{MC}_\chi(\mathrm{MT}_\mathcal{L}(\mathcal{F}))$ has strictly smaller rank than $\mathcal{F}$.

Also, we can arrange $\mathcal{L}$ and $\mathcal{L}_\chi$ to have local monodromies contained in the local monodromies of $\mathcal{F}$. So the local monodromies of $\mathcal{G}$ is contained in the local monodromies of $\mathcal{F}$. In particular, if all th eigenvalues of the local monodromies of $\mathcal{F}$ are roots of unity, then we can arrange the same for $\mathcal{G}$.

Proof For $\alpha\in D$, we write $$\mathcal{F}|_{I(\alpha)^\mathrm{tame}}=\bigoplus_{\chi: \pi_1(\mathbb{G}_m)^\mathrm{tame}\rightarrow \overline{\mathbb{Q}_\ell}^\times}\mathcal{L}_{\chi(x-\alpha)}\otimes \mathrm{Unip}(\alpha,\chi,\mathcal{F}).$$ Similarly, at $\infty$, we write $$\mathcal{F}|_{I(\infty)^\mathrm{tame}}\cong\bigoplus_{\chi:\pi_1(\mathbb{G}_m)^\mathrm{tame}\rightarrow \overline{\mathbb{Q}_\ell}^\times}\mathcal{L}_\chi \otimes \mathrm{Unip}(\infty,\chi,\mathcal{F}).$$ For $s\in S$, we write $e_i(s,\chi,\mathcal{F})$ to be the number of Jordan blocks of length $\ge i$ in the unipotent matrix $\mathrm{Unip}(s,\chi,\mathcal{F})$ (which can be viewed as the dual partition of $\mathrm{Unip}(s,\chi,\mathcal{F})$).

To guess what to do, we look at the rank formula (proved via Fourier transform by Katz): for any nontrivial $\chi$, $$\rank\mathrm{MC}_\chi(\mathcal{F})=|D|\rank\mathcal{F}-\sum_{a\in D} e_1(\alpha,\mathbf{1},\mathcal{F})-\dim(\mathcal{F}\otimes \mathcal{L}_\chi)^{I(\infty)}.$$ If we want to drop the rank, we want to maximize the number of eigenvalues 1 by twisting and then take the middle convolution with respect the $\chi$ that maximizes $\dim(\mathcal{F}\otimes \mathcal{L}_\chi)^{I(\infty)}$.

For each $\alpha\in D$, we choose $\chi_\alpha$ such that $e_1(\alpha,\chi_\alpha,\mathcal{F})$ is maximal. We form $\mathcal{L}\in\mathcal{T}_\ell$ of rank 1 such that $\mathcal{L}|_{I(\alpha)^\mathrm{tame}}\cong\chi_\alpha^{-1}$, i.e., $$\mathcal{L}=\bigotimes_{\alpha\in D} \mathcal{L}_{\chi_\alpha^{-1}}(x-\alpha).$$ Then $\mathrm{MT}_\mathcal{L}(\mathcal{F})$ has larger $e_1(\alpha,\mathbf{1})$ than $\mathcal{F}$ for any $\alpha\in D$.

We replace $\mathcal{F}$ by the resulting twist $\mathrm{MT}_\mathcal{L}(\mathcal{F})$ and choose $\chi$ such that $\dim (\mathcal{F} \otimes \mathcal{L}_\chi)^{I(\infty)}$ is maximal. In order to apply Katz's rank formula, we claim that any such $\chi$ is nontrivial. Assume this claim is true. By the rank formula and the Euler characteristic formula, we have 
  \rank\mathrm{MC}_\chi(\mathcal{F})&=|D|\rank\mathcal{F}-\sum_{\alpha\in D}e_1(\alpha,\mathbf{1},\mathcal{F})-\dim(\mathcal{F} \otimes \mathcal{L}_\chi)^{I(\infty)}\\
  &=-\chi(\mathbb{P}^1,j_*\mathcal{F})+\rank\mathcal{F}+e_1(\infty,\mathbf{1},\mathcal{F})-\dim(\mathcal{F} \otimes \mathcal{L}_\chi)^{I(\infty)}.
So to see the rank actually drops, we just need to show that $$\chi(\mathbb{P}^1,j_*\mathcal{F})+\dim (\mathcal{F}\otimes \mathcal{L}_\chi)^{I(\infty)}-e_1(\infty,\mathbf{1},\mathcal{F})>0.$$ For this we need the cohomological rigidity $$2=\chi(\mathbb{P}^1,j_*\End\mathcal{F})=(1-|D|)(\rank \mathcal{F})^2+\sum_{s\in S}\dim(\End\mathcal{F})^{I(s)}.$$ The last term can be written as $$\sum_s\sum_\chi\sum_i e_i(s,\chi,\mathcal{F})^2.$$ By the maximality of $\chi_s$, we know the last term is at most $$\sum_s\sum_\chi\sum_i e_1(s,\chi_s,\mathcal{F})e_i(s,\chi,\mathcal{F})=\sum_se_1(s,\chi_s,\mathcal{F})\rank\mathcal{F}.$$ So the cohomological rigidity implies that 
0<2&=\chi(\mathbb{P}^1,j_*\End\mathcal{F})\le \rank\mathcal{F}((1-|D|)\rank\mathcal{F}+\sum_se_1(s,\chi_s,\mathcal{F}))\\&=\rank\mathcal{F}(\chi(\mathbb{P}^1,j_*\mathcal{F})-e_1(\infty,\chi_\infty,\mathcal{F})+e_1(\infty,\mathbf{1},\mathcal{F}))
which implies what we wanted since $$e_1(\infty,\chi_\infty,\mathcal{F})=\dim(\mathcal{F}\otimes \mathcal{L}_\chi)^{I(\infty)},$$ where $\chi=\chi_\infty^{-1}$ by definition.

It remains to prove the claim. Assume that $\chi$ is trivial, then the same argument implies that $$2=\rank\mathcal{F}\cdot\chi(\mathbb{P}^1,j_*\mathcal{F}).$$ But since $\mathcal{F}$ is irreducible, we know that $\chi(\mathbb{F}^1,j_*\mathcal{F})\le0$, a contradiction.


Since $\mathrm{MC}_\chi$ and $\mathrm{MT}_\mathcal{L}$ are both reversible operations,

Corollary 20 Given a tuple of monodromies (at each $s\in S$), we can apply Katz's algorithem above to determine whether this tuple actually arises as local monodromies of an irreducible cohomologically rigid local system. This solves the Deligne-Simpson problem in the cohomologically rigid irreducible case.
Example 29 Consider the Dwork family over $\mathbb{P}^1-\{\infty,\mu_{n+1}\}$, $$\mathcal{X}_t: X_0^{n+1}+\cdots X_n^{n+1}=(n+1)tX_0\cdots X_n.$$ The group $$\Gamma=\{(\zeta_0,\ldots\zeta_n)\in\mu_{n+1}^{n+1}:\prod\zeta_i=1\}$$ acts on each fiber (where the diagonal acts trivially). Consider the local system of rank $n$, $$\mathcal{F}=(R^{n-1}\pi_* \overline{\mathbb{Q}_\ell})^\Gamma.$$ $\mathcal{F}$ is not rigid but there exists a rigid local system $\mathcal{G}$ on $\mathbb{P}^1-\{0,1,\infty\}$ such that $[n+1]^*\mathcal{G}\cong\mathcal{F}$. $\mathcal{G}$ is actually hypergeometric with local monodromies:
  1. regular nilpotent $U(n)$ at $\infty$,
  2. a pseudo relfection at 1,
  3. $\diag\{\zeta,\zeta^2,\ldots,\zeta^n\}$ where $\zeta$ is a primitive $n+1$ root of unity.
Exercise 1 Starting with the above local monodromies, apply Katz's algorithem to reduce to the rank 1 case.

TopLocal systems of type $G_2$

Let $k$ be an algebraically closed field of characteristic not $\ell$ (or 2).

Theorem 23
  • Fix $\alpha_1\ne\alpha_2\in\mathbb{A}^1(k)$. Let $\phi,\eta:\pi_1(\mathbb{G}_m)^\mathrm{tame}\rightarrow \overline{\mathbb{Q}_\ell}^\times$ such that $\phi,\eta,\phi\eta,\phi\eta^2,\eta\phi^2,\phi\eta^{-1}\ne-\mathbf{1}$. Then there exists an irreducible cohomologically rigid local system $\mathcal{F}=\mathcal{F}(\phi,\eta)\in\mathcal{T}_\ell$ of rank 7 with the local monodromies:
  1. at $\alpha_1$: $-\mathbf{1}^{\oplus 4} \oplus \mathbf{1}^{\oplus 3}$,
  2. at $\alpha_2$: $U(3) \oplus U(2) \oplus U(2)$,
  3. at $\infty$: any of the following (determined by the conditions on $\phi$ and $\eta$),
    1. $U(7)$,
    2. $\phi U(3) \oplus \phi^{-1}U(3) \oplus \mathbf{1}$,
    3. $\phi U(2) \oplus \phi^{-1}U(2) \oplus \phi^2 \oplus \phi^{-2} \oplus \mathbf{1}$,
    4. $\phi U(2) \oplus \phi^{-1}U(2) \oplus U(3)$,
    5. $\phi \oplus \eta \oplus \phi\eta \oplus \phi\eta^{-1} \oplus \eta^{-1} \oplus \phi^{-1} \oplus \mathbf{1}$.

In each case, the monodromy group (the Zariski closure of the image of the monodromy representation) is $G_2\subseteq GL_7$.

  • Let $\mathcal{F}\in \mathcal{T}_\ell$ be cohomologically rigid, ramified at $\infty$ and have monodromy group $G_2$. Then $\mathcal{F}$ is ramified at exactly two points $\alpha_1,\alpha_2$ of $\mathbb{A}^1$. Moreover, up to permuting $\{\alpha_1,\alpha_2,\infty\}$, $\mathcal{F}$ is conjugate to one of the local systems above.
Remark 58 There are more $G_2$ local systems which are wildly ramified.
Proof We start with the second part. Suppose $\mathcal{F}$ is lisse exactly on $\mathbb{A}^1-D$, we want to know how big $D$ is. Since $\mathcal{F}$ is cohomologically rigid, we have $$\sum_{s\in D\cup\{\infty\}}\dim\mathrm{Cent}(\mathcal{F}|_{I(S)})=(|D|-1)\cdot 49+2.$$ We look at the table of the centralizers of conjugacy classes of $G_2$ in $GL_7$ (copied from Dettweiler-Reiter), we see the largest dimension is 29.

So $(|D|+1)\cdot 29\ge (|D|-1)\cdot 49+2$, hence $|D|\le3$. When $|D|=3$, then the four centralizer adds to dimension 100. We look at the table again and there are only the following three cases

  1. $(25,25,25,25)$,
  2. $(29,29,29,13)$,
  3. $(29,29,25,17)$.

We can rule out all these three cases due to the necessary criterion for irreducibility: $\chi(\mathbb{P}^1,j_*\mathcal{F})\le0$. Namely, the Euler characteristic formula tells us that $$(1-|D|)\cdot7 +\sum_s e_1(s,\mathbf{1},\mathcal{F})\le0,$$ or $$\sum_se_1(s,\mathbf{1},\mathcal{F})\le14.$$ For example, $(25,25,25,25)$ case is ruled out because then the Jordan form is $-1^{\oplus 4}\oplus 1^{\oplus 3}$ by the table: if we twist by the character with local monodromies $-1,-1,-1$ at the three finite points (and $-1$ at $\infty$), we then get a new local system with local monodromies $1^{\oplus 4}\oplus (-1)^{\oplus 3}$, which violates the above irreducible criterion. Other cases are similar.

Now $|D|=2$. Again the cohomological rigidity implies that the sum of dimensions of the three centralizers is 51. The table implies that the possibilities are

  1. $(29,13,9)$,
  2. $(29,11,11)$,
  3. $(25,19,7)$,
  4. $(25,17,9)$,
  5. $(25,13,13)$,
  6. $(19,19,13)$,
  7. $(17,17,17)$.

The necessary criterion of irreducibility implies that $7\ge\sum e_1(s,\mathbf{1},\mathcal{F})$. This excludes the cases a), d), e), g). Since the monodromy group is assumed to be all of $G_2$ (so far we only used the monodromy group is contained in $G_2$), the (14 dimensional) adjoint representation of $\pi_1\rightarrow G_2\rightarrow GL(\mathfrak{g}_2)$ is also irreducible (the adjoint representation is irreducible for general simple reductive groups). Applying the irreducibility criterion gives $$(1-|D|)\cdot 14+\sum\dim\mathrm{Cent}_{G_2}(\mathcal{F}|_{I(s)})\le0.$$ So the sum of dimensions of the three centralizers in $G_2$ is less than 14. This excludes the cases b), f). The corresponding $G_2$ centralizers for case c) must be $(6,6,2)$. The local monodromies are

  • $-\mathbf{1}^{\oplus 4} \oplus \mathbf{1}^{\oplus 3}$ at $\alpha_1$,
  • $U(3) \oplus U(2)\oplus U(2)$ at $\alpha_2$,
  • several possibilities at $\infty$.

The first part the follows by checking which of these possibilities can arise using Katz's algorithm.

For example, take the $U(7)$ case at the third point $\infty$. Write $\mathcal{L}(\chi_1,\chi_2)$ for the rank 1 local system with local monodromies $\chi_1$ at $\alpha_1$ and $\chi_2$ at $\alpha_2$. Twisting by $\mathcal{L}(-1,1)$ we get $1^{\oplus 4}\oplus (-1)^{\oplus 3}$, $U(3)\oplus U(2)\oplus U(2)$ and $-U(7)$. Then the rank formula together with the table tells us that the rank of the middle convolution $\mathrm{MC}_{-\mathbf{1}}$ is equal to $14-4-3-1=6$, with local monodromies $U(2)^{\oplus 3}$, $-U(2)\oplus -1^{\oplus 2} \oplus 1^{\oplus 2}$, $U(6)$. Twisting by $\mathcal{L}(1,-1)$ and take middle convolution, we obtain local monodromies $(-1^{\oplus 3}\oplus 1^{\oplus 2})$, $-1 \oplus U(2)^{\oplus 2}$, $U(5)$ (rank 5),... until we get down to $U(2)$, $U(2)$, $U(2)$ (rank 2) and $1,-1,-1$ (rank 1). The last rank 1 local system actually exists since $(-1)\cdot(-1)=1$. Now running the algorithm reversely proves that the original local system also exists.

The final thing to do is to prove the monodromy group is actually $G_2$. First notice that our monodromy representation $\rho$ is orthogonal. The dual representation has the same local monodromies (up to $GL_7$-conjugacy); since $\rho$ is physically rigid, this implies that $\rho\cong\rho^\vee$. Since the dimension 7 is odd, $\rho$ must be orthogonal. So $\rho$ maps into $O_7(\overline{\mathbb{Q}_\ell})$.

We use the fact that an irreducible subgroup $G$ of $O_7(\overline{\mathbb{Q}_\ell})$ lies inside an $O_7$-conjugate of $G_2\subseteq O_7$ if and only if $(\bigwedge^3 \overline{\mathbb{Q}_\ell}^7)^G\ne0$. So we need to show that $H^0(U, \bigwedge^3\mathcal{F})\ne0$, i.e., $H^0(\mathbb{P}^1,j_*\bigwedge^3\mathcal{F})\ne0$. By the Poincare duality, this is equivalent to $H^2(\mathbb{P}^1,j_*\bigwedge^3 \mathcal{F})\ne0$. We compute the Euler characteristic, $$\chi(\mathbb{P}^1,j_*\bigwedge^3\mathcal{F})=-\rank \bigwedge^3\mathcal{F}+\sum \dim (\bigwedge^3\mathcal{F})^{I(s)}=-35+\dim(\bigwedge^3 \overline{\mathbb{Q}_\ell}^7)^{U(7)}+\cdots.$$

  • At $\infty$: the $U(7)$ is the image of $U(2)$ of $SL_2$ in $SL_7$ via the 6-th symmetric power $$\bigwedge^3\circ \Sym^6=\Sym^{12} \oplus \Sym^8 \oplus \Sym^6 \oplus \Sym^4 \oplus \mathbf{1}.$$ Since $\dim(\bigwedge^3 \overline{\mathbb{Q}_\ell}^7)^{U(7)}$ is the same as the number of irreducible constituents of this $SL_2$-representation, we know $\dim(\bigwedge^3 \overline{\mathbb{Q}_\ell}^7)^{U(7)}=5$.
  • At $\alpha_1$: the local monodromy is semisimple and it is easy to see that the dimension ${4\choose 2}\cdot 3+1=19$ (either two $-1$'s or no $-1$'s).
  • At $\alpha_2$: the local monodromy is the image of $U(2)$ of $SL_2$ via $\Sym^2 \oplus \Std \oplus \Std $. The number of irreducible constituents of $\bigwedge^3 (\Sym^2 \oplus \Std \oplus \Std)$ is 13.

So the Euler characteristic is $-35+19+5+13=2>0$. Hence $H^0(U,\bigwedge^3\mathcal{F})\ne0$. Therefore our $\rho$ lands inside $G_2$. The fact (going back to Dynkin) is that an irreducible subgroup of $G_2$ containing a regular unipotent element must be either $SL_2$ or $G_2$. The $U(3)\oplus U(2)\oplus U(2)$ rules out the possibility of $SL_2$.

Remark 59 One reason that the same realization of $E_6$ is harder: an irreducible subgroup of $E_6$ containing a regular unipotent element can be $F_4\subseteq E_6$. Another special feature about $G_2$ is that these rank 7 rigid local systems are also rigid when viewed as $G_2$-local systems (via the adjoint representation).


Remark 60 We wrap up with group theoretic consideration of the conjugacy classes of $G_2$. In particular, we explain why the three conjugacy classes constructed as local monodromies actually lies inside $G_2$.

Take a basis of simple roots $\Delta=\{\alpha_1,\alpha_2\}$. So $X^\cdot(T)=\mathbb{Z}\alpha_1 \oplus \mathbb{Z}\alpha_2$ (simply-connected and adjoint). Take the dual basis $\beta_1,\beta_2$, so $X_\cdot(T)=\mathbb{Z} \beta_1 \oplus \mathbb{Z} \beta_2$. The fundamental weights are $w_1=2\alpha_1+\alpha_2$ and $w_2=3\alpha_1+2\alpha_2$. Using fact that $(w_i,\alpha_j^\vee)=2(w_i,\alpha_j)/(\alpha_j,\alpha_j)=\delta_{ij}$ ($w_i$ are fundamental weights), one can find $\alpha_1^\vee=2\beta_1-3\beta_2$, $\alpha_2^\vee=-\beta_1+2\beta_2$. The 7-dimension representation $V_{w_1}$ has weights $\{\pm(2\alpha_1+\alpha_2,\pm(\alpha_1+\alpha_2),\pm\alpha_1,0\}$ (the nonzero weights form a single Weyl orbit, so it is quasi-minuscule).

The semisimple case is easy: $\langle\{\beta_2,\{\text{weights of }V_{w_1}\}\rangle=\{\pm1,\pm1,0,0,0\}$ gives the torus $\diag\{a,a,a^{-1},a^{-1},1,1,1\}$. Taking $a=-1$ gives exactly $(-1)^{\oplus 4}\oplus 1^{\oplus 3}$.

$U(7)$ is the regular nilpotent orbit (the unique maximal nilpotent orbit) in $\mathfrak{sl}_7$. In general $\mathfrak{g}_1\rightarrow \mathfrak{g}_2$ does not necessarily take regular nilpotents to regular nilpotents, but this is the case for $\mathfrak{g}_2\rightarrow \mathfrak{gl}_7$. In fact, to compute where the principle $\mathfrak{sl}_2$-triple $\{X,H,Y\}$ in $\mathfrak{g}_2$ goes in $\mathfrak{g}_2\xrightarrow{\rho_{w_1}}\mathfrak{gl}_7$, just compute the pairings $\langle H,\text{weights of }V_{w_1}\rangle=\{\pm6,\pm4,\pm2,0\}$ (since $\alpha(H)=2$ for any simple root $\alpha$). It follows that the composite $\mathfrak{sl}_2\rightarrow \mathfrak{g}_2\rightarrow \mathfrak{gl}_7$ is the 6-th symmetric power, hence maps $X$ to the regular nilpotent class.

The last case is $U(3)\oplus U(2)\oplus  U(2)$. To any subset $\Theta\subseteq \Delta$, let $\mathfrak{p}_\Theta$ be the corresponding parabolic $H_\Theta$. For $\mathfrak{g}_2$, we have two parabolic subgroups $\mathfrak{b}\subseteq \mathfrak{p}_{\alpha_1},\mathfrak{p}_{\alpha_2}\subseteq \mathfrak{g}_2$. Each $\mathfrak{p}_{\alpha}$ has Levi subgroup $\mathfrak{b}_{\alpha}=\mathfrak{t}\oplus \mathfrak{g}_{\alpha} \oplus \mathfrak{g}_{-\alpha}$ with semisimple part a $\mathfrak{sl}_2$. To compute the image of the regular nilpotent in $\mathfrak{sl}_2\xrightarrow{\rho_{\alpha}} \mathfrak{g}_2\xrightarrow{\rho_{w_1}} \mathfrak{gl}_7$, just to compute $\langle H_\alpha, \text{weights of }V_{w_1}\rangle=\langle\text{weights of }V_{w_1},\alpha^\vee\rangle$, which is $\{\pm1,\pm1,\pm2,0\}$ for $\alpha=\alpha_1$. So the composite is the map $\Sym^2 \oplus \Std \oplus \Std$ as desired. (Similar, it gives $U(2)\oplus U(2)\oplus  1^{\oplus3}$ when $\alpha=\alpha_2$.)

TopUniversal rigid local systems

Let $k$ be an algebraically closed field, $\ell\ne\Char(k)$. Fix an order of quasi-unipotence $N\in \mathbb{Z}_{\ne0}$ ($\ell\ne N$) and fix $\zeta_N$ a primitive $N$-th root of unity.

Let $\alpha_1,\ldots\alpha_n\in\mathbb{A}^1(k)$ and $\mathcal{F}_k\in \mathcal{T}_\ell$ be a cohomologically rigid local system, lisse on $\mathbb{A}^1-\{\alpha_1,\ldots,\alpha_n\}$ with eigenvalues contained in $\mu_N$. We will show how to produce a local system over the arithmetic configuration space of $n$ points, whose geometric fibers over $k'$ are cohomologically rigid objects in $\mathcal{T}_\ell(k')$ lisse away from $n$ and one of which gives the original $\mathcal{F}_k $.

Definition 44 Let $R_{N,\ell}=\mathbb{Z}[\zeta_N,1/N \ell]$ and fix a nonzero map $R_{N,\ell}\rightarrow k$. The configuration space over $R_{N,\ell}$ is defined to be ($\Spec$ of) $$S_{N,n,\ell}=R_{N,\ell}[T_1,\ldots, T_n][1/\prod_{i\ne j}(T_i-T_j)].$$

The universal rigid local system will live on $\mathbb{A}^1_{S_{N,n,\ell}}-\{T_1,\ldots,T_n\}$, i.e., $$\Spec R_{N,\ell}[T_1,\ldots,T_n,X][1/\prod(T_i-T_j)\prod(X-T_i)].$$ Notice that one can specialize to $k$ via $T_i\mapsto\alpha_i$.

Theorem 24 Fix $R_{N,\ell}\hookrightarrow \overline{\mathbb{Q}_\ell}$ (inducing $\mathrm{Frac}(R_{N,\ell})_\lambda=E_\lambda\hookrightarrow \overline{\mathbb{Q}_\ell}$).
  1. There exists a lisse $E_\lambda$-sheaf $\mathcal{F}$ on $\mathbb{A}^1_{S_{N,n,\ell}}-\{T_1,\ldots,T_n\}$, which, after specializing along $S_{N,n,\ell}\rightarrow k$, recovers $\mathcal{F}_k|_{\mathbb{A}^1-\{\alpha_1,\ldots,\alpha_n\}}$.
  2. Let $j: \mathbb{A}^1_{S_{N,n,\ell}}-\{T_1,\ldots,T_n\}\hookrightarrow \mathbb{A}^1_{S_{N,n,\ell}}$. The restriction of $j_*\mathcal{F}$ to any geometric fiber is a cohomologically rigid object in the corresponding category $\mathcal{T}_\ell$ (i.e., specialization preserves index of rigidity, tameness, irreducibility, and in some sense preserves the local monodromies as $\mathcal{F}_k $).
  3. $\mathcal{F}$ is pure of some integer weight. The characteristic polynomials of the Frobenius (when specializing to a finite field) lies in $\mathbb{Z}[\mu_N]$.
  4. For any other prime $\ell'$ and $\lambda': \mathbb{Z}[\mu_N]\hookrightarrow \overline{\mathbb{Q}_\ell}$. There exists a lisse $E_{\lambda'}$-sheaf $\mathcal{F}_{\lambda'}$ on $\mathbb{A}^1_{S_N,n,\ell'}-\{T_1,\ldots,T_n\}$ satisfying for any $\psi: \mathbb{A}^1_{S_{N,n,\ell,\ell'}}-\{T_1,\ldots,T_n\}\rightarrow\mathbb{F}_q$, the characteristic polynomials of $\Frob_\psi$ on $\mathcal{F}_{\lambda'}$ is equal to the characteristic polynomial of $\Frob_\psi$ on $\mathcal{F}_\lambda$ in $\mathbb{Z}[\mu_N]$. In other words, we get a compatible system of $\ell$-adic representations.
Proof We will need to make sense of middle convolution in this relative setting to run Katz's algorithm. Admitting that there is a well-behaved notion of middle convolution on $\mathbb{A}^1_R$ for some reasonable ring $R$ (e.g, $R=S_{N,n, \ell}$), we can induct on $\rank\mathcal{F}_k$.

When $\rank(\mathcal{F}_k)=1$. Let $\chi_i: I(\alpha_i)\rightarrow \overline{\mathbb{Q}_\ell}^\times$ be the local monodromy character, of order dividing $N $; so $\mathcal{F}_k\cong \bigotimes_{i=1}^n\mathcal{L}_{\chi_i(x-\alpha_i)}$. Now we can spread this out by interpreting $\mathcal{L}_{\chi_i(x-\alpha_i)}$ as associated to $$\{y^N=x-\alpha_i\}\rightarrow \mathbb{A}^1-\{\alpha_i\},$$ a Galois covering with Galois group $\mu_N(k)$. From the fixed map $R_{N,n,\ell}\rightarrow k$, we can identify $\mu_N(S_{N,n,\ell})=\mu_N(k)$, so we can view $\chi: \mu_N(k)\rightarrow \overline{\mathbb{Q}_\ell}^\times$ as a character of the Galois group of the covering $$\{y^N=X-T_i\}\rightarrow \mathbb{A}_{S_{N,n,\ell}}^1-\{T_i\}.$$ In this way we obtain a lisse sheaf $\mathcal{L}_{\chi_i(X-T_i)}$ on $\mathbb{A}_{S_{N,n,\ell}}^1-\{T_i\}$. Then $$\mathcal{F}=\bigotimes_{i=1}^n\mathcal{L}_{\chi_i(X-T_i)}$$ works.

By Katz's algorithm, we can find $\mathcal{L}_k$ of rank 1 lisse on $\mathbb{A}^1-\{\alpha_1,\ldots,\alpha_n\}$ and a nontrivial character $\chi:\pi_1(\mathbb{G}_m)^\mathrm{tame}\rightarrow \overline{\mathbb{Q}_\ell}^\times$ such that $\mathcal{G}_k=\mathrm{MC}_\chi(\mathrm{MT}_{\mathcal{L}_k})(\mathcal{F}_k)$ has rank strictly smaller than $\mathcal{F}_k $. We can now again spread $\mathcal{L}_k$ to $\mathcal{L}$ as above and by induction we can spread out $\mathcal{G}_k $ to $\mathcal{G}$. Now invoke the middle convolution with parameters $$\mathcal{F}=\mathcal{L}^{-1} \otimes j^*(j_*\mathcal{G}[ 1 ]*_\mathrm{mid} j_{0,*}\mathcal{L}_\chi[ 1 ])$$ is what we are looking for, where $j_0:\mathbb{G}_{m,S_{N,n,\ell}}\hookrightarrow\mathbb{A}^1_{S_{N,n,\ell}}$. In the next section, we will make sense of $*_\mathrm{mid}$ in a way that commutes with base change to $R$ and for $R=k$ specializes to the old notion of middle convolution.


TopMiddle convolution with parameters

Let $R$ be a normal domain, finite type over $\mathbb{Z}$. Let $D\subseteq \mathbb{A}^1_R$ be a divisor given by the equation $g_D(X)=\prod(X-r_i)=0$, where $r_i\in R^\times$ are all distinct. Let $D'\subseteq \mathbb{A}^1_R$ be another divisor given by $g_{D'}+\prod (X-r_i')=0$.

Remark 61 For the application, $R=S_{N,n,\ell}$, $D$ is union of $T_i$-hyperplanes, given by $g_D(X)=\prod(X-T_i)=0$; $D'{}=\{0\}$, given by $X=0$.

The middle convolution is an operation $$\mathrm{LS}(\mathbb{A}^1-D)\times \mathrm{LS}(\mathbb{A}^1-D')\rightarrow \mathrm{LS}(\mathbb{A}^1-D*D'),$$ where $D*D'{}$ is given by $\prod (X-(r_i+r_j'))$. For the application, $D*D'{}=D$ since $D=\{0\}$.

Definition 45 Define $$\mathbb{A}(2)=\Spec R[X_1,X_2][1/g_D(X_1)g_{D*D'}(X_2)g_{D'}(X_2-X_1)].$$

Then we have the following diagram $$\xymatrix{ &  \mathbb{A}(2) \ar[ld]_{\pr_1} \ar[d]^{d} \ar[rd]^{\pr_2}&  \\ \mathbb{A}^1_R-D & \mathbb{A}_R^1-D' & \mathbb{A}_R^1-D*D'},$$ where $\pr_1,\pr_2$ are the projections and $d$ is the difference morphism.

We denote $j:\mathbb{A}(2)\rightarrow \mathbb{P}^1\times (\mathbb{A}^1-D*D')$. Then the compactified projection $\overline{\pr}_2: \mathbb{P}^1\times(\mathbb{A}^1-D*D')\rightarrow \mathbb{A}^1-D*D'$ is proper smooth.

Definition 46 Let $\mathcal{F}\in\mathrm{LS}(\mathbb{A}^1_R-D)$, $\mathcal{F}'\in\mathrm{LS}(\mathbb{A}^1_R-D')$, we define the middle convolution $$\mathcal{F}*_\mathrm{mid}\mathcal{F}'{}=R^1\overline{\pr}_{2,*}(j_*(\pr_1^*\mathcal{F} \otimes d^*\mathcal{F}'))$$ and the naive convolution $$\mathcal{F}*_\mathrm{naive}\mathcal{F}'{}=R^1\pr_{2,!}(\pr_1^*\mathcal{F}\otimes d^*\mathcal{F}').$$
Proposition 13 Assume either that everything is tame or $R$ has a generic point of characteristic 0.
  1. $\mathcal{F}*_\mathrm{naive}\mathcal{F}'$ and $\mathcal{F}*_\mathrm{mid}\mathcal{F}'$ are lisse and tame.
  2. Assume $\mathcal{F}$ is pure of weight $w$ and $\mathcal{F}'$ is pure of weight $w'$. Then $\mathcal{F}*_\mathrm{naive}\mathcal{F}'$ is mixed of weights $\le w+w'+1$; $\mathcal{F}*_\mathrm{mid}\mathcal{F}'$ is pure of weight $w+w'+1$. Even better, the middle convolution $$\mathcal{F}*_\mathrm{mid}\mathcal{F}'{}=\Gr_{w+w'+1}^W(\mathcal{F}*_\mathrm{naive}\mathcal{F'}),$$ the top graded piece in the weight filtration of the naive convolution.
  3. Assume $\mathcal{F}$ or $\mathcal{F}'$ is geometrically irreducible and nonconstant. Write $\mathcal{G}=\pr_1^*\mathcal{F}\otimes d^*\mathcal{F}'$ for short. Then $$R^i\pr_{2,!}\mathcal{G}=0,\quad R^i\overline{\pr}_{2,*}(j_*\mathcal{G})=0,$$ unless $i=1$.
Proof (Ideas of the proof)

a. Let us take $D'{}=\{0\}$. Then $\mathbb{A}(2)$ is the complement of hyperplanes in the $X_1$ and $X_2$ directions and the diagonal. Recall that $R^i\overline{\pr}_{2,*}j_*\mathcal{G}$ is the sheaf associated to $U\mapsto H^i(\overline{\pr}_2^{-1}(U), j_*\mathcal{G})$. Since the projection $\overline{\pr}_2$ is trivialized with respect to the stratification $\mathbb{A}(2)\subseteq \mathbb{P}^1\times (\mathbb{A}^1-D)$, it can be computed as $H^i(\pr_2^{-1}(u_0)\times U, j_{u_0,*}\mathcal{G}|_{\pr_2^{-1}(u_0)} \boxtimes \overline{\mathbb{Q}_\ell}_{,U})$. It follows that the middle convolution is lisse. For a formal proof, see Katz, Sommes exponentielles, Section 4.7.

b. Recall that if $\mathcal{F}$ is an $\overline{\mathbb{Q}_\ell}$-sheaf on $Y$ (a scheme of finite type over $\mathbb{Z}$), we say that $\mathcal{F}$ is pure of weight $w\in \mathbb{Z}$ if for any closed points $s: \Spec \mathbb{F}_q\rightarrow Y$, $s^*\mathcal{F}$ is pure of weight $w$ in the familiar sense. We say $\mathcal{F}$ is mixed if it admits a filtration by subsheaves such that $\Gr_i\mathcal{F}$ are all pure.

Theorem 25 (Weil II) Suppose $f:X\rightarrow Y$ is a morphism of schemes of finite type over $\mathbb{Z}$ ($\ell$ is invertible on $X,Y$). If $\mathcal{F}$ is mixed of weight $\le w$ on $X$ then $R^if_!\mathcal{F}$ is mixed of weight $\le w+i$.

It follows that the naive convolution is mixed of weight $w+w'+1$. The purity of the middle convolution follows from the analogous statement for curves over finite fields:

Theorem 26 If $j:U\hookrightarrow X$ is a smooth curve, $\mathcal{F}$ lisse on $U$ pure of weight $w$, then $H^1(X,j_*\mathcal{F})$ is pure of weight $w+1$.

This result is less surprising by noticing that $H^1(X, j_*\mathcal{F})=\im(H^1_c(U,\mathcal{F})\rightarrow H^1(U,\mathcal{F}))$ (the source is mixed of weight $\le w+1$ and the target is mixed of weight $\ge w+1$). To show the graded piece statement, we use the short exact sequence of sheaves $$0\rightarrow j_!\mathcal{G}\rightarrow j_*\mathcal{G}\rightarrow i_*i^*j_*\mathcal{G}\rightarrow0,$$ which gives a long exact sequence $$R^0\overline{\pr}_{2,*}(i_*i^*j_*\mathcal{G})\rightarrow \mathcal{F}\otimes_\mathrm{naive}\mathcal{F}'\rightarrow \mathcal{F}\otimes_\mathrm{mid}\mathcal{F}'\rightarrow R^1\overline{\pr}_{2,*}(i_*i^*j_*\mathcal{G}).$$ The last term is zero since $i_*i^*j_*\mathcal{G}$ is punctual. Since $j_*$ is an open immersion, we know that $j_*\mathcal{G}$ has weights $\le w+w'$ by 1.8.9 in Weil II. So we are done since $R^0$-term is mixed of weight $\le w+w'$.

c. We prove the ! version. We can check on geometric fibers and it suffices to show for $i=2$. Notice that $\mathcal{G}|_{s\text{-fiber}}$ is lisse on $\mathbb{A}^1-D\cup\{s\}$ and $H^2_c(\mathbb{A}^1-D,\mathcal{G}|_{s\text{-fiber}})$ is the coinvariants under the local monodromy at $s$. In our case, $\pr_1^*\mathcal{F}$ is unramified along $X_1=X_2$, but $d^*\mathcal{L}_\chi$ is indeed ramified, and hence the coinvariants is 0.

Remark 62 We claim that Definition 46 does recover the old definition of the middle convolution (Definition 41) when $R$ is an algebraically closed field. This finalizes the description of the middle convolution algorithm in the universal context: it produces local systems on $\mathbb{A}^1_{S,N,\ell}-\{T_1,\ldots,T_n\}$ that specializes to cohomologically rigid tame irreducible local systems on $\mathbb{A}^1_k-\{\alpha_1,\ldots,\alpha_n\}$ for $k$ algebraically closed.

Notice by definition $$K*_!K_\chi=R \pr_{2,!}(\pr_1^*K \otimes d^* K_\chi),$$ where $K=j_{D,*}\mathcal{F}[ 1 ]$ and $K_\chi=j_{0,*}\mathcal{L}_\chi[ 1 ]$. Since one can replace $j_{!*}$ by $j_*$ (Example 26) and hence the claim follows from $$K*_\mathrm{mid}K_\chi=R\overline{\pr}_{2,*}(j_{!*}(\pr_1^*K \otimes d^*K_\chi).$$ This is a special case of the following theorem (take $U=\mathbb{A}(2)$, $f=\pr_2$, $K=\pr_1^*K \otimes d^*K_\chi$).

Theorem 27 Suppose $j: U\rightarrow X$ is affine open, $S$ is separated finite type of an algebraically closed field $k$. $\bar f: X\rightarrow S$ is proper and $f=\bar f|_U$ is finite. Suppose $K\in\mathrm{Perv}(U)$ such that $Rf_!K$ and $Rf_*K$ are perverse. Then $$\im(Rf_!K\rightarrow Rf_*K)=R\bar f_*(j_{!*}K)$$ in $\mathrm{Perv}(S)$.
Proof We have two exact sequences in $\mathrm{Perv}(X)$, $$0\rightarrow\ker\rightarrow j_!K\rightarrow j_{!*}K\rightarrow0,$$ $$0\rightarrow j_{!*}K\rightarrow Rj_*K\rightarrow\coker \rightarrow 0.$$ The kernel and cokernel are supported on $D$, applying $R\bar f_*$, we obtain two distinguished triangle on $S$, four out of the six terms are perverse by assumption. One can check that $R\bar f_*j_{!*}K$ is perverse by taking the long exact sequence on cohomology, hence the two distinguished triangles are indeed short exact sequences of perverse sheaves. Splicing together we obtain the result.
Remark 63 Our next goal is to show that these universal cohomologically rigid local systems $\mathcal{F}$ have the following geometric realization: there exists a smooth family $\pi: X\rightarrow \mathbb{A}^1_{S_{N,n,\ell}}-\{T_1,\ldots,T_n\}$ with a finite group $\Gamma$ action and an idempotent $e\in \overline{\mathbb{Q}_\ell}[\Gamma]$ such that $$\mathcal{F}=\Gr_r^W(e R^r\pi_! \overline{\mathbb{Q}_\ell}).$$ Moreover, for any $s\ne r$, $eR^s\pi_! \overline{\mathbb{Q}_\ell}=0$.