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 be a field with algebraic closure (so ). Consider smooth projective varieties over (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 (a -vector space), the singular cohomology of the topological space .
- the de Rham realization (a -vector space with a Hodge filtration), the algebraic de Rham cohomology = the hypercohomology of the sheaf of algebraic differential forms on .
- the -adic realization (a -vector space with -action), the -adic etale cohomology.
It is not even clear a priori that these -vector space, -vector space and -vector have the same dimension. But miraculously there are comparison isomorphisms between them. For example,
(Comparison for B-dR)
There are isomorphisms
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")
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.
By the early 60's, one knew that if
is a smooth projective variety, then it follows from Hodge theory that
naturally carries a pure Hodge structure
, i.e. a
with a bi-grading
is the complex conjugation with respect to
. On the other hand, Weil had conjectured that for a smooth projective variety
, in the sense that the eigenvalues
of the geometric Frobenius
are algebraic numbers and for each embedding
into the complex numbers,
When is smooth but not projective, people played with examples and found that can be filtered (the weight filtration) such that the Frobenius eigenvalue is pure on each graded piece.
be a smooth projective curve,
be a finite set of points and
. Then we have an exact sequence
is pure of weight 1 and
-adic cyclotomic character) is pure of weight 2 and
is also pure of weight 2. Therefore one obtains an increasing weight filtration on
The above mentioned -adic intuition (generalized to higher dimension) lead Deligne to mixed Hodge theory. To give a mixed Hodge structure for not smooth projective, the key point is to find a spectral sequence such that its term is (conjecturally) pure of weight .
In Hodge II, Deligne treated the case of smooth but no longer projective varieties . The (-adic analogue of the) spectral sequence is the Leary spectral sequence for , where a smooth compactification of with is a union of smooth divisors with normal crossings, One can explicitly compute the sheaf where is smooth. Therefore is pure of weight .
Let us look at the differential : notice both the target and the source are pure of weight (all are pure of weight ), nothing is weired. But on the -page, , where the source has weight and the target has weight respectively. The mismatching of the weight of the Frobenius eigenvalues implies that for . Therefore the Leray spectral sequence degenerates at -page. One can compute that The Betti analogue (of maps of pure Hodge structure) is provided by the reinterpretation that and the differentials 's are simply Gysin maps ( = Poincare dual to pullbacks), which are also maps of pure Hodge structures.
The upshot is that the -adic Leray spectral sequence gives the weight filtration (= the Leary filtration up to shift), and the graded piece is pure of weight . The Betti Leray sequence also gives a weight (defined to be) filtration on such that we already know that are naturally pure Hodge structures.
Another motivation for considering the category of pure motives is toward a motivic Galois formalism.
be a field. The classical Galois theory establishes an equivalence between finite etale
-schemes with finite sets with
-actions. Linearizing a finite set
-action gives finite dimensional
with the continuous
. The linearization of finite etale
-schemes are the Artin motives
(motives built out of zero dimension motives). The equivalence between the two linearized categories is then given by
Generalizing to higher dimension: the category of finite etale -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 , which is a extension of classical Galois theory 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 as a quotient of (the recent work of Dettweiler-Reiter and Yun).
We now formulate the notion of Weil cohomology, in the frame work of motives.
be a field. Let
be smooth projective (not assumed to be connected) variety over
be the category of such varieties. Then
is a symmetric monoidal via the fiber product
with the obvious associative and commutative constraints and the unit
be a field. Let
be the category of finite dimensional graded
-vector spaces in degrees
with the usual tensor operation
It is endowed with a graded commutative constraint via
A Weil cohomology
(a field of characteristic 0) on
is a tensor functor
comes with a functorial (Kunneth) isomorphisms
respecting the symmetric monoidal structure. Notice the monoidal structure induces a cup product
a graded commutative
-algebra. We require it to satisfy the following axioms.
- (normalization) . In particular, is invertible in . We define the Tate twists (this is well motivated by -adic cohomology).
- (trace axiom) For any of (equi-)dimension , there is a trace map satisfying
- Under , one has .
- and the cup product induces a perfect duality (Poincare duality)
- (cycle class maps) Let be the -vector spaces with a basis consisting of integral closed schemes of codimension . Then there are cycle class maps satisfying
- factors through the Chow group (modulo the rational equivalence).
- is contravariant in , i.e., for a morphism and a cycle of codimension , we have whenever this makes sense. This will always make sense after passing to the Chow group. In general, one cannot always define on . But if is flat , then one can: in fact, by flatness has all its components of codimension in (but is not necessarily integral). Let be the (reduced structure) of the irreducible components. One then associates a cycle where , the length of the local ring at . We also require it to be compatible with pushforward (defined in Definition 10) that
- For , . Notice is not necessarily a combination of integral closed subscheme (e.g., is nonreduced), the cycle should be understood as the reduced structure with multiplicity.
- (pinning down the trace) the composite sends to , where are closed points.
(Trace in Betti cohomology)
be smooth projective of dimension
to be the composite
is the smooth de Rham complex with
is an isomorphism because the sheafy singular cochain complex is a flasque resolution of
is an isomorphism because
is a fine resolution of
. The choice
will chancel out the choice of the orientation we made on complex manifold when we do integration and one can check that
Algebraic de Rham cohomology
Suppose is a field of characteristic 0 and smooth (not necessarily projective).
We define the algebraic de Rham cohomology
to be the hypercohomology of the sheaf
of algebraic differential forms on
is a Weil cohomology.
We now selectively check a few of the axioms.
is a tensor functor.
Given a morphism
be an injective resolution of
be an injective resolution of
is quasi-isomorphic to
. The map
induces the pull-back on
The Kunneth isomorphism is explicitly given by , where and are the natural projections.
is affine, we have
(this follows from the vanishing of
quasi-coherent; in particular, ), which makes the computation feasible. For example, the de Rham complex for
. So taking cohomology gives
. This also gives an example in characteristic
is infinite dimensional because
, so one don't really want to work with the algebraic de Rham cohomology in characteristic
In general, one covers by open affines . For any quasi-coherent sheaf on , one then obtains the Cech complex , a resolution of by acyclic sheaves, defined by Now we have a double complex whose columns are acyclic resolutions of . The general formalism implies that the cohomology of the global sections of the total complex. Recall the total complex is defined by where .
be a covering of
. The Cech double complex looks like
The total complex is thus
where the two differentials are given by
One can easily compute
is 1-dimensional generated by
. Indeed one sees the computation really shows
This is an instance of the Hodge to de Rham spectral sequence.
is projective, then the Hodge to de Rham spectral sequence degenerates at the
To define the trace for the algebraic de Rham cohomology, we proceed in two steps. We first show that is abstractly the right thing, i.e., and is if is geometrically connected. Then we pin down that actual map 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 By Serre duality ( is the dualizing sheaf), one has the trace map So we want to say that the map is an isomorphism. By the Hodge de Rham spectral sequence, it is enough to show that (or because is a free -module of rank 1). This can be checked bare-handed by reduction to : choose a finite flat map to get the trace map (so ). It follows that is injective. It suffices to prove that which boils down to the direct computation that using the Hodge to de Rham spectral sequence.
To pin down , we need to choose carefully a generator of and set i.e., .
De Rham cycle class maps (via Chern characters)
We seek a cycle class map such that as follows.
- Define the Chern class of line bundles.
- Define the Chern class of vector bundles.
- Define the Chern character of vector bundles on .
- One knows that factors through , Grothendieck group of vector bundles on . Using the fact that is smooth, the latter can be identified with , the Grothendieck group of coherent sheaves on .
- For a codimension cycle, makes sense and we define the cycle class map
Now we describe each step in details.
We want a group homomorphism . Identify . The map induces a map and hence induces the desired map .
Let be a vector bundle of rank on . Denote the projective bundle . Notice on has a tautological line subbundle . Let . The fact (the Leray-Hirsch theorem, a special case of the Leray spectral sequence) is that is a free module over with basis . We then define by Notice this agrees with the previous definition of of line bundles and is functorial in .
Define the total Chern class The key is the following multiplicative property.
For any sort exact sequence of vector bundles
To show this, one first show that if
is a direct sum of line bundles, then
Then reduce the general to the first case by showing that there exists a map
splits as direct sum of line bundles and
is injective on
(the splitting principle
For the first case, since the statement is invariant under twist, one can assume each is very ample of the form and reduce to the case to the case of being a product of projective spaces. Notice that each gives a section and by definition . Write . Pullback the defining relation for along each , we obtain the relation So the polynomial in has roots . But has the advantage of being like a polynomial ring, Assume , then the defining relation must be , which shows that is the -th symmetric polynomial of as desired.
To reduce the general case to the first case, arrange so that has a full flag of subbundles by iterating the projective bundle construction, then split the extension by further pullback: if one has a surjection of vector bundles, then the sections form an affine bundle over ; pulling back along this affine bundle splits and induces an isomorphism on cohomology.
Using the multiplicative property, we can define formally the Chern roots of so that . Here the Chern roots don't not make sense but their the symmetric polynomials do make sense in cohomology. Define the Chern character This makes sense in cohomology. Now we have the additivity in exact sequences. Moreover . Therefore we obtain a ring homomorphism
When is smooth, one can form finite locally free resolutions of any coherent sheaves on , and taking the alternating sum of the terms in the resolutions induces the inverse of natural map . Thus (see Hartshorne, Ex III.6.8).
For a codimension cycle, makes sense and we define the cycle class map In particular, our choice of the basis for is given by for any closed point of , This is the choice we made to normalize the trace map. We need to check that is independent on the choice of (this follows from connecting two points by a curve in and the invariance of in a flat family). We also need to check that . This reduce to the case of projective spaces. Let be a closed point. One can put in a chain Using the short exact sequences of the form (given a choice of a section of ), for each , it follows that in , we have Applying the Chern character we obtain that for , which is nonzero.
Formalism of cohomological correspondences
Let be a Weil cohomology.
Given a morphism
, we define the Gysin map
to be the transpose of
under the Poincare duality. At the level of cycles,
and zero otherwise (this matches the degree shift in
A cohomological correspondence
is an element
interpreted (using the Poincare duality and the Kunneth formula) as a linear map
. Explicitly, if
(extended to be zero away from top degree). Let
be the natural projections. Then another way of writing
, intersect with
, then pushforward to
. One can check that
by the projection formula.
is defined to be the image of
. One can check that
(Composition of correspondences)
, we define
. The claim then follows from the associativity
of composition of correspondences. For details, see Fulton, Intersection theory, Chapter 16.
be the graph of the morphism
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.
Formal consequences of a Weil cohomology
Let be a Weil cohomology.
(Lefschetz fixed point)
is algebraically closed. Let
be connected. If
are of degree
; similar for
We compute by each Kunneth component so let
be a basis of
be a dual basis of
. So we can write
So the left hand side is equal to
introduces another sign
which cancels out the sign
. So the left hand side is equal to
To compute the trace on the right hand side, we notice that
Since we care only about the
-term when taking the trace, this matches the left hand side.
Let be a cohomological correspondence so that on . Write where is the cohomological correspondence . So .
is of degree zero). Then
Taking and using , we obtain the following refinement.
Now let and be the (absolute) Frobenius morphism. Then is the fixed point of for any .
(Grothendieck and others)
There exists a Weil cohomology on
To interpret the left hand side as the fixed points of , we need the following lemma.
intersect properly: every irreducible component of
is of codimension
(i.e., the codimensions add). So
can be computed as a sum of local terms, one for each point in
. Moreover, the local terms are multiplicity-free (by computing the tangent space intersection
at an intersection point
Therefore we conclude that
The Weil conjecture (expect the Riemann hypothesis) the follows.
The zeta function
can be computed as
The previous corollary of the Lefschetz fixed point theorem and the easy linear algebra identity
proves the claim.
Combining this cohomological expression of with the Poincare duality, we also obtain the functional equation of (part of the Weil conjecture).
is the Euler characteristic of
be a smooth quasi-projective variety. For any proper
(this is not serious since we will be working in
), we define pushforward cycles by
and 0 otherwise.
On the other hand, we defined pullback of cycles along a flat morphism (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, by intersecting with the graph of . 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 whose intersection has all components of the right codimension. Then should be a sum of irreducible components of with multiplicities here is the local ring of at the an irreducible component of the intersection . 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 to make the intersection properly meanwhile staying in the same rational equivalence class (moving lemma).
We say two cycles
are rationally equivalence if
is generated by terms of the following form. Let
-dimensional closed subvariety and take its normalization
; these generators are the proper pushfowards
An alternative approach is to consider a dimension closed subvariety. Then the rationally equivalent to zero cycles are generated by , here is the fiber of in .
These two definitions are equivalent. One can check that being rational equivalent is a equivalence relation. We denote it by .
The Chow group
Chow's Moving Lemma then gives a well defined intersection pairing on the Chow groups This makes a graded and commutative unital ring. The proper pushforward descends to the level of Chow groups.
We define the pullback
on Chow groups for
Adequate equivalences on algebraic cycles
We showed last time that
is an adequate relation.
For any Weil cohomology
, the cohomological equivalence
is an adequate relation. Here
. Notice that a priori these cohomological equivalences may not be independent of the choice of
. If two such Weil cohomology theories are related by comparison, e.g.,
, then the corresponding cohomological equivalences are the same.
is numerically equivalent 0
if for all
, here the degree map
(one can think of it as
, for the structure map
is an adequate relation.
- is the finest adequate equivalence relation.
- is the coarsest adequate equivalence relation.
- Let be an adequate relation. We want to show that if , then . By definition, is linear combination . Let and be the projections. Then Suppose we knew that . Then by the definition of adequate relation , we know . So we reduced to show that on . Let (assume for simplicity). Since is adequate, we can find intersecting properly with , i.e., with . We can certainly write down a map such that and . Explicitly, Therefore we have a chain of equivalences as desired.
- The second part is basically a tautology.
be an adequate equivalence relation on
be field of characteristic 0 (e.g.,
). We define
, the ring of cycles on
be the category with objects
(usually write it as
thought as a cohomological object), and
(Think: graphs of homomorphisms
.) This is an
-linear category, with
There is a functor
We want to enlarge 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,
We define the category
(the coefficient field
is implicit) of pure motives over modulo
. Its object is of the form
is an idempotent in
is an integer (Think:
). The morphisms are given by
Here the existence of Tate twists allows one to shift dimensions (e.g, a map
- is pseudo abelian ( = preadditive and every idempotent has a kernel).
- is -linear. The addition is given by (if ) Here we think of as the summand of and identify
- (next time) There is a -structure
Grothendieck conjectured (Standard Conjecture D) that for , for any Weil cohomology . He also conjectured that is abelian. Hence under Conjecture D, is abelian. Conjecture D is still widely open, but in the early 90s, Jannsen proved the following startling theorem.
That means that the numerical equivalence is arguably the "unique" right choice for the theory of motives.
Let be an additive tensor ( = symmetric monoidal) category. One can check for the unit object , then endomorphisms is a commutative ring and becomes an -linear category.
We say a category
if for any
("dual") and morphisms
such that the composite map
and the composite map
be a field. A neutral Tannakian category
is a rigid abelian tensor category with
and for which there exists a fiber functor
. By a fiber functor
, we mean a faithful, exact,
-linear tensor functor. It is neutralized by a choice of such a fiber functor. (Think: the category of locally constant sheaves of finite dimensional
-vector spaces on a topological space
; a fiber functor is given by taking the fiber over
The main theorem of Tannakian theory is the following.
be a neutral Tannakian category over
be a fiber functor. Then the functor on
is represented by an affine group scheme over
is an equivalence of categories.
The Kunneth Standard Conjecture (Conjecture C)
(Conjecture D) and that all Kunneth projectors
are all algebraic cycles (Kunneth). Then
-coefficients) is an a neutral Tannakian category over
By Jannsen's theorem,
is abelian. We saw last time that
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
-graded via the projectors
, i.e., for any
, we get a weight decomposition
Now for any
, we define the modified commutativity constraint
is a fiber functor.
- For any , and an abelian variety, is true.
- For a finite field, then is true for any (with respect to any Weil cohomology satisfying weak Lefschetz). This is a theorem of Katz-Messing. Deligne's purity theorem on allows one to distinguish different degrees. Katz-Messing shows that for any Weil cohomology with weak Lefschetz, the characteristic polynomial of the Frobenius on agrees with that on the -adic cohomology. Choose a polynomial such that (for ) and , then is algebraic (as the combinations of the graphs of ) and is the projection onto .
The Lefschetz Standard Conjecture (Conjecture B)
be a Weil cohomology. We say
satisfies the hard Lefschetz theorem if for any
, any ample line bundle
is an isomorphism. Here
, this is part of Hodge theory. For any
, this is proved by Deligne in Weil II.
The hard Lefschetz gives the primitive decomposition of .
(this depends on the choice of
One should think of as a nilpotent operator on , then the Jacobson-Morozov theorem implies that this action can be extended to a representation of . The primitive parts are exactly the lowest weight spaces for this -action.
Let . Then is semisimple and sends to . So applying Jacobson-Morozov gives a unique -triple (the name comes from Hodge theory). Moreover, it follows that .
A more convenient operator, the Hodge star
, can be extracted as follows. The
gives rise to a representation of
is the weight
is in the
is not quite an involution. So we renormalize and define
Another variant is the Lefschetz involution
as well. It differs from
from certain rational coefficients on each primitive component.
Now we have the following cohomological correspondences:
- , , , , ( is the inverse to on the image of ),
- Kunneth projectors ,
- Primitive projectors :
- For , for and 0 on ;
- For , for (so it satisfies ).
The following lemma is immediate.
are all given by universal (noncommutative) polynomials in
One can show that
Now we can state various versions of the Lefschetz Standard conjecture.
(Weak form )
is an isomorphism (i.e., it is surjective).
(Strong form )
is algebraic. Namely, it equals to the cohomology class a cycle in
Because , we know that
is algebraic and induces an isomorphism
is also algebraic (see Remark 14
gives a map
, this map is algebraic and an isomorphism. Hence
is an algebraic and an isomorphism. Therefore
is algebraic by Remark 14
is also algebraic.
is independent of the choice of the ample line bundle
giving rise to
is given by another ample line bundle
. Then the hard Lefschetz tells us that
is an algebraic isomorphism (notice the correspondence
is equal to
is algebraic when
is algebraic). Hence its inverse is also algebraic by the previous corollary. Now use e) of the previous proposition.
The 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 . For any , carries a pure -Hodge structure of weight . More fundamental in algebraic geometry is the polarizable -Hodge structure.
an ample line bundle, we have
can be thought of as the Kahler form in
(valid for general Kahler manifolds). Define
-linearly we define the sesquilinear pairing
We would like to study the positivity properties of by reducing to particular pieces of the bigrading and the primitive decomposition.
(Hodge index theorem)
is definite of sign
On a curve
On a surface
, and sign
is negative definite on
, positive definite on
and positive definite on
. For example, if
is a K3 surface, then
and has signature
This theorem is the source of polarization in Hodge theory.
) is polarizable
if there exists a morphism of Hodge structures
is positive definite.
So the Hodge index theorem has the following corollary.
is a polarizable
-Hodge structure. A polarization is given by
. We need to show that
is positive on
Now using the Hodge index theorem we see the sign cancels out and takes value in
Now we would like a (weak) version of this that makes sense for any field and any Weil cohomology satisfying hard Lefschetz (so the primitive cohomology still makes sense). Inside there is -vector subspace .
(Hodge Standard Conjecture )
, the pairing on
is positive definite.
By Corollary 10(take ),
We now explain that for , the Hodge Standard conjecture implies the Riemann hypothesis. A more convenient reformulation of is that the pairing is positive definite. It follows that there is a positive involution on (acting on ) given by Explicitly, (which is algebraic under Lefschetz).
So we want that the eigenvalues of the Frobenius on are pure of weight . We renormalize the Frobenius (acting on ) as Under Lefschetz, . We want all eigenvalues of has absolute value 1 for all complex embeddings. This can be obtained by realizing as a unitary operator on the inner product space (). We notice that commutes with and , so We claim that , so that is -invariant. This follows from the following more general lemma. One can check that ( is the chosen ample line bundle), so the following lemma applies to .
b), c) implies that
is invertible. In fact, for
has nonzero trace, so
It follows that is unitary with respect to the inner product (the positivity follows from and the fact that ). In particular, the eigenvalues acting on have all absolute values 1. Hence by Cayley-Hamilton, the roots of characteristic polynomials of on have all absolute values 1, as desired.
Absolute Hodge cycles
Our next goal is to construct a modified category of pure motives such that
- Under the standard conjectures, .
- has all categorical properties we want: (say ) it is -linear, semisimple, neutral Tannakian (this gives unconditional motivic Galois formalism).
- 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 motivated cycles (Andre) absolute Hodge cycles (Deligne) Hodge cycles
An absolute Hodge cycle
has characteristic 0 and finite transcendence degree) is a class
, such that for all
, the pullback class
comes from a Hodge cycle in
-vector space) via the comparison isomorphisms.
Any Hodge cycle on an abelian variety (
) 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,
Any Hodge cycle on an abelian variety (
) is motivated.
One classical application of absolute Hodge cycles is the algebraicity of (products of) special values of the
(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
For an algebraic cycle
) and a differential form
, then one obtains a period
The same principle applies for
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 be a number field and a smooth projective variety. Let 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 is equal to the dimension of the motivic Galois group (when one makes sense of it). For the motive (defined by absolute Hodge cycles), we have . Deligne's theorem implies that the later is equal , the dimension of the Mumford-Tate group (the Hodge theoretic analogy of the motivic Galois group).
-Hodge structure. The Mumford-Tate group
-Zariski closure of the image of
(i.e., the smallest
Notice a priori, one only knows the inequality (since absolute Hodge cycles Hodge cycles).
Here is another application due to Andre. Suppose
is finitely generated. Let
are K3 surfaces over
with polarizations (the important fact is that
for K3 surfaces). Then any isomorphism of
arises from a
-linear combination of motivated
cycles. Also the Mumford-Tate conjecture is true for
is equal to the connected component of the Zariski closure of the image of
. 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
be the orthogonal complement of Hodge cycles (the transcendence lattice
which is 21 dimensional generically). Then
is a field because
, 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
and in the CM case a unitary group over
, with the pairing coming from the polarization.
Let be a Weil cohomology with hard Lefschetz. Fix a subfield (e.g. ).
is defined to be the subset of elements of
of the form
algebraic cycles. Here
is the Lefschetz involution associated to a product polarization
. 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.
The basic calculation (with the above remark) shows the following.
- is an -subalgebra of (with respect to the cup product).
As for algebraic cycles, we define the motivated correspondences similarly.
with the similar composition law (the target is correct by the previous lemma). Then
is a graded
-algebra. We also have a formalism of
and projection formulas for
(analogue of Jannsen's theorem)
is a finite dimensional semisimple
is semisimple abelian.
We define an analogue of numerical equivalence:
is called to be motivated numerically equivalent to 0 if for any
. Then Jannsen's argument shows that
is semisimple. But since
holds for motivated cycles by construction, the motivated equivalence is the same as the motivated numerical equivalence (Remark 22
(Properties of )
- is pro-algebraic, even pro-reductive over .
- splits over the maximal CM extension of (i.e., for any and , is isomorphic to for some , where is the maximal CM subfield).
The Motivated variational Hodge conjecture
Source of motivated cycles: the motivated analogue of variational Hodge conjecture.
(Variational Hodge conjecture)
be a smooth projective morphism, let
is algebraic for some
, then for any
is also algebraic.
The variational Hodge conjecture holds with "motivated" in place of "algebraic".
Let us review the theorem of the fixed part and the necessary background in mixed Hodge theory.
is smooth projective and
is smooth. Let
be a smooth compactification. Then
is surjective. In other words, the image
is the fixed part under the monodromy, i.e.,
The above maps are given by
- is the edge map in the Leary spectral sequence associated to . By a theorem of Deligne, when is smooth projective, the Leary spectral sequence degenerates at . So is surjective.
- and have the same image. Since is injective, it follows that and have the same image. Hence is surjective.
More generally, if is smooth projective (applied to ), is smooth, Then the image of the composite map is the same as the image of the latter map. The reason is that each of these cohomology groups has a weight filtration such that is a pure Hodge structure of weight . Since , are smooth, their weight filtration looks like Since is smooth but not projective, its weight filtration looks like 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 of mixed Hodge structure, then for any , Now the strictness implies that it suffices to check the images on each are the same. The results then follows from . To see this, it essentially follows from the definition of the weight filtration as the shift of the Leray filtration associated to : and by definition is the whole thing.
be a motivated cycle such that a finite index subgroup of
acts trivially on
, then all parallel transport of
are still motivated.
Apply the previous theorem after a finite base change.
be an abelian variety. Then the Hodge cycles on
are known to be motivated, due to Deligne-Andre. The idea of the proof is to put
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
has the form
, where we can take
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
For any abelian variety
This follows from Hodge cycles on abelian varieties are motivated and that the product of abelian varieties are still abelian varieties:
- Let be the stabilizer of . Then is a sub Hodge structure if and only if stabilizes if and only if factors through if and only if .
- Apply the first part to the subspace .
The natural functor from
to the category of
-Hodge structures is fully faithful and realize
as the Tannakian group of
(as a subcategory
The full subcategory of polarized
is a (connected) reductive group.
The connectedness follows from the definition. To show that
is reductive, we only need to exhibit a faithful and completely reducible representation of
. The standard representation
works: the subrepresentations exactly corresponds to the sub Hodge structures of
, whose complete reducibility is ensured by the previous lemma.
is exactly the subgroup that fixes all Hodge tensors.
This follows from the following general results. Let
be a reductive group and
be a subgroup of
is reductive, then
a priori). The claim follows from taking
. For any
(reductive or not), by the theorem of Chevalley, there exists a representation
and a line
is the stabilizer of
is further reductive, there exists a
consists of the elements fixing any generator of this line. So
giving rises to
is exactly the subgroup of
fixing all motivated cycles in all tensor constructions.
Because motivated cycles Hodge cycles,
Does even arises as for some polarized -Hodge structure ? This is at least necessary for it to be a motivic Galois group.
A semisimple adjoint group
is a Mumford-Tate group of a polarizable
-Hodge structure if and only if
contains a compact maximal torus.
Let explain the case when is simple with compact maximal torus over . Write . Let be a compact maximal torus, fixed by some Cartan involution of . The Cartan involution is essential for the polarization. Namely, is an involution on satisfying the following positive condition: is positive definite. Decompose into the and eigenspaces for . Here matches up with the Lie algebra of the maximal compact subgroup . Now any yields a polarizable -Hodge structure on if and only if is a Cartan involution on .
Let us write down . Choose a cocharacter such that for any compact roots and for any noncompact roots . Notice such cocharacters is in bijections with . Extend (trivial on ) to obtain Then acts on the root space by , which is 1 when is compact and when is noncompact. Now use is negative definite on and positive definite on . One knows that gives a polarization on . Using this framework, it is easy to check can't arise as .
Consider the split form of
. The two compact roots are
After the break we will construct 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 and ). We will focus on the former, since the latter needs more machinery from geometric Langlands.
Applications of the motivated variational Hodge conjecture
The Kuga-Satake construction is "motivated", i.e., for
a projective K3 surface, the attached abelian variety
which is a priori a morphism of
-Hodge structure, is indeed a motivated cycle (i.e., a morphism in
). This implies the Mumford-Tate conjecture for K3 surfaces, etc..
Now let us give the motivated analogue of a refinement of this assertion. So we need a notion of a family of motivated motives.
- Let the exceptional locus does not contain the image of a finite index subgroup of . Then is contained in a countable union of closed analytic subvarieties of .
- (Refinement) There exists a countable collection of algebraic subvarieties such that is contained in the union of . (In Hodge theory, this continues to hold for arbitrary -polarized variational Hodge structure. This "algebraicity of the Hodge loci" is a strong evidence for the Hodge conjecture.)
- There exists a local system of algebraic subgroups of such that
- for any .
- for all .
- contains the image of a finite index subgroup of (notice the latter is a purely topological input!)
Rigid local systems
Let be a smooth projective connected curve. Let be a finite set of points. Let . For the time being, we work with the associated complex analytic spaces (so implicitly).
A local system
-vector spaces on
is a locally constant sheaf of
is most interesting for our purpose. Here
is a free group on
Given a local system
, when does come from geometry
By coming from geometry, we mean there exists a smooth projective family such that for some (notice itself is a local system).
One necessary condition for 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:
-variational Hodge structure over a punctured disc
has quasi-unipotent monodromy.
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 .
There are several notions which you may want to call rigid local systems.
-local system on
is physically rigid
if for local system
such that for any
. In terms of the monodromy representation: if the generators are conjugate (possibly by different matrices), then they are globally conjugate.
A slight variant:
is physically semi-rigid
if there exists finitely many local systems
such that if
is locally isomorphic to
(as in the previous definition), then
These two notions are very intuitive but extremely hard to check. The following definition provides a numerical condition and is easier to check.
is cohomologically rigid
is still a local system on
is no longer a local system on
The following lemma gives a very useful numerical criterion for cohomologically rigidity.
be an irreducible local system of rank
is cohomologically rigid if and only if
, if and only if
We denote the Jordan block of length
with the eigenvalue
. They give a local system on
. They have Jordan forms
, all are quasi-unipotent. It actually comes from geometry (classically known) as the local monodromies of the Legendre family
Namely, it comes from the local system
(e..g, one can see these matrices by Picard-Lefschetz). Moreover, it is cohomologically rigid by the previous lemma:
Hypergeometric local systems are irreducible.
If not, let
be a subrepresentation and
be the corresponding quotient. Since
is a pseudo-reflection, we know that it must acts trivially on one of
on one of them, which contradicts the assumption
Given , one can write down the explicit matrix description for the local monodromies.
- It suffices to show that is pseudo-reflection: indeed has rank 1.
- Given such an , Set , . Let . Then has dimension since is a pseudo-reflection). Hence has dimension at least one; let be a nonzero vector of this space. Thus . Therefore , , ..., . We claim that the span is the whole space. In fact, by Caylay-Hamilton stabilize on this span, so it must be the whole space by the irreducibility. In this basis, , , have the desired form.
- What are (the Zariski closure) of the monodromy group of hypergeometric local systems? For example, does appear?
- Are those with roots of unity always geometric?
- We saw that hypergeometric local systems are both physically rigid and cohomologically rigid. What is the relationship between physical and cohomological rigidity in general?
- Not . Beukers-Heckman computed all possibilities: , , and some specific finite groups.
- Yes, by Katz's theory.
Now let us come to the third question in more detail.
be an irreducible local system on
is cohomologically rigid, then
is physically rigid.
be a local system with the same local monodromy as
has the same local monodromies, the Euler characteristic formula implies that
and by Poincare duality,
So at least one of the local systems
has a global section. Since
is irreducible and
, this global section gives an isomorphism
For the other direction, we will use a transcendental argument. This direction is not known in the -adic setting (knowing local monodromy matrices is not enough in the -adic setting: one needs to know continuity).
be an irreducible local system on
is physically rigid, then
is cohomologically rigid.
We know a prior that
. We need to show it is
be the local generators around the punctures. Suppose
is given by matrices
is given by matrices
is physically rigid, if there exists
, then there exists
. We want to show that
Consider the map
corresponds to the local systems with the same local monodromies as
. The group
acts on the domain and codomain by
acts on the fiber
is physically rigid if and only if
, it follows that
which gives the desired inequality!
be a field. For a separated finite type
, we have a triangulated category
derived category of constructible
-adic sheaves on
) equipped with a standard
-structure such that there is an equivalence of categories
a morphism, we have adjoint pairs
. We also have adjoint pairs
The triangulated category is defined to be the colimit of . The latter triangulated category is hard to define (it is not defined as the derived category of -sheaves, which do not have enough injectives). There are 3 approaches to define ..
- Use the pro-etale topology introduced by Bhatt-Scholze ( becomes a genuine sheaf).
- Taking limit is well-behaved for the stable -category version of . The triangulated limit comes for free.
- Deligne's classical approach: replace with the full subcategory of very well-behaved complexed (these are quasi-isomorphic to bounded complexes of constructible -flat sheaves). Call this full subcategory . Then is naturally triangulated: is a distinguished triangle if is a distinguished triangle for any .
if for any
are both semi-perverse, where
is the Verdier dual of
is smooth of dimension
is perverse (since
) 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 for singular proper varieties satisfying the Poincare duality and purity. Another major motivation for us is the following function-sheaf dictionary.
. We define for any
For example, when
produces the trace of the Frobenii on the cohomology of the fibers of the morphism
. Generalizing this, for any
, we define
The key thing is that these functions interact nicely with the sheaf-theoretic operations. For example,
- When is a distinguished triangle, we have .
- For , we have for .
- For , we have (think: 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:
are two semisimple perverse sheaves. Then
are isomorphic if and only if
How do we produce more perverse sheaves from the "lisse on smooth" case (Example 25)?
- Suppose is an affine morphism, the preserves semi-perversity (but not perversity).
- Suppose is a quasi-finite morphism, then preserves semi-perversity (but not perversity).
is both affine and quasi-finite (e.g.,
is an affine immersion), then both
is perverse, then
is semi-perverse (by the previous theorem). Now
(by duality). Since
is perverse (by definition), hence
is also semi-perverse (by the previous theorem).
Here comes the key construction: intermediate extensions. Let be a locally closed immersion. For simplicity, let us assume that is affine, so is affine and quasi-finite. If . Then both and lie in . There is a natural map .
Define the intermediate extension
(or middle extension
(in the abelian category
- is fully faithful.
- preserves simple objects, injections and surjections.
Any simple perverse sheaf
is of the form
for some smooth affine
locally closed subvariety of
, for some lisse sheaf
to be the closure of
such that the constructible sheaves
become lisse when restricted to
. This works.
Interesting things happen when extending to the boundary of .
The middle convolution
Today we will introduce the key operation on perverse sheaves in Katz's classification of rigid local systems: the middle convolution.
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
Here the parameter
determines the local monodromies. More generally,
is the solution of
This integral looks like
namely the (additive) convolution of
. The function
corresponds to the rank 1 Kummer sheaf associated to the representation
. Similarly, the function
corresponds to a tensor product of (translated) Kummer sheaves. So rigid local system
can be expressed in terms of the convolution
of simpler objects.
Here is the precise construction of the convolution.
be an algebraically closed field. Let
be a connected smooth affine algebraic group. Let
be the multiplication map. For
, we can define two kinds of convolutions
such that for all
are perverse. We define the middle convolution
to be the image of
in the abelian category of perverse sheaves.
be the local system on
associated to a nontrivial character
. Then the middle convolution
makes sense: both
preserve perversity, by the following proposition.
be irreducible such that its isomorphism class is not translation invariant. Then
both preserve perversity.
- The statement follows from the statement: because is also perverse and not translation invariant, so is perverse; taking dual implies that is perverse.
- For , then is perverse if and only if is semi-perverse: is semi-perverse since is affine.
- If is perverse, then the followings are equivalent:
- is perverse for any ;
- is perverse for any irreducible .
In fact, because is an abelian category with all objects having finite length, we can induct on the length of . A distinguished triangle (with lower lengths) gives a distinguished triangle ; the long exact sequence in cohomology then implies that
- So we reduce to the case of irreducible perverse sheaves . We now use the assumption that . Namely, we need to check that By Example 26, an irreducible perverse sheaf is either punctual or an intermediate extension . If either or is punctual, then is a translate of or , hence is perverse. So we can assume that there exists and lisse on such that and . The stalk of at a geometric point is This vanishes for since . It remains to check that for , this vanishes for at most finitely many . Now we need to use the assumption that is not translation invariant. Notice the fiber , so for , we have Since both and are lisse on , it is equal to Since both source and target are irreducible, this is zero unless there is an isomorphism . Since the right hand side does not depend on , either we win or there exists infinitely many such that there is such an isomorphism. Since these lie in the support of a constructible sheaf on a curve, the same would happen for in an open dense subset . Let , then the isomorphism class of is translation invariant under . Thus we obtain a subgroup containing , which must be the whole group, under which is translation invariant. A contradiction!
Henceforth we take .
Now we can state the main results (slightly specialized) about the middle involution.
Let us explain some of the ideas of the proof without going into details.
For any algebraic closed field and any separated and of finite type, we define the subcategory of middle extensions consisting of , where is lisse for some . We have an operation
If has characteristic and . It turns out that the category on satisfying is equivalent to via the Fourier transform. The middle convolution on then corresponds to on (as in the classical Fourier theory: the Fourier transform of the convolution is the product of the Fourier transforms).
We now define the Fourier transform, which is a functor
. Fix an additive character
and denote the associated the Artin-Schrier sheaf on
be the two projections
. Motivated by the classical Fourier transform
is the pullback of
via the multiplication map. Similarly define
. It turns out that
and we denote it by
for short. It follows that
since projection maps are affine and the duality switches
is an auto-equivalence of
Using Theorem 21, we can prove the Katz's classification theorem of tamely ramified cohomological rigid local systems. Besides , we also need a simpler twisting operation: If is rank 1 lisse on , we define The index of rigidity is easily seen to be preserved under .
Given an irreducible tame cohomological rigid local system on , our aim is to apply a series of and 's (these are all invertible operations) to obtain a rank 1 object (which is easy to understand).
is lisse on
) and cohomologically rigid. Then there exists a generic rank 1
and a nontrivial character
smaller rank than
Also, we can arrange and to have local monodromies contained in the local monodromies of . So the local monodromies of is contained in the local monodromies of . In particular, if all th eigenvalues of the local monodromies of are roots of unity, then we can arrange the same for .
, we write
, we write
, we write
to be the number of Jordan blocks of length
in the unipotent matrix
(which can be viewed as the dual partition of
To guess what to do, we look at the rank formula (proved via Fourier transform by Katz): for any nontrivial , 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 that maximizes .
For each , we choose such that is maximal. We form of rank 1 such that , i.e., Then has larger than for any .
We replace by the resulting twist and choose such that is maximal. In order to apply Katz's rank formula, we claim that any such is nontrivial. Assume this claim is true. By the rank formula and the Euler characteristic formula, we have
So to see the rank actually drops, we just need to show that For this we need the cohomological rigidity The last term can be written as By the maximality of , we know the last term is at most So the cohomological rigidity implies that
which implies what we wanted since where by definition.
It remains to prove the claim. Assume that is trivial, then the same argument implies that But since is irreducible, we know that , a contradiction.
Since and are both reversible operations,
Given a tuple of monodromies (at each
), 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.
Starting with the above local monodromies, apply Katz's algorithem to reduce to the rank 1 case.
Local systems of type
Let be an algebraically closed field of characteristic not (or 2).
- Fix . Let such that . Then there exists an irreducible cohomologically rigid local system of rank 7 with the local monodromies:
- at : ,
- at : ,
- at : any of the following (determined by the conditions on and ),
In each case, the monodromy group (the Zariski closure of the image of the monodromy representation) is .
- Let be cohomologically rigid, ramified at and have monodromy group . Then is ramified at exactly two points of . Moreover, up to permuting , is conjugate to one of the local systems above.
We start with the second part. Suppose
is lisse exactly on
, we want to know how big
is cohomologically rigid, we have
We look at the table of the centralizers of conjugacy classes of
(copied from Dettweiler-Reiter), we see the largest dimension is 29.
So , hence . When , then the four centralizer adds to dimension 100. We look at the table again and there are only the following three cases
We can rule out all these three cases due to the necessary criterion for irreducibility: . Namely, the Euler characteristic formula tells us that or For example, case is ruled out because then the Jordan form is by the table: if we twist by the character with local monodromies at the three finite points (and at ), we then get a new local system with local monodromies , which violates the above irreducible criterion. Other cases are similar.
Now . Again the cohomological rigidity implies that the sum of dimensions of the three centralizers is 51. The table implies that the possibilities are
The necessary criterion of irreducibility implies that . This excludes the cases a), d), e), g). Since the monodromy group is assumed to be all of (so far we only used the monodromy group is contained in ), the (14 dimensional) adjoint representation of is also irreducible (the adjoint representation is irreducible for general simple reductive groups). Applying the irreducibility criterion gives So the sum of dimensions of the three centralizers in is less than 14. This excludes the cases b), f). The corresponding centralizers for case c) must be . The local monodromies are
- at ,
- at ,
- several possibilities at .
The first part the follows by checking which of these possibilities can arise using Katz's algorithm.
For example, take the case at the third point . Write for the rank 1 local system with local monodromies at and at . Twisting by we get , and . Then the rank formula together with the table tells us that the rank of the middle convolution is equal to , with local monodromies , , . Twisting by and take middle convolution, we obtain local monodromies , , (rank 5),... until we get down to , , (rank 2) and (rank 1). The last rank 1 local system actually exists since . 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 . First notice that our monodromy representation is orthogonal. The dual representation has the same local monodromies (up to -conjugacy); since is physically rigid, this implies that . Since the dimension 7 is odd, must be orthogonal. So maps into .
We use the fact that an irreducible subgroup of lies inside an -conjugate of if and only if . So we need to show that , i.e., . By the Poincare duality, this is equivalent to . We compute the Euler characteristic,
- At : the is the image of of in via the 6-th symmetric power Since is the same as the number of irreducible constituents of this -representation, we know .
- At : the local monodromy is semisimple and it is easy to see that the dimension (either two 's or no 's).
- At : the local monodromy is the image of of via . The number of irreducible constituents of is 13.
So the Euler characteristic is . Hence . Therefore our lands inside . The fact (going back to Dynkin) is that an irreducible subgroup of containing a regular unipotent element must be either or . The rules out the possibility of .
Universal rigid local systems
Let be an algebraically closed field, . Fix an order of quasi-unipotence () and fix a primitive -th root of unity.
Let and be a cohomologically rigid local system, lisse on with eigenvalues contained in . We will show how to produce a local system over the arithmetic configuration space of points, whose geometric fibers over are cohomologically rigid objects in lisse away from and one of which gives the original .
and fix a nonzero map
. The configuration space over
is defined to be (
The universal rigid local system will live on , i.e., Notice that one can specialize to via .
Middle convolution with parameters
Let be a normal domain, finite type over . Let be a divisor given by the equation , where are all distinct. Let be another divisor given by .
The middle convolution is an operation where is given by . For the application, since .
Then we have the following diagram where are the projections and is the difference morphism.
We denote . Then the compactified projection is proper smooth.
, we define the middle convolution
and the naive convolution
(Ideas of the proof)
a. Let us take . Then is the complement of hyperplanes in the and directions and the diagonal. Recall that is the sheaf associated to . Since the projection is trivialized with respect to the stratification , it can be computed as . It follows that the middle convolution is lisse. For a formal proof, see Katz, Sommes exponentielles, Section 4.7.
b. Recall that if is an -sheaf on (a scheme of finite type over ), we say that is pure of weight if for any closed points , is pure of weight in the familiar sense. We say is mixed if it admits a filtration by subsheaves such that are all pure.
is a morphism of schemes of finite type over
is invertible on
is mixed of weight
is mixed of weight
It follows that the naive convolution is mixed of weight . The purity of the middle convolution follows from the analogous statement for curves over finite fields:
is a smooth curve,
pure of weight
is pure of weight
This result is less surprising by noticing that (the source is mixed of weight and the target is mixed of weight ). To show the graded piece statement, we use the short exact sequence of sheaves which gives a long exact sequence The last term is zero since is punctual. Since is an open immersion, we know that has weights by 1.8.9 in Weil II. So we are done since -term is mixed of weight .
c. We prove the ! version. We can check on geometric fibers and it suffices to show for . Notice that is lisse on and is the coinvariants under the local monodromy at . In our case, is unramified along , but is indeed ramified, and hence the coinvariants is 0.
is affine open,
is separated finite type of an algebraically closed field
is proper and
is finite. Suppose
are perverse. Then
We have two exact sequences in
The kernel and cokernel are supported on
, we obtain two distinguished triangle on
, four out of the six terms are perverse by assumption. One can check that
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.