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**Cohomologically proper algebraic stacks:**

We argue that the standard notion of a proper (i.e. "compact")
algebraic stack is too restrictive, and we provide an alternative
notion of *cohomological properness*: the stack should satisfy
the 'Grothendieck existence' theorem and the 'coherent pushforward'
theorem. Although these are usually regarded as theorems, taking them
as the definition of properness leads to a notion which is well-behaved
and generalizes the more geometric definition of properness in Champs
algébriques in useful ways.

**Equivariant Hodge theory:**

For any dg-category, there is a spectral sequence starting with the Hochschild homology and converging to the periodic cyclic homology. When the category is $Perf(X)$ for a scheme, this can be identified with the classical Hodge-to-deRham spectral sequence, and it degenerates when $X$ is smooth and proper. The non-commutative HdR sequence degenerates for many cohomologically proper smooth stacks and fails to do so for many smooth stacks which are not cohomologically proper. This provides further evidence that cohomologically proper is a good generalization of the notion of properness to Artin stacks.

**Mapping stacks:**

If $\fX$ and $\fY$ are stacks, then one can form a *mapping stack*,
$Map(\fX,\fY)$, which by definition is the moduli functor
parameterizing families of maps from $\fX$ to $\fY$. When $\fX$ and
$\fY$ are algebraic spaces (locally finitely presented over a
Noetherian base) and $\fX$ is proper and flat, then it is a classical
result that $Map(\fX,\fY)$ is in fact represented by an algebraic
space. When $\fX$ is an Artin stack, this is still true as long as
$\fY$ is geometric and $\fX$ is *cohomologically proper*.

**Theta stratifications:**

If $X$ is a symplectic manifold admitting a Hamiltonian action by a compact group, $K$, then one has a $K$-equivariant stratification of $X$ by the gradient descent flow of the norm-squared of the moment map. When $X$ is a projective variety, then this stratification has an alternative, purely algebraic, description in terms of the Hilbert-Mumford numerical criterion in GIT, and one can think of this as a stratification of the algebraic stack $\fX = X/K_{\bC}$. Theta stratifications provide a generalization of this to stacks which are not global quotients stacks, such as the non finite-type stacks appearing in many moduli problems. The theory provides a framework for studying "stability" of algebro geometric objects generalizing GIT and many other commonly studied notions of stability.

*Key idea*: the strata have canonical modular interpretations
-- they parameterize maps $f : \bC/\bC^\ast \to \fX$ which exhibit
"optimally destabilizing" data for the unstable point $f(1) \in \fX$.

**Reductive stack: **

Given an algebraic stack representing a certain moduli problem, one can ask what data is required to define a Theta-stratification. For reductive stacks, all one needs is a class in $H^2(\fX;\bQ)$ and $H^4(\fX;\bQ)$ satisfying a "boundedness" hypothesis.

*Simplest example:* $X/G$, where $X$ is affine and $G$ is
reductive

*Non-example:* $X/G$ where $X$ is projective and $G$ is
reductive

*Interesting examples:* Moduli of objects in the heart of a
t-structure on the derived category of coherent sheaves of a projective
variety

*Formal definition:* the map of "evaluation at 1"
$Map(\bC/\bC^\ast,\fX) \to \fX$ should be proper on connected components

**Generalized buildings:**

A key construction in the theory of Theta-stability assigns to any
point in an algebraic stack $ p \in \fX $, a topological space
$\sD(\fX,p) $, called the *degeneration space*. A map
$\bC/\bC^\ast \to \fX$ along with an isomorphism $f(1) \simeq p$
determines a point of $\sD(\fX,p)$, and points of this form are dense
in $\sD(\fX,p)$.

When $\fX = BG$ for a semisimple group $G$, then $ \sD(\fX,p) $ is homeromorphic to the spherical building of $G$, and when $\fX = X/T$ for a normal toric variety $X$, the degeneration space of a generic point of $X$ is homeomorphic to $(|\Sigma| - \{0\}) / \bR^\times_{\geq 0}$, where $|\Sigma|$ denotes the support of the fan defining $X$. Thus these degeneration spaces can be thought of as "generalized buildings," and they connect the theory of buildings in representation theory with toric geometry.

**Categorification of Kirwan surjectivity:**

Kirwan surjectivity states that for a GIT quotient of a smooth variety, $X^{ss} / G \subset X/G$, the restriction map on cohomology $H^\ast(X/G) \to H^\ast(X^{ss}/G)$ is surjective. There is a "categorification" of this result to a statement about derived categories of coherent sheaves. The restriction functor $D^b Coh (X/G) \to D^b Coh(X^{ss}/G)$ is always surjective on the level of objects, so that's not the right categorification. But it turns out there is a subcategory $G_w \subset D^b Coh(X/G)$ such that the restriction functor gives an equivalence $G_w \simeq D^b Coh(X^{ss}/G)$. This implies that the restriction functor admits a section $D^b Coh(X^{ss}/G) \to D^b Coh(X/G)$, and so for any invariant which can be extracted from the derived category, the restriction functor from $X/G$ to $X^{ss}/G$ is surjective.

**Topological invariants of derived categories:**

A result of Feigin and Tsygan holds that the cohomology of (the analytification of) an affine variety, $X$, over $\bC$ is isomorphic to the periodic-cyclic homology of the coordinate ring. This agrees with the periodic-cyclic homology of the category of perfect complexes on $X$, so some topological information can be extracted directly from the derived category. For equivariant categories, Thomason showed that the topological equivariant K-theory modulo a prime power can be recovered from the derived category of equivariant coherent sheaves. It turns out that the equivariant K-theory itself can be recovered from the derived category of equivariant coherent sheaves. This result allows one to de-categorify categorical Kirwan surjectivity and recover classical Kirwan surjectivity.

**Virtual non-abelian localization theorems:**

When $X$ is a compact manifold and $T$ is a torus acting on $X$, localization theorems in equivariant cohomology provide a method for reducing the integrals of equivariant cohomology classes on $X$ to integrals over the fixed locus $X^T$ (which could be just a sum over a finite set of points). There is another, closely related, flavor of localization theorems for stacks with a $\Theta$-stratification. The integral $\int_X \omega$ is replaced by the K-theoretic integral \[\chi(X/G,F) := \Sigma (-1)^p \dim R^p \Gamma(X,F)^G,\] where $F \in Perf(X/G)$, and the localization formula expresses $\chi(X/G)$ as a sum of $\chi(X^{ss}/G,F)$ and "correction terms" coming from each stratum. In some cases the correction terms vanish, leading to an identification between the integral over $X$ and over $X^{ss}$, and in other cases $X^{ss} = \emptyset$, leading to an formula for $\chi(X/G,F)$ in terms of the fixed locus as in the cohomological version. Thus the $K$-theoretic localization theorem is a little more flexible and has the advantage of working for non-abelian G, and the cohomological localization formula can be recovered for classes of the form $ch(F)$.

The non-abelian localization theorem was developed by Teleman and Woodward in the case of smooth global (and local) quotient stacks. In fact the non-abelian localization formula is intrinsic to any stack with $\Theta$-stratification, and does not require a local quotient description. Furthermore, using a little bit of derived algebraic geometry, it continues to hold of stacks which are not necessarily smooth, but quasi-smooth, leading to a "virtual" non-abelian localization theorem.

**Structure theorems for equivariant derived categories:**

In addition to identifying a subcategory $G_w \subset D^bCoh(X/G)$
with $D^bCoh(X^{ss}/G)$, the categorical version of Kirwan surjectivity
realizes $G_w$ as one piece of an infinite *semiorthogonal
decomposition* of $D^bCoh(X/G)$. More precisely there are other
subcategories of $D^bCoh(X/G)$, consisting of objects supported on the
unstable locus, and each can be identified (more-or-less) with the
derived category of the GIT quotient of certain closed subvarieties of
$X$ by reductive subgroups of $G$.

The main application is to a variation of GIT quotient: as the GIT quotient changes, $X^{ss}_- \leftrightarrow X^{ss}_+$, the stratification changes and thus the subcategories $G^\pm_w \subset D^bCoh(X/G)$ differ. In nice situations, the category corresponding to one GIT quotient contains the category corresponding to the other GIT quotient, $G_w^- \subset G_w^+$. Using the full structure theorem, one can identify the semiorthogonal complement of $G^-_w$ in $G^+_w$ explicitly.

**'Mapping stacks and categorical notions of properness' (with
Anatoly Preygel): **

We develop the notion of *cohomologically proper* stacks. We
show that the mapping stack from a flat cohomologically proper stack to
a geometric stack (locally of finite type) is algebraic, generalizing
known results for proper algebraic spaces and DM stacks. We introduce a
notion of *cohomologically projective* morphism and show that
such a morphism is cohomologically proper, leading to many examples. We
also develop new results showing that $D^- Coh$ satisfies derived
h-descent, as well as derived descent along morphisms which are
cohomologically proper.

**Key technical tools:** Tannakian formalism, Artin's
criteria for algebraicity, derived h-descent

**Connections:** The algebraicity of mapping stacks is
key to the modular interpretation of Theta-stratifications

**'Tannaka duality revisited' (with Bhargav Bhatt):**(in
prep)

Any map between algebraic stacks $f: \fX \to \fY$ yields a symmetric monoidal functor between derived categories of quasicoherent sheaves $f^\ast : QC(\fY) \to QC(\fX)$. Jacob Lurie showed that when $\fY$ is geometric (meaning quasicompact with affine diagonal), $f$ can be uniquely recovered from $f^\ast$, and the symmetric monoidal functors arising in this way are those satisfying certain hypotheses (continuous, preserving connective objects and flat objects). We generalize this result, showing that for many stacks, it is not necessary to show that $f$ preserves flat objects. This seemingly minor modification allows for a much wider range of applications of this "Tannakian formalism."

**'Equivariant Hodge theory' (with Daniel Pomerleano):**(in
prep)

We show that the non-commutative Hodge-to-deRham sequence degenerates for $Perf(X/G)$ when $X$ is smooth and projective-over-affine, $G$ is reductive, and $X/G$ is cohomologically proper. (More generally, we show degeneration for categories of matrix factorizations for a function on $X/G$ with cohomologically proper critical locus.) We canonically identify the periodic cyclic homology with $K_{G^c}(X^{an}) \otimes \bC$, the complexified equivariant (w.r.t. a maximal compact subgroup) K-theory spectrum in the sense of Atiyah-Segal. Thus the equivariant K-theory carries a weight-0 Hodge structure, and we identify the associated graded, the Hochschild homology of $Perf(X/G)$, with the space of functions on the derived inertia stack. We also discuss explicit complexes computing these invariants.

**Key technical tools:** categorical Kirwan
surjectivity, the construction (due to Anthony Blanc) of a topological
K-theory spectrum for a dg-category

**Connections:** The computation of the periodic cyclic
homology the derived category of a quotient stack allows us to recover
classical Kirwan surjectivity from the categorical version.

**'On the structure of instability in moduli theory':**

We develop the theory of $\Theta$-stability. We define the degeneration space, $\sD(\fX,p)$, for a point in an algebraic stack $p \in \fX$. The stack $\Theta := \bA^1 / \bG_m$ plays a central role: points in $\sD(\fX,p)$ correspond to maps $\Theta \to \fX$ with $f(1) \simeq p$. We give a general method of constructing a "numerical invariant" from cohomology classes on $\fX$, which gives a continuous function on $\sD(\fX,p)$ for all $p \in |\fX|$. The central question of the paper is when such a numerical invariant leads to a $\Theta$-stratification of $\fX$, and in particular when every "unstable" point $p \in \fX$ admits an essentially unigue map $f: \Theta \to \fX$ with $f(1) \simeq p$ which maximizes the numerical invariant.

**Main result:** For weakly reductive stacks, the the
uniqueness of a maximizer is automatic, and existence amounts to
checking a certain "boundedness" principle. For reductive stacks, one
obtains a $\Theta$-stratification under mild hypotheses whenever a
(slightly stronger) boundedness principle holds

**Main example:** The main example is the moduli stack
of flat families of objects in the heart of a $t$-structure on
$D^bCoh(X)$, where $X$ is a projective variety. We show that this stack
is (weakly) reductive and analyze its degeneration space.

**'The reconstruction problem for finite orbit stacks: (in
progress)'**

This paper will develop some consequences of the degeneration space (see $\Theta$-stability) framework in toric geometry.

**'The derived category of a GIT quotient'**

We establish the categorical version of Kirwan surjectivity for quotient stacks $X/G$ with a "KN" stratification (a notion generalized by that of a $\Theta$-stratification). We prove the structure theorem under special hypotheses on the stratification -- properties (L+) and (A) -- which hold automatically when $X$ is smooth.

Applications developed in this paper: studying variation of GIT quotient, proving that all GIT quotients (with generic linearization) of an equivariantly Calabi-Yau projective-over-affine manifold by a torus are derived equivalent, showing that categorical Kirwan surjectivity applies to (many) hyperkaehler quotients, and showing that categorical Kirwan surjectivity applies to categories of singularities.

**'Autoequivalences of derived categories via geometric
invariant theory'**

Under a particularly simple kind of variation of GIT quotient
(VGIT), $X^{ss}_- \leftrightarrow X^{ss}_+$, categorical Kirwan
surjectivity can be used to identify both $D^b Coh(X^{ss}_+/G)$ and
$D^bCoh(X^{ss}_- /G)$ with the *same* subcategory of
$D^bCoh(X/G)$. However, the subcategory produced by categorical Kirwan
surjectivity depends on an (integer) parameter, so there are in fact
many equivalences $D^b Coh(X^{ss}_+/G) \to D^bCoh(X^{ss}_- /G)$, and
this allows us to produce non-trivial autoequivalences of
$D^bCoh(X^{ss}_+ / G)$. This paper describes these autoequivalences as
a special kind of autoequivalence called a generalized spherical twist
(or a twist by a spherical functor). In fact, we show that all
generalized spherical twists arise from semiorthogonal decompositions
analogous to those arising from a VGIT quotient. We also introduce the
notion of "fractional grade restriction rules" in the context of VGIT
and show how this leads to predictions about the poles and monodromy of
the quantum connection of $X^{ss}_+/G$.

**'$\Theta$-stratifications and derived categories' (in
preparation)**

This paper will develop categorical Kirwan surjectivity and the virtual non-abelian localization theorem for locally finite type quasi-geometric derived stacks with a $\Theta$-stratification.

**'Remarks on $\Theta$-stratifications and derived categories'**

This is a preliminary version of the paper. It develops the main structure theorem for derived local quotient stacks along the lines of the 'derived category of a GIT quotient' paper. In this case, derived stacks can be locally regarded as a sheaf of cdga's on a classical quotient stack, so many of the previous methods apply. This note also describes the virtual non-abelian localization theorem in the case of local quotient stacks.