Θ-stratifications, Θ-reductive stacks, and applications

These are expanded notes on a lecture of the same title at the 2015 AMS summer institute in algebraic geometry. We give an introduction and overview of the “beyond geometric invariant theory” program for analyzing moduli problems in algebraic geometry. We discuss methods for analyzing stability in general moduli problems, focusing on the moduli of coherent sheaves on a smooth projective scheme as an example. We describe several applications: a general structure theorem for the derived category of coherent sheaves on an algebraic stack; some results on the topology of moduli stacks; and a “virtual non-abelian localization formula” in K-theory. We also propose a generalization of toric geometry to arbitrary compactifications of homogeneous spaces for algebraic groups, and formulate a conjecture on the Hodge theory of algebraic-symplectic stacks. We present an approach to studying moduli problems in algebraic geometry which is meant as a synthesis of several different lines of research in the subject. Among the theories which fit into our framework: 1) geometric invariant theory, which we regard as the “classification” of orbits for the action of a reductive group on a projective-over-affine scheme; 2) the moduli theory of objects in an abelian category, such as the moduli of coherent sheaves on a projective variety and examples coming from Bridgeland stability conditions; 3) the moduli of polarized schemes and the theory of K-stability. Ideally a moduli problem, described by an algebraic stack X, is representable by a quasi-projective scheme. Somewhat less ideally, but more realistically, one might be able to construct a map to a quasi-projective scheme q : X→ X realizing X as the good moduli space [A] of X. Our focus will be on stacks which are far from admitting a good moduli space, or for which the good moduli space map q, if it exists, has very large fibers. The idea is to construct a special kind of stratification of X, called a Θ-stratification, in which the strata themselves have canonical modular interpretations. In practice each of these strata is closer to admitting a good moduli space. Given an algebraic stack X, our program for analyzing X and “classifying” points of X is the following: (1) find a Θ-reductive enlargement X ⊂ X′ of your moduli problem (See Definition 2.3), (2) identify cohomology classes ` ∈ H2(X′;Q) and b ∈ H4(X′;Q) for which the theory of Θ-stability defines a Θ-stratification of X′ (See §1.2), (3) prove nice properties about the stratification, such as the boundedness of each stratum. We spend the first half of this paper (§1 & §2) explaining what these terms mean, beginning with a detailed review of the example of coherent sheaves on a projective scheme. Along the way we discuss constructions and results which may be of independent interest, such as a proposed generalization of toric geometry which replaces fans in a vector space with certain collections of rational polyhedra in the spherical building of a reductive group G (§2.2). In the second half of this paper we discuss applications of Θ-stratifications. In (§3 & §4) we discuss how to use derived categories to categorify Kirwan’s surjectivity theorem for cohomology (See Theorem 3.1), and several variations on that theme. Specifically, we discuss how methods of derived algebraic geometry and the theory of Θ-stratifications can be used to establish structure theorems (Theorem 3.17,Theorem 3.22) for derived categories of stacks with a Θ-stratification, and we use this to prove a version of Kirwan surjectivity for Borel-Moore homology (Corollary 4.1). As an application we show (Theorem 4.3) that the Poincare polynomial for the Borel-Moore homology of