Theta-stratifications, Theta-reductive stacks, and applications

Abstract

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 a toy 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.

25 pages, 2015 AMS Summer Institute in Algebraic Geometry

References in corpus (5)

• The equivariant Verlinde formula on the moduli of Higgs bundles
• Remarks on Theta-stratifications and derived categories
• The Derived Marsden-Weinstein Quotient is Symplectic
• Stratifications for moduli of sheaves and moduli of quiver representations
• Wall crossing for derived categories of moduli spaces of sheaves on rational surfaces