Speaker: Andrés Ibáñez Núñez (Columbia University)

Title: Motivic enumerative invariants of algebraic stacks

Abstract: The Euler characteristic of a complex algebraic variety is the alternating sum of its Betti numbers. However, moduli spaces in algebraic geometry are often not varieties but Artin stacks X, and for them the Euler characteristic is undefined, since there may be infinitely many nonzero Betti numbers.

When X parametrises objects in an abelian category, Joyce defined a meaningful notion of Euler characteristic of X using the motivic Hall algebra, a structure that heavily depends on the underlying abelian category. Joyce--Song used this idea, in combination with the Behrend function, to define Donaldson--Thomas invariants in the presence of strictly semistable sheaves.

I will explain how to define motivic Hall algebra like structures for general stacks and how this yields a definition of Euler characteristic for an Artin stack and of Donaldson--Thomas invariant for a (-1)-shifted symplectic stack. The theory thus applies to nonlinear moduli problems, like G-bundles, G-local systems or quotient stacks. The starting point is the stack of filtrations of X, defined by Halpern--Leistner.

This is joint work with Chenjing Bu and Tasuki Kinjo.