Enumerative geometry can be seen as the study of invariants associated to moduli stacks X. For a Deligne-Mumford stack X, we can talk about the Euler characteristic of X and, in some cases, about the virtual Euler characteristic of X, which gives Donaldson–Thomas invariants and Gromov–Witten invariants.

When the stack X has positive dimensional stabilisers, the naive Euler characteristic or virtual Euler characteristic can be infinity or undefined. For moduli stacks of objects in an abelian category, this problem was solved by Joyce and Joyce–Song using the motivic Hall algebra, a structure that heavily depends on the underlying abelian category.

In this talk, I will explain how to define motivic Hall algebra like structures for general stacks and how this yields a definition of Euler characteristic and of Donaldson–Thomas invariants that is valid for nonlinear moduli problems, like G-bundles or G-local systems. A crucial ingredient is the stack of filtrations of X, defined by Halpern–Leistner.

This is joint work with Chenjing Bu and Tasuki Kinjo.