Enumerative Geometry on KSBA moduli spaces

TL;DR

This paper surveys two new compactification methods for KSBA moduli spaces of surfaces of general type, enabling the definition of virtual classes and tautological invariants for enumerative geometry.

Contribution

It introduces two novel compactification techniques for KSBA moduli spaces that admit perfect obstruction theories, facilitating enumerative geometric studies.

Findings

Existence of virtual fundamental classes on the new moduli spaces

Definition of tautological invariants on KSBA moduli spaces

Potential for enumerative geometry applications

Abstract

We survey two new compactification methods for the KSBA moduli space of general type surfaces so that both of them admit a perfect obstruction theory. Virtual fundamental classes exist on these two moduli spaces, and tautological invariants can be defined on KSBA moduli spaces. This is the starting point to do enumerative geometry on KSBA moduli spaces, and we include some discussions in this direction.