Kappa classes on KSBA spaces

TL;DR

This paper introduces kappa classes on moduli spaces of KSBA stable varieties, generalizing known classes from curves, and computes them for specific surfaces where the virtual fundamental class is established.

Contribution

It defines and computes kappa classes on KSBA moduli spaces, extending the concept from curves to higher-dimensional varieties and pairs.

Findings

Kappa classes are defined on KSBA moduli spaces.

Explicit computations for Burniat and Campedelli surfaces.

Chow ring of a specific GIT quotient related to Campedelli surfaces.

Abstract

We define kappa classes on moduli spaces of KSBA stable varieties and pairs, generalizing the Miller-Morita-Mumford classes on moduli of curves, and compute them in some cases where the virtual fundamental class is known to exist, including Burniat and Campedelli surfaces. For Campedelli surfaces, an intermediate step is finding the Chow (same as cohomology) ring of the GIT quotient $(\mathbb P^2)^7//SL(3)$.