The virtual fundamental class for the moduli space of surfaces of general type

Abstract

We prove that the moduli stack of index-one covers of semi-log-canonical surfaces of general type is isomorphic to the KSBA moduli stack of stable general type surfaces. Using the index-one covering Deligne-Mumford stack of a semi-log-canonical surface, we define the $\lci$ cover. The $\lci$ cover, as a Deligne-Mumford stack, has only locally complete intersection singularities. We then construct the moduli stack of $\lci$ covers so that it admits a proper map to the moduli stack of surfaces of general type. Next, we construct a perfect obstruction theory on this stack and a virtual fundamental class in its Chow group. We then pushforward the virtual fundamental class from the moduli stack of lci covers to the KSBA moduli space. Thus, our construction proves Donaldson's conjecture on the existence of a virtual fundamental class for KSBA moduli spaces. A tautological invariant is defined by integrating a power of the first Chern class of the CM line bundle over the virtual fundamental class. This serves as a generalization of the tautological invariants defined by integrating tautological classes over the moduli space $\overline{M}_g$ of stable curves to the moduli space of stable surfaces.