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

Abstract: We suggest a construction of obstruction theory on the moduli stack of index one covers over semi-log-canonical surfaces of general type. Based on the index one covering Deligne-Mumford stack of a semi-log-canonical surface, we define the $\lci$ covering Deligne-Mumford stack. The $\lci$ covering Deligne-Mumford stack only has locally complete intersection singularities. We construct the moduli stack of $\lci$ covers such that it admits a proper map to the moduli stack of surfaces of general type. We then construct a perfect obstruction theory on the moduli stack of lci covers and a virtual fundamental class on the Chow group of the moduli stack. Thus, our construction proves a conjecture of Sir Simon Donaldson for the existence of virtual fundamental class on the KSBA moduli spaces. A tautological invariant is defined by the integration of the power of the first Chern class for the CM line bundle of the moduli stack over the virtual fundamental class. This can be taken as a generalization of the tautological invariants defined by the integration of tautological classes over the moduli space $\overline{M}_g$ of stable curves to the moduli space of stable surfaces.