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We suggest a construction of obstruction theory on the moduli stack of indexone covers over semi-log-canonical surfaces of general type. Comparing with theindex one covering Deligne-Mumford stack of a semi-log-canonical surface, wedefine the $\lci$ covering Deligne-Mumford stack. The $\lci$ coveringDeligne-Mumford stack only has locally complete intersection singularities.  We construct the moduli stack of $\lci$ covers over the moduli stack ofsurfaces of general type and a perfect obstruction theory. The perfectobstruction theory induces a virtual fundamental class on the Chow group of themoduli stack of surfaces of general type. Thus our construction proves aconjecture of Sir Simon Donaldson for the existence of virtual fundamentalclass. A tautological invariant is defined by taking integration of the powerof first Chern class for the CM line bundle on the moduli stack over thevirtual fundamental class. This can be taken as a generalization of thetautological invariants defined by the integration of tautological classes overthe moduli space $\overline{M}_g$ of stable curves to the moduli space ofstable surfaces.

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