  We suggest a construction of obstruction theory on the moduli stack of index
one covers over semi-log-canonical surfaces of general type. Comparing with 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 over the moduli stack of
surfaces of general type and a perfect obstruction theory. The perfect
obstruction theory induces a virtual fundamental class on the Chow group of the
moduli stack of surfaces of general type. Thus our construction proves a
conjecture of Sir Simon Donaldson for the existence of virtual fundamental
class. A tautological invariant is defined by taking integration of the power
of first Chern class for the CM line bundle on 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.
