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We introduce a method of constructing the virtual cycle of any schemeassociated with a tangent-obstruction complex. We apply this method toconstructing the virtual moduli cycle of the moduli of stable maps fromn-pointed genus g curves to any smooth projective variety. As an application,we give an algebraic definition of GW-invariants for any smooth projectivevariety.

Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties