2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/148366We introduce a method of constructing the virtual cycle of any scheme associated with a tangent-obstruction complex. We apply this method to constructing the virtual moduli cycle of the moduli of stable maps from n-pointed genus g curves to any smooth projective variety. As an application, we give an algebraic definition of GW-invariants for any smooth projective variety.68 pages, Amstex. Revised version. It appeared in J. of Amer. Math. Soc. Jan 98Algebraic Geometrytext