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In this paper, we define the virtual moduli cycle of moduli spaces with perfect tangent-obstruction theory. The two interesting moduli spaces of this type are moduli spaces of vector bundles over surfaces and moduli spaces of stable morphisms from curves to projective varieties. As an application, we define the Gromov-Witten invariants of smooth projective varieties and prove all its basic properties.

In this paper, we prove that there exists a K~ihler-Einstein metric, abbreviated as K-E metric, on a m-dimension Fermat hypersurface with degree greater than m-l. In particular, a Fermat cubic surface admits such a K-E metric. By standard Implicit function theorem, it also implies that there are a lot of mdimension hypersurfaces with degree __>m, which admit K-E metrics. The problem of K-E metric on a K~ihler manifold with definite first Chern class was raised by Calabi The invariant e(M) (resp. eG(M)) plays a role in the study of K-E metric more and less same as the Moser-Trundinger constant does in the study of prescribed curvature problem on S 2. It would be an interesting problem to determine how large ~(M) is. A local problem, which is relevant to ~(M), was considered by Bombieri [2] and Skoda [11]. Precisely, they proved that given a plurisubharmonic function q~, if the Lelong number of ~b is small enough, then ~b is locally integrable, c~(M) is regarding to the properties of anticanonical bundle of M and the families of holomorphic curves of smaller degree with respect to the polarization given by CI(M). We guess that e(M) has a lower bound only depending on the dimension m. It is pointed out by Professor Yau that this will result in a upper bound of (--KM) m.