9703884 Zheng This project lies in the area of complex differential geometry. The investigator earlier showed that the ration of the two Chern numbers of a nonpositively curved compact Kahler surface is always between two and three; he now wishes to generalize that work to higher dimensional settings. In addition, the investigator is to pursue the remaining classification in the famous Kodaira-Enriques classification for complex surfaces. Finally, the investigator would like to construct a certain class of nonpositively curved Kahler manifolds. Kahler manifolds generalize curved surfaces, and are based on complex numbers. The Chern numbers of a Kahler manifold are certain integers that are topologically invariant, i.e., they are quantities that are unchanged under small deformation of the manifold. Kahler manifolds have found applications in other parts of mathematics as well as theoretical physics. For example, the famous Yau-Calabi spaces used in the string model of the universe are Kahler manifolds.