This project will focus on mathematical problems of complex analysis and geometry. Specifically, work will be done on questions in complex manifold theory and Kahler geometry. Several major areas are mapped out in the proposal, including that of superrigidity of compact quotients of irreducible Riemannian symmetric manifolds. By this one understands manifolds with the property that any smooth transformation to a nonpositively curved manifold must be homotopic to an isometry. The goal of this work is to prove superrigidity of quotients of symmetric manifolds other than balls. Another direction will involve the global nondeformability of compact Hermitian symmetric manifolds. This work will build on a recent proof of the nondeformability of complex projective space using a technique which counts the number of zeros of holomorphic vector fields. Additionally, work will be done on the Hartshorne conjecture concerning the nonempty intersection of two complex submanifolds with positive normal bundle in a projective algebraic manifold. Current research suggests such intersections must always be nonempty if the sum of their dimensions is at least the dimension of the ambient manifold. At least three different approaches are planned.
