Mathematical Sciences: Complex Manifold Theory and Kaehler Geometry

这个项目将专注于复杂分析和几何的数学问题。具体地说，将在复流形理论和Kahler几何的问题上做工作。文中列出了几个主要领域，包括不可约黎曼对称流形紧商的超刚性。由此可以理解流形的性质，即到非正曲线流形的任何光滑变换一定与等距同伦。这项工作的目的是证明除球以外的对称流形的商的超刚性。另一个方向将涉及紧致厄米特对称流形的整体不可变形性。这项工作将建立在最近用一种计算全纯向量场的零点数的方法证明复射影空间的不可变形性的基础上。此外，还将对射影代数流形中两个复子流形与正法丛的非空交的HartShorne猜想进行研究。目前的研究表明，如果这些交叉口的尺寸之和至少是环境流形的尺寸，那么这些交叉口必须总是非空的。至少计划了三种不同的方法。

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.