Mathematical Sciences: Complex Manifold Theory and Kaehler Geometry

这个项目将侧重于复杂分析和几何的数学问题。具体来说，工作将在复杂流形理论和卡勒几何的问题上完成。提出了几个主要的研究领域，包括不可约黎曼对称流形紧商的超刚性问题。由此可以理解流形具有这样的性质：任何到非正弯曲流形的光滑变换都必须与等距同伦。本文的目的是证明非球对称流形商的超刚性。另一个方向将涉及紧致厄米对称流形的全局不可变形性。这项工作将建立在最近的复射影空间的不可变形性证明的基础上，使用一种计算全纯向量场的零个数的技术。此外，还研究了射影代数流形中两个正正规束复子流形的非空交的Hartshorne猜想。目前的研究表明，如果这些交叉点的维数之和至少等于周围流形的维数，那么它们必须总是非空的。计划中至少有三种不同的方法。

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.