Mathematical Sciences: Complex Manifold Theory and Kaehler Geometry

这个项目将集中在复杂的数学问题 分析和几何。 具体而言，将开展以下工作： 复流形理论和Kahler几何中的问题。 建议书列出几个主要范畴，包括 不可约紧代数的超刚性 黎曼对称流形 由此可知， 流形的性质，任何光滑变换到一个 非正曲流形必与等距同伦。 本工作的目的是要证明超刚性的子 球以外的对称流形 另一个方向将 涉及到紧致厄米特的整体不可变形性 对称流形 这项工作将建立在最近的证据， 复射影空间的不可变形性 计算全纯向量零点个数的技巧 领域的 此外，还将在哈茨霍恩号上开展工作 关于两个复形的非空交的猜想 射影空间中具有正法丛子流形 代数流形 目前的研究表明， 必须总是非空的，如果它们的维数之和为 至少是环境歧管的尺寸。 至少三 计划采取不同的办法。

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.