Singular Hermitian Metrics on Vector Bundles and Codimension Two Subvarieties of Quadrics

小行星9701779 本项目主要研究两个问题：（1）向量丛上的奇异厄米特度量和（2）二次曲面的余维两个子簇。 对于第一个问题，主要研究者将确定复流形上全纯向量丛的奇异厄米度量理论。 其目的是发展对它们的基本理解，特别是积极性。对于第二个问题，主要研究者将研究维度至少为5的余维两个子簇。 他将致力于分类的低程度子流形;分类的子流形进行特殊结构;的作用，向量丛在上述分类;大小的希尔伯特计划的余维两个子流形不一般类型;的合理性和uniratification前面提到的品种;和adjunction理论性质的这些子流形。 这是代数几何领域的研究。 代数几何是现代数学中最古老的部分之一，但在过去的四分之一个世纪里，它已经有了革命性的发展。 在其起源，它处理的数字，可以定义在平面上的最简单的方程，即多项式。 如今，该领域不仅使用代数方法，还使用分析和拓扑学方法，相反，这些方法在这些领域以及物理学，理论计算机科学和机器人学中也得到了应用。

9701779 de Cataldo This project is concerned with two problems: (1) the study of singular hermitian metrics on vector bundles and (2) codimension two subvarieties of quadrics. For the first problem, the principal investigator will determine a theory of singular hermitian metrics for holomorphic vector bundles on complex manifolds. The aim is to develop a basic understanding of them with special regards to positivity. For the second problem, the principal investigator will study codimension two subvarieties of dimension at least five. He will work on the classification of low degree submanifolds; the classification of submanifolds which carry special structure; the role of vector bundles in the above classification; the size of the Hilbert scheme of codimension two submanifolds not of general type; the rationality and unirationality of the earlier mentioned varieties; and the adjunction theoretic properties of these submanifolds. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.