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

本课题涉及两个问题：(1)向量束上奇异厄米度量的研究；(2)二次曲面的余维两个子变种。对于第一个问题，主要研究者将确定复流形上全纯向量束的奇异厄密度量理论。其目的是培养对它们的基本理解，特别是对积极性的理解。对于第二个问题，首席研究员将研究至少五维的协维两个子变种。他将研究低次子流形的分类；具有特殊结构的子流形的分类；向量束在上述分类中的作用；非一般型余维子流形Hilbert格式的大小；上述品种的合理性与不合理性；以及这些子流形的附加理论性质。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。在它的起源中，它处理的图形可以用最简单的方程，即多项式，在平面上定义。如今，该领域不仅使用代数的方法，还使用分析和拓扑的方法，相反，它在这些领域以及物理学、理论计算机科学和机器人技术中也得到了应用。

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.