Subvarieties of Hypersurfaces and Local Positivity of Ample Line Bundles

超曲面的子变体和充足线束的局部正性

• 批准号: 9700491
• 负责人: Geng Xu
• 金额: $ 8.45万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9700491&HistoricalAwards=false
• 关键词: Subvarieties; Hypersurfaces; Local; Positivity; Ample

Xu 9700491 This project is concerned with research in algebraic geometry. Based on the mirror symmetry principle, physicists proposed several years ago a formula to predict the numbers of rational curves of various degrees on a general quintic threefold. Recently, Givental showed that the numbers suggested by the physicists agree with the numbers of J-holomorphic curves of genus 0 of various degrees on a general quintic threefold for a generic almost complex structure J. The principal investigator will work on Clemens' conjecture about the finiteness of rational curves on a general quintic threefold in each degree. He is also working towards an understanding of the local positivity of ample line bundles on a smooth projective variety through the study of Seshadri constants, which is related to a conjecture of Nagata on a relation between the singularity of a plane algebraic curve and the degree of the curve. In addition, the principal investigator will study the algebraic hyperbolicity of the complement of a general hypersurface and the enumeration of algebraic curves on smooth projective surfaces. Finally, he will work on the quantum cohomology algebras of Fano manifolds, to see whether the quantum cohomology algebras of Fano hypersurfaces are semi-simple in the sense of Dubrovin. 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.

徐9700491 这个项目与代数几何的研究有关。 根据镜像对称原理，物理学家在几年前提出了一个公式，可以预测一般五次三次曲线上不同次数的有理曲线的个数。 最近，Givental表明，物理学家提出的数字与一般几乎复杂结构J的三重一般五次曲线上各种度数亏格0的J-全纯曲线的数量一致。首席研究员将致力于克莱门斯关于有理曲线的有限性的猜想。一般五次曲线的每个度数都是三倍。 他还致力于了解当地的积极性充足的线丛上顺利的投影品种通过研究Seshadri常数，这是有关的猜想永田之间的关系的奇异性的一个平面代数曲线和程度的曲线。 此外，主要研究者将研究一般超曲面的补的代数双曲性和光滑射影曲面上代数曲线的计数。 最后，他将研究法诺流形的量子上同调代数，看看法诺超曲面的量子上同调代数是否是杜布罗文意义上的半单。 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。