Subvarieties of Hypersurfaces and Local Positivity of Ample Line Bundles

Xu 9700491这个项目是关于代数几何的研究。几年前，物理学家根据镜像对称原理，提出了一个公式来预测一般五次三次曲线上不同程度的有理曲线的数量。最近，Givental证明了物理学家提出的对于一般几乎复杂结构J.的一般五次三重上不同次的j -全纯曲线的数目是一致的。主要研究者将对克莱门斯关于一般五次三重上每次有理曲线有限的猜想进行研究。他还致力于通过研究Seshadri常数来理解光滑射影变化上充足线束的局部正性，这与Nagata关于平面代数曲线奇点与曲线度之间关系的猜想有关。此外，主要研究者将研究一般超曲面补的代数双曲性和光滑射影曲面上代数曲线的枚举。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在杜布罗文意义上是半简单的。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。

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.