Gromov-Witten理论与超Kaehler簇上的代数闭链

Algebraic cycles are one of the key concepts in algebraic geometry. Originated in the algebraic structure of a variety, algebraic cycles have deep connections with the topology and geometry of the variety. It is a crucial task to understand these connections...Hyper-Kaehler varieties form an important class of varieties. Alongside abelian varieties and Calabi-Yau varieties, they are the building blocks of varieties with trivial canonical bundle. Hyper-Kaehler varieties enjoy many beautiful properties thanks to their extra symmetry. The study of hyper-Kaehler varieties has attracted broad attention in recent years. This includes the conjecture of Beauville and Voisin regarding a mysterious decomposition on the ring of algebraic cycles of a hyper-Kaehler variety...This project aims at bringing new aspects and tools to the understanding of the Beauville-Voisin conjecture. More concretely, we try to construct the Beauville-Voisin decomposition via virtual cycle classes in Gromov-Witten theory. In abstract terms, we hope to view this decomposition as a form of mirror symmetry, and develop a theory of algebraic mirror symmetry for hyper-Kaehler varieties. This project will relate a wide range of subjects in algebraic geometry.

代数闭链是代数几何的核心概念之一。代数闭链源自代数簇的代数结构，却与代数簇的拓扑与几何有着千丝万缕的联系。如何理解“代数”与“几何”的联系是本学科的一个主要议题。..超Kaehler簇是一类重要代数簇。它们以及阿贝尔簇和Calabi-Yau簇是典范丛平凡代数簇的三大基石。超Kaehler簇上丰富的对称赋予其更多优秀性质。近年来对超Kaehler簇的研究引发了广泛关注，这其中包括Beauville和Voisin提出的关于超Kaehler簇上代数闭链环的某种奇妙分解的猜想。..本项目将引入新观点和工具来理解并尝试证明Beauville-Voisin猜想。具体来说，我们利用Gromov-Witten理论中的虚拟闭链类来构造Beauville-Voisin分解。抽象来说，我们期望把此种分解理解成某种镜对称现象，并发展一套关于超Kaehler簇的代数镜对称理论。本项目将联结代数几何多个分支。

本项目的主要研究对象是代数几何中的超Kähler流形。超Kähler流形是典范丛平凡代数簇的三大组成部分之一，近年来受到了广泛关注。本项目在超Kähler流形的特殊子簇，代数链的周环分解，以及Lagrange纤维化的拓扑与Hodge理论等问题的研究上取得一系列进展。