Kähler曲面上Hermitian-Yang-Mills联络的收敛性与镜像对称性
Convergence of Hermitian-Yang-Mills Connections on Kähler Surfaces and mirror symmetry
摘要：
本文的目的是展示一个自然的构造复几何和辛几何的镜像对称的思想。设给定一族二维Kähler环面对及其上的稳定全纯向量丛$(\hat M_{\ep}, E_{\ep})$，\ep \in（0，1] $, and each has structure of a Lagrangian torus fibration $ π：\hat M_{\ep} \to B $ whose fibers are of diameter $ O（\ep）$, and let $ A_{\ep} $ be a family of hermitian Yang-Mills(HYM) connections on $ E_{\ep} $. As $ \ep $ goes to zero, $ A_{\ep} $ will, modulo possible bubbles, converge to a connection which is flat on each fiber. Since each fiber is a torus, limit connection will determine elements of the dual torus, which are points of the fiber of the mirror $ M_1$.这些点聚集在一起构成（特殊的）拉格朗日簇。
英文摘要：
The purpose of this paper is to exhibit a natural construction between complex geometry and symplectic geometry following the idea of mirror symmetry. Suppose we are given a family of pairs of 2-dimensional Kähler tori and stable holomorphic vector bundles on them $(\hat M_{\ep}, E_{\ep})$, \ep \in (0, 1]$, and each has structure of a Lagrangian torus fibration $π:\hat M_{\ep} \to B$ whose fibers are of diameter $O(\ep)$, and let $A_{\ep}$ be a family of hermitian Yang-Mills(HYM) connections on $E_{\ep}$. As $\ep$ goes to zero, $A_{\ep}$ will, modulo possible bubbles, converge to a connection which is flat on each fiber. Since each fiber is a torus, limit connection will determine elements of the dual torus, which are points of the fiber of the mirror $M_1$. These points gather to make up (special) Lagrangian variety.
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