Convergence of Hermitian-Yang-Mills Connections on Kähler Surfaces and mirror symmetry

Takeo Nishinou

paper

Convergence of Hermitian-Yang-Mills Connections on Kähler Surfaces and mirror symmetry

arXiv:math/0301324

Abstract

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] $, an d e a c hha ss t r u c t u r eo f a L a g r an g ian t or u s f ib r a t i o n$ π:\hat M_{\ep} \to B $w h ose f ib er s a r eo f d iam e t er$ O(\ep) $, an d l e t$ A_{\ep} $b e a f ami l y o f h er mi t ianY an g - M i l l s (H Y M) co nn ec t i o n so n$ E_{\ep} $. A s$ \ep $g oes t oz er o,$ A_{\ep} $w i l l, m o d u l o p oss ib l e b u bb l es, co n v er g e t o a co nn ec t i o n w hi c hi s f l a t o n e a c h f ib er . S in cee a c h f ib er i s a t or u s, l imi t co nn ec t i o n w i l l d e t er min ee l e m e n t so f t h e d u a l t or u s, w hi c ha r e p o in t so f t h e f ib er o f t h e mi r r or$ M_1$. These points gather to make up (special) Lagrangian variety.

Title changed. Extensive revision