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

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

2003-01-28

arxiv.org/abs/math/0301324v2

Mathematics

Symplectic Geometry

Title changed. Extensive revision

Scientific paper

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 K\"ahler 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 $\pi:\hat M_{\ep} \to B$ 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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