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

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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 (M, E), ∈ (0, 1], and each has structure of a Lagrangian torus fibration π: M → B whose fibers are of diameter O(), and let A be a family of hermitian Yang-Mills(HYM) connections on E. As goes to zero, A 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 M₁. These points gather to make up (special) Lagrangian variety.