The SYZ conjecture suggests a folklore that "Lagrangian multi-sections are mirror to holomorphic vector bundles". In this talk, we prove this folklore for Lagrangian multi-sections inside the cotangent bundle of a vector space, which are equivariantly mirror to complete toric varieties by the work of Fang-Liu-Treumann-Zaslow. We will also introduce the notion of tropical Lagrangian multi-sections and the Lagrangian realization problem. The latter asks whether one can construct an unobstructed Lagrangian multi-section with asymptotic conditions prescribed by a given tropical Lagrangian multi-section. In dimension 2, we solve the realization problem for those tropical Lagrangian multi-sections that satisfy the so-called N-generic condition. As an application, we show that every rank 2 toric vector bundle on the projective plane is mirror to a simply connected Lagrangian multi-section. This is a joint work Yong-Geun Oh.

 

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