Lagrangian multi-sections and their toric equivariant mirror

TL;DR

This paper proves the SYZ conjecture for Lagrangian multi-sections in cotangent bundles, introduces the Lagrangian realization problem, and demonstrates the mirror correspondence for rank 2 toric vector bundles on the projective plane.

Contribution

It establishes the mirror relationship for Lagrangian multi-sections in cotangent bundles and solves the realization problem for certain tropical Lagrangian multi-sections.

Findings

Proves the SYZ folklore for Lagrangian multi-sections in cotangent bundles.

Solves the Lagrangian realization problem for 2D tropical multi-sections under N-generic conditions.

Shows all rank 2 toric vector bundles on the projective plane are mirror to Lagrangian multi-sections.

Abstract

The SYZ conjecture suggests a folklore that "Lagrangian multi-sections are mirror to holomorphic vector bundles". In this paper, 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 also introduce the Lagrangian realization problem, which asks whether one can construct an unobstructed Lagrangian multi-section with asymptotic conditions prescribed by a tropical Lagrangian multi-section. We solve the realization problem for tropical Lagrangian multi-sections over a complete 2-dimensional fan that satisfy the so-called $N$-generic condition with $N\geq 3$. As an application, we show that every rank 2 toric vector bundle on the projective plane is mirror to a Lagrangian multi-section.