Holomorphic line bundles on projective toric manifolds from Lagrangian sections of their mirrors by SYZ transformations

TL;DR

This paper explores the SYZ mirror symmetry for projective toric manifolds, establishing a correspondence between Lagrangian sections in the mirror Landau-Ginzburg model and hermitian metrics on line bundles, including special Hermitian-Einstein metrics.

Contribution

It introduces a new class of Lagrangian submanifolds in the mirror and demonstrates their correspondence to hermitian metrics on line bundles over toric manifolds, linking geometric and algebraic structures.

Findings

Lagrangian sections correspond to hermitian metrics on line bundles

Mirror of Hermitian-Einstein metrics identified as special Lagrangian sections

SYZ transformation explicitly relates Lagrangian geometry to line bundle metrics

Abstract

The mirror of a projective toric manifold $X_\Sigma$ is given by a Landau-Ginzburg model $(Y,W)$. We introduce a class of Lagrangian submanifolds in $(Y,W)$ and show that, under the SYZ mirror transformation, they can be transformed to torus-invariant hermitian metrics on holomorphic line bundles over $X_\Sigma$. Through this geometric correspondence, we also identify the mirrors of Hermitian-Einstein metrics, which are given by distinguished Lagrangian sections whose potentials satisfy certain Laplace-type equations.