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

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.

v2: 20 pages; Definition 3.1 modified, a couple of examples added; to appear in IMRN

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• SYZ mirror of Hirzebruch surface ${\mathbb{F}}_k$ and Morse homotopy
• Homological mirror symmetry of ${\mathbb{F}}_1$ via Morse homotopy
• Homological mirror symmetry for local Calabi-Yau manifolds via SYZ
• Delzant type theorem for torus-equivariantly embedded toric hypersurfaces