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

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

2009-03-06

arxiv.org/abs/0903.1164v2

Int. Math. Res. Not. 2009, no. 24, 4686-4708

Mathematics

Symplectic Geometry

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

Scientific paper

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.

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