Lagrangian torus fibration and mirror symmetry of Calabi-Yau hypersurface in toric variety

TL;DR

This paper constructs Lagrangian torus fibrations for Calabi-Yau hypersurfaces in toric varieties using gradient flow, and proves a symplectic version of the SYZ mirror conjecture, including singular fibers.

Contribution

It provides a new construction method for Lagrangian fibrations and a proof of the SYZ mirror conjecture for generic Calabi-Yau hypersurfaces in toric varieties.

Findings

Constructed Lagrangian torus fibrations via gradient flow.

Proved symplectic topological SYZ mirror conjecture.

Formulated SYZ conjecture including singular fibers and duality.

Abstract

In this paper we give a construction of Lagrangian torus fibration for Calabi-Yau hypersurface in toric variety via the method of gradient flow. Using our construction of Lagrangian torus fibration, we are able to prove the symplectic topological version of SYZ mirror conjecture for generic Calabi-Yau hypersurface in toric variety. We will also be able to give precise formulation of SYZ mirror conjecture in general (including singular locus and duality of singular fibres).