Homological mirror symmetry for local Calabi-Yau manifolds via SYZ

TL;DR

This paper surveys how SYZ transforms are used to understand Kontsevich's homological mirror symmetry for specific local Calabi-Yau manifolds, connecting geometric and categorical aspects of mirror symmetry.

Contribution

It demonstrates the application of SYZ transforms to prove cases of homological mirror symmetry for local Calabi-Yau manifolds, advancing the geometric understanding of HMS.

Findings

SYZ transforms relate symplectic and complex geometry in local Calabi-Yau cases

Establishment of HMS for certain local Calabi-Yau manifolds

Integration of geometric and categorical mirror symmetry insights

Abstract

This is a write-up of the author's talk in the conference "Algebraic Geometry in East Asia 2016" held at the University of Tokyo in January 2016. We give a survey on a series of papers of the author and his collaborators Daniel Pomerleano and Kazushi Ueda where we show how Strominger-Yau-Zaslow (SYZ) transforms can be applied to understand the geometry of Kontsevich's homological mirror symmetry (HMS) conjecture for certain local Calabi-Yau manifolds.