Homogeneous coordinate rings and mirror symmetry for toric varieties

Abouzaid, Mohammed

doi:10.2140/gt.2006.10.1097

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Homogeneous coordinate rings and mirror symmetry for toric varieties

Mohammed Abouzaid

Geometry & Topology 10 (2006) 1097–1156

DOI: 10.2140/gt.2006.10.1097

arXiv: math.SG/0511644

Abstract

Given a smooth toric variety $X$ and an ample line bundle $O (1)$ , we construct a sequence of Lagrangian submanifolds of ${(ℂ^{⋆})}^{n}$ with boundary on a level set of the Landau–Ginzburg mirror of $X$ . The corresponding Floer homology groups form a graded algebra under the cup product which is canonically isomorphic to the homogeneous coordinate ring of $X$ .

Keywords

homological mirror symmetry, toric varieties, tropical geometry

Mathematical Subject Classification 2000

Primary: 14J32

Secondary: 53D40

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Publication

Received: 26 November 2005

Revised: 3 May 2006

Accepted: 1 June 2006

Published: 24 August 2006

Proposed: Jim Bryan

Seconded: Lothar Göttsche, Simon Donaldson

Authors

Mohammed Abouzaid
Department of Mathematics University of Chicago Chicago, IL 60637 USA

Volume 10, issue 2 (2006)