On the quantum homology algebra of toric Fano manifolds

2008/04/02 by Yaron Ostrover, Ostrover, Yaron, Ilya Tyomkin +1 · 1 citation
Mathematics · #14J45 #14M25 #53D40 #53D45 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Symplectic Geometry (math.SG) #math.AG #math.SG #msc:14J45 #msc:14M25 #msc:53D40 #msc:53D45

paper · pdf · doi:10.48550/arxiv.0804.0270

Revised version. Two corollaries have been added

openalex publication_date 2008/04/02 · arxiv created 2008/06/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we study certain algebraic properties of the quantum homology algebra for the class of symplectic toric Fano manifolds. In particular, we examine the semi-simplicity of the quantum homology algebra, and the more general property of containing a field as a direct summand. Our main result provides an easily-verified sufficient condition for these properties which is independent of the symplectic form. Moreover, we answer two questions of Entov and Polterovich negatively by providing examples of toric Fano manifolds with non semi-simple quantum homology algebra, and others in which the Calabi quasi-morphism is non-unique.

Citations

• Quantum Structures for Lagrangian Submanifolds
• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties

Cited by

• A-D-E diagrams, Hodge--Tate hyperplane sections and semisimple quantum cohomology

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