TY - JOUR
T1 - On the quantum homology algebra of toric fano manifolds
AU - Ostrover, Yaron
AU - Tyomkin, Ilya
N1 - Funding Information:
We thank D. Auroux, L. Polterovich, P. Seidel, and M. Temkin for insightful comments and discussions. We are grateful to D. McDuff for important remarks and suggestions, and for pointing out some inaccuracies in the first draft of this paper. We also wish to thank the anonymous referees for their comments, which helped us to improve the presentation of the text. The first named author was supported by NSF grant DMS-0706976.
PY - 2009/6
Y1 - 2009/6
N2 - We study certain algebraic properties of the small quantum homology algebra for the class of symplectic toric Fano manifolds. In particular, we examine the semisimplicity of this algebra, and the more general property of containing a field as a direct summand. Our main result provides an easily verifiable 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-semisimple quantum homology, and others in which the Calabi quasi-morphism is not unique.
AB - We study certain algebraic properties of the small quantum homology algebra for the class of symplectic toric Fano manifolds. In particular, we examine the semisimplicity of this algebra, and the more general property of containing a field as a direct summand. Our main result provides an easily verifiable 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-semisimple quantum homology, and others in which the Calabi quasi-morphism is not unique.
KW - Quantum homology
KW - Semisimplicity
KW - Toric fano manifolds
UR - http://www.scopus.com/inward/record.url?scp=67650309308&partnerID=8YFLogxK
U2 - 10.1007/s00029-009-0526-9
DO - 10.1007/s00029-009-0526-9
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AN - SCOPUS:67650309308
VL - 15
SP - 121
EP - 149
JO - Selecta Mathematica, New Series
JF - Selecta Mathematica, New Series
SN - 1022-1824
IS - 1
ER -