On the quantum homology algebra of toric Fano manifolds

Abstract

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. (2000). Primary 53D45; Secondary 14M25. Mathematics Subject Classification

Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms

“…Meanwhile (i) generalizes results that have been proven for a variety of toric Fano manifolds; in particular in Ostrover and Tyomkin [48] and Fukaya, Oh, Ohta and Ono [22], it is shown that if M is toric and Fano then QH.M; !/ 0 is semisimple for generic choices of the toric symplectic form ! .…”

“…This follows from the Batyrev-Givental formula for the quantum homology of such a manifold as re-expressed by eg Fukaya, Oh, Ohta and Ono [22] and Ostrover and Tyomkin [48]; in particular, in light of Proposition 7.11 above, we can simply read off this conclusion from [48,Theorem 4.1].…”

“…. For example, in [48,Section 5] Ostrover and Tyomkin produce a toric Fano 4-fold M such that QH.M; !/ 0 fails to be semisimple when ! is the monotone toric symplectic form but is semisimple for generic toric symplectic forms !…”

“…A reader who still prefers to work with undeformed (ie Á D 0) quantum homology may take solace in the following, which is somewhat reminiscent of Ostrover and Tyomkin [48,Theorem 4.1] and Fukaya, Oh, Ohta and Ono [22,Proposition 8.8]:…”

“…In Ostrover and Tyomkin [48], the authors use a slightly different convention for the Novikov ring, in that they consider the ring K # in which the exponents are allowed to be arbitrary real numbers rather than being restricted to the period group D ! .…”