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Symplectic quasi-states and semi-simplicity of quantum homology. (English) Zbl 1146.53066
Harada, Megumi (ed.) et al., Toric topology. International conference, Osaka, Japan, May 28–June 3, 2006. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4486-1/pbk). Contemporary Mathematics 460, 47-70 (2008).
Summary: We review and streamline our previous results and the results of Y. Ostrover on the existence of Calabi quasi-morphisms and symplectic quasistates on symplectic manifolds with semi-simple quantum homology. As an illustration, we discuss the case of symplectic toric Fano 4-manifolds. We present also new results due to D. McDuff: she observed that for the existence of quasi-morphisms/quasi-states it suffices to assume that the quantum homology contains a field as a direct summand, and she showed that this weaker condition holds true for one point blow-ups of non-uniruled symplectic manifolds.
For the entire collection see [Zbl 1140.57001].

MSC:
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
53D40 Symplectic aspects of Floer homology and cohomology
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