Symplectic quasi-states and semi-simplicity of quantum homology

TL;DR

This paper reviews and extends results on the existence of symplectic quasi-states and Calabi quasi-morphisms in symplectic manifolds with semi-simple quantum homology, including new insights from D. McDuff.

Contribution

It simplifies previous proofs and introduces weaker conditions for the existence of quasi-morphisms and quasi-states, broadening their applicability.

Findings

Existence of Calabi quasi-morphisms and symplectic quasi-states on semi-simple quantum homology manifolds.

Weaker conditions involving a field as a direct summand suffice for quasi-morphism existence.

Application to one point blow-ups of non-uniruled symplectic manifolds.

Abstract

We review and streamline our previous results and the results of Y.Ostrover on the existence of Calabi quasi-morphisms and symplectic quasi-states 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.