Relative family Gromov-Witten invariants and symplectomorphisms

Abstract

We study the symplectomorphism groups $G_{\lambda }=Sym{p}_0(M,{\omega }_{\lambda })$ of an arbitrary closed manifold M equipped with a 1-parameter family of symplectic forms ${\omega }_{\lambda }$ with variable cohomology class. We show that the existence of nontrivial elements in ${\pi }_{\ast }(\mathcal{A},{\mathcal{A}}^{\prime })$, where $(\mathcal{A},{\mathcal{A}}^{\prime })$ is a suitable pair of spaces of almost complex structures, implies the exiarxiv.org stence of families of nontrivial elements in ${\pi }_{\ast -i}{G}_{\lambda }$, for $i=1$ or 2. Suitable parametric Gromov Witten invariants detect nontrivial elements in ${\pi }_{\ast }(\mathcal{A},{\mathcal{A}}^{\prime })$. By looking at certain resolutions of quotient singularities we investigate the situation $(M,{\omega }_{\lambda })=(S^2\times S^2\times X,{\sigma }_F\oplus \lambda {\sigma }_B\oplus {\omega }_{st})$, with $(X,{\omega }_{st})$ an arbitrary symplectic manifold. We find families of nontrivial elements in ${\pi }_k(G_{\lambda }^X)$, for countably many $k$ and different values of $\lambda$. In particular we show that the fragile elements $w_{\ell }$ found by Abreu-McDuff in ${\pi }_{4\ell }(G_{\ell +1}^{pt})$ do not disappear when we consider them in $S^2\times S^2\times X$.

23 pages