Relative family Gromov-Witten invariants and symplectomorphisms

Abstract

We study the symplectomorphism groups $G_λ=Symp_0(M,ω_λ)$ of an arbitrary closed manifold M equipped with a 1-parameter family of symplectic forms $ω_λ$ with variable cohomology class. We show that the existence of nontrivial elements in $π_*({\cal A},{\cal A}')$, where $({\cal A},{\cal A}')$ is a suitable pair of spaces of almost complex structures, implies the exiarxiv.org stence of families of nontrivial elements in $π_{*-i}G_λ$, for $i=1$ or 2. Suitable parametric Gromov Witten invariants detect nontrivial elements in $π_*({\cal A},{\cal A}')$. By looking at certain resolutions of quotient singularities we investigate the situation $(M,ω_λ)= (S^2 \times S^2 \times X,σ_F \oplus λσ_B \oplus ω_{st})$, with $(X,ω_{st})$ an arbitrary symplectic manifold. We find families of nontrivial elements in $π_k(G_λ^X)$, for countably many $k$ and different values of $λ$. In particular we show that the fragile elements $w_{\ell}$ found by Abreu-McDuff in $π_{4 \ell}(G_{\ell+1}^{pt})$ do not disappear when we consider them in $S^2 \times S^2 \times X$.