Mapping-space Gromov--Witten classes and the Fano variety of lines

Abstract

Let $Y$ be a smooth cubic 3-fold. We show that $S^2 \times Y$ admits an infinite sequence of cohomologous symplectic forms which are all deformation equivalent but pairwise non-isotopic; McDuff showed the corresponding result on $(S^2)^2 \times T^2$, but these appear to be the first simply-connected and monotone examples. The results are obtained from McDuff's bordism-flavoured Gromov--Witten invariant, re-interpreted cohomologically in terms of an evaluation map valued in the space of maps from the 2-sphere to a symplectic target. The application then relies on classical properties of the Fano space of lines on $Y$. We also discuss higher-dimensional generalisations.