On the group of symplectic automorphisms of $\C P^m \times \C P^n$

Abstract

Let M be the product of \C P^m and \C P^n, with the standard integral symplectic form. We prove that the inclusion map from the group of symplectic automorphisms of M to its diffeomorphism group is not surjective on homotopy groups. More precisely, it is not surjective on \pi_j for all odd j \leq \max\{2m-1,2n-1\}. This is a weak higher-dimensional analogue of Gromov's results for \C P^1 \times \C P^1. The proof uses parametrized Gromov-Witten invariants in a new (?) way. We also give some information about the symplectic automorphism groups of M with differently weighted product symplectic structures.