  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.
