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

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

1998-03-19

arxiv.org/abs/math/9803085v1

In: Northern California Symplectic Geometry Seminar, AMS Translations vol. 196, 1999, pp. 237-250

Mathematics

Differential Geometry

16 pages, LaTex2e with xy-pic

Scientific paper

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.

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