On the group of symplectic automorphisms of \C P^m × \C Pⁿ

1998/03/19 by Paul Seidel
Mathematics · #math.DG

paper · pdf

published as In: Northern California Symplectic Geometry Seminar, AMS Translations vol. 196, 1999, pp. 237-250 · 16 pages, LaTex2e with xy-pic

arxiv created 1998/03/19 · arxiv updated 2009/11/30

Abstract

Let M be the product of \C P^m and \C Pⁿ, 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 π_j for all odd j ≤ max\2m-1,2n-1\. This is a weak higher-dimensional analogue of Gromov's results for \C P¹ × \C P¹. 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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