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Diffeomorphisms, symplectic forms and Kodaira fibrations

Claude LeBrun

Geometry & Topology 4 (2000) 451–456

arXiv: math.SG/0005195

Abstract

As was recently pointed out by McMullen and Taubes [Math. Res. Lett. 6 (1999) 681–696], there are 4–manifolds for which the diffeomorphism group does not act transitively on the deformation classes of orientation-compatible symplectic structures. This note points out some other 4–manifolds with this property which arise as the orientation-reversed versions of certain complex surfaces constructed by Kodaira [J. Analyse Math. 19 (1967) 207–215]. While this construction is arguably simpler than that of McMullen and Taubes, its simplicity comes at a price: the examples exhibited herein all have large fundamental groups.

Keywords
symplectic manifold, complex surface, Seiberg–Witten invariants
Mathematical Subject Classification 2000
Primary: 53D35
Secondary: 14J29, 57R57
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Publication
Received: 11 June 2000
Accepted: 21 November 2000
Published: 26 November 2000
Proposed: Ronald Stern
Seconded: Yasha Eliashberg, Ronald Fintushel
Authors
Claude LeBrun
Department of Mathematics
SUNY at Stony Brook
Stony Brook
New York 11794-3651
USA