Symplectic (−2)-spheres and the symplectomorphism group of small rational 4-manifolds

Let $(X, ω)$ be a symplectic rational surface. We study the space of tamed almost complex structures $𝒥_{ω}$ using a fine decomposition via smooth rational curves and a relative version of the infinite dimensional Alexander–Pontrjagin duality. This decomposition provides new understandings of both the variation and the stability of the symplectomorphism group $Symp (X, ω)$ when deforming $ω$ . In particular, we compute the rank of $π_{1} (Symp (X, ω))$ with $χ (X) \leq 7$ in terms of the number $N_{ω}$ of $(- 2)$ -symplectic sphere classes.

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Keywords

symplectomorphism group, almost complex structure, rational symplectic manifold, Lagrangian root system

Mathematical Subject Classification 2010

Primary: 32Q65, 53C15, 53D35, 57R17

Secondary: 53D05, 57S05

Milestones

Received: 22 March 2019

Revised: 3 July 2019

Accepted: 31 August 2019

Published: 12 February 2020

Authors

Jun Li
Department of Mathematics University of Michigan Ann Arbor, MI United States

Tian-Jun Li
School of Mathematics University of Minnesota Minneapolis, MN United States

Vol. 304, No. 2, 2020

Jun Li and Tian-Jun Li

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification 2010

Milestones

Authors