Vol. 304, No. 2, 2020

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

Jun Li and Tian-Jun Li

Vol. 304 (2020), No. 2, 561–606

Let (X,ω) be a symplectic rational surface. We study the space of tamed almost complex structures 𝒥ω 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 π1(Symp(X,ω)) with χ(X) 7 in terms of the number Nω of (2)-symplectic sphere classes.

symplectomorphism group, almost complex structure, rational symplectic manifold, Lagrangian root system
Mathematical Subject Classification 2010
Primary: 32Q65, 53C15, 53D35, 57R17
Secondary: 53D05, 57S05
Received: 22 March 2019
Revised: 3 July 2019
Accepted: 31 August 2019
Published: 12 February 2020
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