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

DOI: 10.2140/pjm.2020.304.561

Abstract

Let $\left(X,\omega \right)$ be a symplectic rational surface. We study the space of tamed almost complex structures ${\mathcal{𝒥}}_{\omega }$ 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 $Symp\left(X,\omega \right)$ when deforming $\omega$. In particular, we compute the rank of ${\pi }_1\left(Symp\left(X,\omega \right)\right)$ with $\chi \left(X\right)\le 7$ in terms of the number $N_{\omega }$ of $\left(-2\right)$-symplectic sphere classes.

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