Symplectic mapping class groups of blowups of tori

TL;DR

This paper proves that the symplectic mapping class group of a one-point blowup of a 4-torus with an irrational Kaehler form is infinitely generated, revealing complex structure in symplectic topology.

Contribution

It demonstrates that under certain irrationality conditions, the symplectic mapping class group of the blowup is infinitely generated, a novel result in symplectic topology.

Findings

The symplectic mapping class group is infinitely generated.

The result applies to blowups of 4-tori with irrational Kaehler forms.

The irrationality condition can be achieved by small perturbations.

Abstract

Let $\omega$ be a Kaehler form on the real $4$-torus $T^4$. Suppose that $\omega$ satisfies an irrationality condition which can be achieved by an arbitrarily small perturbation of $\omega$. This note shows that the smoothly trivial symplectic mapping class group of the one-point symplectic blowup of $(T^4,\omega)$ is infinitely generated.