Family Seiberg-Witten equation on Kahler surface and $\pi_i(\Symp)$ on multiple-point blow ups of Calabi-Yau surfaces

Abstract

Let $\omega$ be a Kahler form on $M$, which is a torus $T^4$, a $K3$ surface or an Enriques surface, let $M\#n\overline{\mathbb{CP}^2}$ be $n-$point Kahler blowup of $M$. Suppose that $\kappa=[\omega]$ satisfies certain irrationality condition. Applying techniques related to deformation of complex objects, we extend the guage-theoretic invariant on closed Kahler suraces developed by Kronheimer\cite{Kronheimer1998} and Smirnov\cite{Smirnov2022}\cite{Smirnov2023}. As a result, we show that even dimensional higher homotopy groups of $\Symp(M\#n\overline{\mathbb{CP}^2},\omega)$ are infinitely generated.