Family Seiberg-Witten equation on Kahler surface and ${\pi }_i(\text{Symp})$ on multiple-point blow ups of Calabi-Yau surfaces

Let $\omega$ be a Kahler form on $M$, which is a torus $T^4$, a $K3$ surface or an Enriques surface, let $M\mathrm{\#}n\overline{{\mathbb{C}\mathbb{P}}^2}$ 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 $\text{Symp}(M\mathrm{\#}n\overline{{\mathbb{C}\mathbb{P}}^2},\omega )$ are infinitely generated.