Resolving symplectic orbifolds with applications to finite group actions

Abstract

We associate to each symplectic $4$-orbifold $X$ a canonical smooth symplectic resolution $\pi: \tilde{X}\rightarrow X$, which can be done equivariantly if $X$ comes with a symplectic $G$-action by a finite group. Moreover, we show that the resolutions of the symplectic $4$-orbifolds $X/G$ and $\tilde{X}/G$ are in the same symplectic birational equivalence class; in fact, the resolution of $\tilde{X}/G$ can be reduced to that of $X/G$ by successively blowing down symplectic $(-1)$-spheres. To any finite symplectic $G$-action on a $4$-manifold $M$, we associate a pair $(M_G,D)$, where $\pi: M_G\rightarrow M/G$ is the canonical resolution of the quotient orbifold and $D$ is the pre-image of the singular set of $M/G$ under $\pi$. We propose to study the group action on $M$ by analyzing the smooth or symplectic topology of $M_G$ as well as the embedding of $D$ in $M_G$. In this paper, an investigation on the symplectic Kodaira dimension $\kappa^s$ of $M_G$ is initiated. In particular, we conjecture that $\kappa^s(M_G)\leq \kappa^s(M)$. The inequality is verified for several classes of symplectic $G$-actions, including any actions on a rational surface or a symplectic $4$-manifold with $\kappa^s=0$.