Resolving symplectic orbifolds with applications to finite group actions

TL;DR

This paper develops a canonical method to resolve symplectic 4-orbifolds into smooth symplectic manifolds, preserving group actions, and explores how these resolutions relate to the original orbifolds and their symplectic Kodaira dimensions.

Contribution

It introduces a canonical equivariant resolution process for symplectic 4-orbifolds and studies the impact on symplectic Kodaira dimension under finite group actions.

Findings

Resolutions are in the same symplectic birational class.

The resolution process reduces to simpler orbifolds via blowing down.

Conjecture: the symplectic Kodaira dimension does not increase under resolution.

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 $\kappa^s=0$.