On the fundamental group of the regular part of Fujiki's compact Kahler symplectic orbifolds
Shun Yamaguchi
TL;DR
The paper resolves Perego's question by computing the fundamental group of the regular part of Fujiki's compact Kahler symplectic orbifolds. It introduces a general principle: for a simply connected $Z$ acted on by a discrete group $\mathcal{G}$ with finite stabilizers and suitably small fixed loci, the fundamental group of an open subset $U$ of the quotient $Y=Z/\mathcal{G}$ equals $\mathcal{G}/N_U$, where $N_U$ is generated by the stabilizers over $\pi^{-1}(U)$. Applying this to Fujiki's construction with $Z=S\times S$ and $G$ built from a symplectic finite abelian group $H$ on $S$ (K3 or torus) yields: when $S$ is a K3 surface, $\pi_1(X_{\mathrm{reg}})=\{1\}$, so the orbifold is irreducible; when $S$ is a complex torus, $\pi_1(X_{\mathrm{reg}})$ is a finite group determined by $H$ (with four torus‑derived cases not irreducible), and the universal cover of $X_{\mathrm{reg}}$ extends to an étale in codimension one cover that is an irreducible symplectic orbifold. Together, these results classify irreducibility within Fujiki's examples and provide explicit fundamental groups for the regular parts.
Abstract
We calculate the fundamental group of the regular part of certain compact Kahler symplectic orbifolds constructed by Fujiki, called Fujiki's examples. We determine which one is an irreducible symplectic orbifold among Fujiki's examples. This answers a question posed by A.Perego.