A remark on isolated symplectic singularities with trivial local fundamental group
Yoshinori Namikawa
TL;DR
This work constructs isolated symplectic singularities with trivial local fundamental group in arbitrary dimensions using a toric hyperkähler framework: a primary family $Y(\mathbf{a},0)$ defined by the moment map $\mu=\sum_i a_i z_i w_i$ on $\mathbf{C}^{2n}$ and the quotient by $\mathbf{C}^*$ yields a $2n-2$-dimensional isolated singularity with $\pi_1(Y(\mathbf{a},0)_{\mathrm{reg}})=1$, and a higher-codimension generalization $Y(A,0)$ gives similar results with dimension $2(n-d)$. Both families admit two crepant partial resolutions that are linked by flops, and distinct data produce non-isomorphic varieties; the maximal-weight discussion shows how to realize arbitrary weights in this setup. The paper also connects these constructions to contact geometry via the LeBrun-Salamon conjecture, describing how conical symplectic varieties arise from contact Fano manifolds and how the converse yields a weighted blow-up picture, motivating open questions about finding new $\mathbb{Q}$-factorial conical symplectic varieties with isolated singularities and trivial local fundamental group. The results provide a flexible, dimension-agnostic framework for examining isolated symplectic singularities and their birational and contact-geometry interplays.
Abstract
Recently, Bellamy et al. constructed an infinite series of 4-dimensional isolated symplectic sngularities with trivial local fundamental group, inspired by a question of Beauville. In this short note, we introduce an easy construction of isolated symplectic singularities (of any dimension) with trivial local fundamental group. We will use a toric hyperkaehler construction.