A new family of isolated singularities with trivial local fundamental group

Gwyn Bellamy (1) , Cédric Bonnafé (2) , Baohua Fu (3) , Daniel Juteau (4, 5) , Paul Levy (6) , Eric Sommers (7)

1 School of Mathematics and Statistics, Mathematics and Statistics Building, University of Glasgow
2 IMAG - Institut Montpelliérain Alexander Grothendieck
3 UCAS - University of Chinese Academy of Sciences [Beijing]
4 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352
5 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
6 Department of Mathematics & Statistics [Lancaster]
7 Department of Mathematics and Statistics of the University of Massachusetts, Amherst

We construct a new infinite family of 4-dimensional isolated symplectic singularities with trivial local fundamental group, answering a question of Beauville raised in 2000. Three constructions are presented for this family: (1) as singularities in blowups of the quotient of 4 by the dihedral group of order 2d , (2) as singular points of Calogero-Moser spaces associated with dihedral groups of order 2d at equal parameters, (3) as singularities of a certain Slodowy slice in the d-fold cover of the nilpotent cone in sl d .

Domaines: Géométrie algébrique [math.AG]; Théorie des représentations [math.RT]

Gwyn Bellamy, Cédric Bonnafé, Baohua Fu, Daniel Juteau, Paul Levy, et al.. A new family of isolated singularities with trivial local fundamental group. Proceedings of the London Mathematical Society, 2023, 126 (5), pp.1496-1521. ⟨10.1112/plms.12513⟩. ⟨hal-03565140⟩