On singular Calogero-Moser spaces

TL;DR

This paper characterizes when generalized Calogero-Moser spaces are singular based on complex reflection groups, showing that only specific groups admit symplectic resolutions, with implications for the structure of these spaces.

Contribution

It provides a complete classification of singular Calogero-Moser spaces for complex reflection groups and establishes the existence or non-existence of symplectic resolutions for these cases.

Findings

Calogero-Moser spaces are singular for all parameters except for specific groups.

Symplectic resolutions exist only for the binary tetrahedral group and wreath product groups.

No symplectic resolution exists for other complex reflection groups.

Abstract

Using combinatorial properties of complex reflection groups, we show that the generalised Calogero-Moser space associated to the centre of the corresponding rational Cherednik algebra is singular for all values of its deformation parameter c if and only if the group is different from the wreath product $S_n\wr C_m$ and the binary tetrahedral group. This result and a theorem of Ginzburg and Kaledin imply that there does not exist a symplectic resolution of the singular symplectic variety h+h*/W outside of these cases; conversely we show that there exists a symplectic resolution for the binary tetrahedral group (Hilbert schemes provide resolutions for the wreath product case).