Birational geometry for the covering of a nilpotent orbit closure

Abstract

Let \({\mathfrak {g}}\) be a complex classical simple Lie algebra and let O be a nilpotent orbit of \({\mathfrak {g}}\). The fundamental group \(\pi _1(O)\) is finite. Take the universal covering \(\pi ^0: X^0 \rightarrow O\). Then \(\pi ^0\) extends to a finite cover \(\pi : X \rightarrow {\bar{O}}\). By using the Kirillov–Kostant form \(\omega _{KK}\) on O, the normal affine variety X becomes a conical symplectic variety. In this article we give an explicit construction of a Q-factorial terminalization of X.