Birational geometry and deformations of nilpotent orbits

Abstract

This is a continuation of [N2], where we have described the relative movable cone for a Springer resolution of the closure of a nilpotent orbit in a complex simple Lie algebra. But, in general, the movable cone does not coincide with the whole space of numerical classes of divisors on the Springer resolution.

The purpose of this article is to describe the remainder. We first construct a deformation of the nilpotent orbit closure in a canonical manner, according to Brieskorn and Slodowy (see [S]), and next describe all its crepant simultaneous resolutions. This construction enables us to divide the whole space into a finite number of chambers.

Moreover, by using this construction, one can generalize the main result of [N2] to arbitrary Richardson orbits whose Springer maps have degree greater than $1$. New Mukai flops, different from those of types $A$, $D$, and $E_6$, appear in the birational geometry for such orbits