Induced nilpotent orbits and birational geometry

In general, a nilpotent orbit closure in a complex simple Lie algebra \g, does not have a crepant resolution. But, it always has a Q-factorial terminalization by the minimal model program. According to B. Fu, a nilpotent orbit closure has a crepant resolution only when it is a Richardson orbit, and the resolution is obtained as a Springer map for it. In this paper, we shall generalize this result to Q-factorial terminalizations when \g$ is classical. Here, the induced orbits play an important role instead of Richardson orbits.Comment: Lemma (1.2.4) is added. A little mistake in the proof of (1.4.3) is corrected. A Dynkin diagram has been missed in the statement of Prop (2.2.1); we add it in the new version. In the version 7, Lemma (1.1.1) and Example (2.3) are added. Version 8: The proof of (1.2.4) is correcte

Birational geometry and deformations of nilpotent orbits

This is a continuation of math.AG/0408274, 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 paper is, to describe the remainder. We shall first construct a deformation of the nilpotent orbit closure in a canonical manner according to Brieskorn and Slodowy, 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 math.AG/0408274 to arbitrary Richardson orbits whose Springer maps have degree > 1. New Mukai flops, different from those of type A,D,E_6, will appear in the birational geometry for such orbits.Comment: 33 pages, revise