Induced nilpotent orbits and birational geometry

TL;DR

This paper explores the birational geometry of nilpotent orbit closures in classical Lie algebras, focusing on Q-factorial terminalizations and the role of induced orbits, extending known results about crepant resolutions.

Contribution

It generalizes the characterization of crepant resolutions to Q-factorial terminalizations for classical Lie algebras using induced orbits.

Findings

Nilpotent orbit closures lack crepant resolutions unless Richardson orbits.

Q-factorial terminalizations always exist via the minimal model program.

Induced orbits are key to understanding these resolutions in classical cases.

Abstract

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.