Birational geometry in codimension 2 of symplectic resolutions

TL;DR

This paper proves that for certain symplectic varieties, any two projective symplectic resolutions are connected through Mukai's elementary transformations in codimension 2, advancing understanding of their birational geometry.

Contribution

It establishes the conjecture for nilpotent orbit closures in classical Lie algebras and certain quotient symplectic varieties, showing their resolutions are related by elementary transformations.

Findings

Resolutions are related by Mukai's transformations in specified cases.

Supports conjecture in the context of classical Lie algebra orbit closures.

Extends understanding of symplectic resolution birational geometry.

Abstract

We prove the conjecture that two projective symplectic resolutions for a symplectic variety $W$ are related by Mukai's elementary transformations over $W$ in codimension 2 in the following cases: (i). nilpotent orbit closures in a classical simple complex Lie algebra; (ii). some quotient symplectic varieties.