Normality of orbit closures in the enhanced nilpotent cone

TL;DR

This paper investigates the geometric properties of orbit closures in the enhanced nilpotent cone, establishing their relation to enhanced quiver varieties and proving their normality in certain cases.

Contribution

It demonstrates that each orbit closure is an invariant-theoretic quotient of an enhanced quiver variety and proves normality for these varieties in specific cases.

Findings

Orbit closures are invariant-theoretic quotients of enhanced quiver varieties.

Enhanced quiver varieties are conjectured to be normal complete intersections.

Orbit closures are shown to be normal in special cases.

Abstract

We continue the study of the closures of $GL(V)$-orbits in the enhanced nilpotent cone $V\times\cN$ begun by the first two authors. We prove that each closure is an invariant-theoretic quotient of a suitably-defined enhanced quiver variety. We conjecture, and prove in special cases, that these enhanced quiver varieties are normal complete intersections, implying that the enhanced nilpotent orbit closures are also normal.