Orbital degeneracy loci II: Gorenstein orbits

TL;DR

This paper extends the theory of orbital degeneracy loci, focusing on cases where the subvariety's coordinate ring is Gorenstein, and explores their canonical bundles and applications to constructing varieties with special canonical properties.

Contribution

It provides a systematic study of orbital degeneracy loci with Gorenstein coordinate rings and determines their canonical bundles in these cases.

Findings

Canonical bundles can be controlled when the coordinate ring is Gorenstein.

Determination of canonical bundles for orbit closure subvarieties.

Construction of low-dimensional varieties with negative or trivial canonical bundle.

Abstract

In [BFMT17] we introduced orbital degeneracy loci as generalizations of degeneracy loci of morphisms between vector bundles. Orbital degeneracy loci can be constructed from any stable subvariety of a representation of an algebraic group. In this paper we show that their canonical bundles can be conveniently controlled in the case where the affine coordinate ring of the subvariety is Gorenstein. We then study in a systematic way the subvarieties obtained as orbit closures in representations with finitely many orbits, and we determine the canonical bundles of the corresponding orbital degeneracy loci in the Gorenstein cases. Applications are given to the construction of low dimensional varieties with negative or trivial canonical bundle.