# Orbital degeneracy loci and applications

## TL;DR

This paper introduces a generalized framework for degeneracy loci based on orbit closures of algebraic groups, enabling the construction of new Calabi-Yau and Fano varieties with controlled geometric properties.

## Contribution

It generalizes degeneracy loci concepts using orbit closures, providing new tools for constructing complex algebraic varieties with specific canonical sheaf properties.

## Key findings

- Constructed new Calabi-Yau threefolds and fourfolds.
- Produced new Fano fourfolds.
- Demonstrated the method's flexibility and effectiveness.

## Abstract

Degeneracy loci of morphisms between vector bundles have been used in a wide variety of situations. We introduce a vast generalization of this notion, based on orbit closures of algebraic groups in their linear representations. A preferred class of our orbital degeneracy loci is characterized by a certain crepancy condition on the orbit closure, that allows to get some control on the canonical sheaf. This condition is fulfilled for Richardson nilpotent orbits, and also for partially decomposable skew-symmetric three-forms in six variables. In order to illustrate the efficiency and flexibility of our methods, we construct in both situations many Calabi--Yau manifolds of dimension three and four, as well as a few Fano varieties, including some new Fano fourfolds.