Crepant resolutions of orbit closures in Quiver representations of type A\_n and D\_4

TL;DR

This paper investigates crepant resolutions of orbit closures in quiver representations of types A extsubscript{n} and D extsubscript{4}, identifying conditions for crepancy and constructing related Fano and Calabi-Yau varieties.

Contribution

It determines when existing resolutions of orbit closures in specific quiver types are crepant and applies these results to construct new Fano and Calabi-Yau varieties.

Findings

Identifies conditions for crepancy in resolutions of A3, An, D4 quivers.

Constructs Fano and Calabi-Yau varieties from orbit degeneracy loci.

Provides criteria for crepant resolutions in quiver representation singularities.

Abstract

We study resolutions of singularities of orbit closures in quiver representations. We consider certain resolutions of singularities which have already been constructed by Reineke, and we determine under which conditions they are crepant. More specifically, we deal with quivers of type A3, An, D4. Then, as an application of the results we have found, we construct several Fano and Calabi-Yau varieties as orbital degeneracy loci.