Four dimensional Fano quiver flag zero loci (with an appendix by T. Coates, E. Kalashnikov, and A. Kasprzyk)

TL;DR

This paper develops methods to compute Gromov--Witten invariants of quiver flag zero loci, determines their ample cones, and identifies at least 141 new four-dimensional Fano manifolds as such loci.

Contribution

It proves the Abelian/non-Abelian Correspondence for quiver flag zero loci and finds numerous new four-dimensional Fano manifolds using these techniques.

Findings

Computed genus zero Gromov--Witten invariants for quiver flag zero loci.

Determined the ample cone of quiver flag varieties, disproving a prior conjecture.

Identified at least 141 new four-dimensional Fano manifolds as quiver flag zero loci.

Abstract

Quiver flag zero loci are subvarieties of quiver flag varieties cut out by sections of homogeneous vector bundles. We prove the Abelian/non-Abelian Correspondence in this context: this allows us to compute genus zero Gromov--Witten invariants of quiver flag zero loci. We determine the ample cone of a quiver flag variety, disproving a conjecture of Craw. In the Appendices, which are joint work with Tom Coates and Alexander Kasprzyk, we use these results to find four-dimensional Fano manifolds that occur as quiver flag zero loci in ambient spaces of dimension up to 8, and compute their quantum periods. In this way we find at least 141 new four-dimensional Fano manifolds.