Four-dimensional Fano quiver flag zero loci

Abstract

Quiver flag zero loci are subvarieties of quiver flag varieties cut out by sections of representation theoretic 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, and disprove a conjecture of Craw. In the appendices (which can be found in the electronic supplementary material), 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.

Fano 3-folds from homogeneous vector bundles over Grassmannians

“…One can directly check using ( 23) and ( 24) that H 0 (K ) ∼ = Sym 4 V 3 and H 1 (K ) ∼ = V 3 . The conclusion follows by considering the direct sum of ( 23) and (22).…”

“…What we plan to do in a series of subsequent works is to classify all Fano in dimension 4 that can be obtained in this way, comparing our results with the already existing known classes of Fano 4-folds ( [1,11,22], to cite a few). We are confident that many new and interesting examples can be found in this way, and the results of this paper are for sure strong motivations.…”

Fano fourfolds of K3 type

“…Plentiful of four-dimensional Fano toric complete intersections were found in [CKP15] (at least 738 families), and a few dozens more (at least 141) were constructed in [Kal19] as quiver flag zero loci. As already mentioned it would be very interesting to compare these families with ours, and find how many families have been found altogether -probably a very significant proportion of the complete, still elusive list of Fano fourfolds.…”

“…For Fano threefolds this is the point made in [DBFT]. In dimension four there is a potential overlap with the list of families of Fano subvarieties of quiver flag varieties found in [Kal19], since the latter can be constructed as towers of Grassmann bundles. This could be clarified by comparing the quantum periods, which for quiver flag zero loci were computed in the Appendix of loc.…”

Smooth Fano 4-Folds in Gorenstein Formats

“…One of the larger sets of examples was constructed by Coates, Kasprzyk and Prince: in [5], they constructed 527 new deformation families of Fano 4-folds as complete intersections in toric varieties and they constructed one more in [6] by using the Laurent inversion. Another collection of 141 deformation families has been given by Coates, Kasprzyk and Kalashnikov [18] as quiver flag zero loci in quiver flag varieties. The smooth Fano 4-folds of Picard rank two with hypersurface Cox ring have been classified by Hausen et al [13], who found 17 new deformation families of smooth Fano 4-folds of index one.…”