Equivariant stable sheaves and toric GIT
Abstract
For (X,L) a polarized toric variety and G a torus of automorphisms of (X,L), denote by Y the GIT quotient X/G. We define a family of fully faithful functors from the category of torus equivariant reflexive sheaves on Y to the category of torus equivariant reflexive sheaves on X. We show, under a genericity assumption on G, that slope stability is preserved by these functors if and only if the pair ((X, L), G) satisfies a combinatorial criterion. As an application, when (X,L) is a polarized toric orbifold of dimension n, we relate stable equivariant reflexive sheaves on (X, L) to stable equivariant reflexive sheaves on certain (n-1)-dimensional weighted projective spaces.
Cite
@article{arxiv.2001.01500,
title = {Equivariant stable sheaves and toric GIT},
author = {Andrew Clarke and Carl Tipler},
journal= {arXiv preprint arXiv:2001.01500},
year = {2021}
}
Comments
28 pages, final version. More details on the construction of the functors, an extra example added in Section 5.2 (rank 2 stable reflexive but not locally free sheaf on (P^1)^3). To appear in Proceedings of the Royal Society of Edinburgh Section A: Mathematics