On the strange duality conjecture for abelian surfaces II

TL;DR

This paper advances the understanding of Le Potier's strange duality conjecture for sheaves on abelian surfaces by establishing new isomorphisms using representation theory, Fourier-Mukai techniques, and degeneration methods.

Contribution

It introduces a third version of the conjecture, proves the isomorphism in rank one, and extends results to product abelian surfaces and general cases via degeneration.

Findings

Isomorphism established in rank one using Heisenberg group representation theory.

Duality shown for sheaves with fiber degree 1 on product abelian surfaces via Fourier-Mukai.

Verlinde sheaves are proven to be locally free over moduli spaces.

Abstract

In the prequel to this paper, two versions of Le Potier's strange duality conjecture for sheaves over abelian surfaces were studied. A third version is considered here. In the current setup, the isomorphism involves moduli spaces of sheaves with fixed determinant and fixed determinant of the Fourier-Mukai transform on one side, and moduli spaces where both determinants vary, on the other side. We first establish the isomorphism in rank one using the representation theory of Heisenberg groups. For product abelian surfaces, the isomorphism is then shown to hold for sheaves with fiber degree 1 via Fourier-Mukai techniques. By degeneration to product geometries, the duality is obtained generically for a large number of numerical types. Finally, it is shown in great generality that the Verlinde sheaves encoding the variation of the spaces of theta functions are locally free over moduli.