Bundles of generalized theta functions over abelian surfaces (Q2312760)

Let \((A,\Theta)\) be a polarized complex abelian surface, with \(\Theta\) a generic symmetric polarization, and let \(\hat{A}\) be the dual of \(A\). In the article under review, the author studies the Verlinde bundles \(\mathbf{E}(v,w)\) over the abelian fourfold \(A \times \hat{A}\). These bundles play an important role in the Abelian surface strange duality conjecture and are associated to a certain class of mutually orthogonal Mukai vectors \(v,w \in H^{2 \bullet}(A,\mathbb{Z})\) [\textit{A. Marian} and \textit{D. Oprea}, Math. Ann. 343, No. 1, 1--33 (2009; Zbl 1156.14036); J. Eur. Math. Soc. (JEMS) 16, No. 6, 1221--1252 (2014; Zbl 1322.14063); \textit{B. Bolognese} et al., J. Algebr. Geom. 26, No. 3, 475--511 (2017; Zbl 1370.14038)]. In more precise terms, the Mukai vector \(v\) is required to be primitive, of positive rank and have positive odd Mukai self pairing. Further, let \(\mathfrak{M}_v\) be the moduli space of \(\Theta\)-semistable sheaves, with Chern character \(v\), and let \(K_v\) be the fibre over the identity of the Albanese morphism \(\mathfrak{M}_v \rightarrow A \times \hat{A}\). With these notations, the Mukai vector \(w\), which is orthogonal to \(v\), determines a line bundle \(\Theta_w\) over \(\mathfrak{M}_v\); it is assumed that the restriction of \(\Theta_w\) to \(K_v\), belongs to \(K_v\)'s movable cone. Within this context, the author's main results are summarized in the following way. First of all, over the moduli space \(\mathfrak{M}_v\), the construction of the theta line bundle \(\Theta_w\) depends on a choice of reference sheaf. The author's first result gives a description of these dependencies. Second, the author determines the splitting type of the bundles \(\mathbf{E}(v,w)\), depending on such a pair of degree zero orthogonal Mukai vectors \((v,w)\), in terms of indecomposable semihomogenous factors. This is achieved after having first established another result, which is of an independent interest and pertains to the action of a certain group of torsion points on the space of generalized theta functions. Finally, the author's results allow for a formulation of the abelian surface strange duality conjecture as a specific isomorphism between the Verlinde bundles and their Fourier Mukai transforms. For instance, if the first Chern classes of such Mukai vectors \(v\) and \(w\), as above, are divisible by their ranks, then the author establishes an isomorphism between the dual of the Verlinde bundle \(\mathbf{E}(v,w)\) and the Fourier-Mukai transform of the Verlinde bundle \(\mathbf{E}(w,v)\). This perspective on the abelian surface strange duality conjecture complements the author's previous work on related topics, including [Math. Nachr. 291, No. 17--18, 2599--2630 (2018; Zbl 1409.14025)].