Vector Bundles and Monads on Abelian Threefolds

Using the Serre construction, we give examples of stable rank 2 vector bundles on principally polarized abelian threefolds (X,Θ) with Picard number 1. The Chern classes (c1, c2) of these examples realize roughly one half of the classes that are a priori allowed by the Bogomolov inequality and Riemann-Roch (the latter gives a certain divisibility condition). In the case of even c1, we study deformations of these vector bundles E , using a second description in terms of monads, similar to the ones studied by Barth–Hulek on projective space. By an explicit analysis of the hyperext spectral sequence associated to the monad, we show that the space of first order infinitesimal deformations of E equals the space of first order infinitesimal deformations of the monad. This leads to the formula dimExt(E , E ) = 1 3 ∆(E ) ·Θ+ 5 (we emphasize that its validity is only proved for special bundles E coming from the Serre construction), where ∆ denotes the discriminant 4c2 − c 2 1 . Finally we show that, in the first nontrivial example of the above construction (where c1 = 0 and c2 = Θ ), the infinitesimal identification between deformations of E and of the monad can be extended to a Zariski local identification: this leads to an explicit description of a Zariski open neighbourhood of E in its moduli space M(0,Θ). This neighbourhood is a ruled, nonsingular variety of dimension 13, birational to a P-bundle over a finite quotient of X ×X X 2 ×X X , where X is considered as a variety over X via the group law.