Vector Bundles and Monads On Abelian Threefolds

Abstract

The purpose of this article is to construct examples of stable rank 2 vector bundles on abelian threefolds and to study their moduli.

More precisely, we consider principally polarized abelian threefolds (X, Θ) with Picard number 1. Using the Serre construction, we obtain stable rank 2 bundles realizing roughly one half of the Chern classes (c ₁, c ₂) that are a priori allowed by the Bogomolov inequality and Riemann–Roch. In the case of even c ₁, we study first order deformations of these vector bundles ℰ, using a second description in terms of monads, similar to the ones used by Barth–Hulek on projective space. We find that all first order deformations of the bundle are induced by first order deformations of the corresponding monad, which leads to the formula

where Δ denotes the discriminant .

In the simplest nontrivial case (where c ₁ = 0 and c ₂ = Θ²), we construct an explicit parametrization of a Zariski open neighbourhood of ℰ in its moduli space: this neighbourhood is a ruled, nonsingular variety of dimension 13, birational to a ℙ¹-bundle over X × X × H, where H is the Hilbert scheme (of Kodaira dimension zero) of two points on the Kummer threefold X/(−1).

Key Words:

• Abelian threefold
• Monad
• Moduli space
• Vector bundle

2000 Mathematics Subject Classification:

• Primary 14D20
• Secondary 14K05
• 14J60
• 14F05

ACKNOWLEDGMENT

The author thanks the Max-Planck-Institut für Mathematik in Bonn for its hospitality and financial support. This work was initiated during a stay as postdoc at the MPIM in 2008–2009.

Notes

Here and elsewhere, whenever f: ℱ₁ → ℱ₂ is a homomorphism of sheaves, we use the same symbol to denote any twist f: ℱ₁(D) → ℱ₂(D).

Communicated by R. Piene.