ULRICH BUNDLES ON ABELIAN SURFACES

ABSTRACT. We prove that any abelian surface admits a rank 2 Ulrich bundle. Let X ⊂ PN be a projective variety of dimension d over an algebraically closed field. An Ulrich bundle on X is a vector bundle E on X satisfying H∗(X,E(−1)) =... = H∗(X,E(−d)) = 0. This notion was introduced in [ES], where various other characterizations are given; let us just mention that it is equivalent to say that E admits a linear resolution as a OPN-module, or that the pushforward of E onto Pd by a general linear projection is a trivial bundle. In [ES] the authors ask whether every projective variety admits an Ulrich bundle. The answer is known only in a few cases: hypersurfaces and complete intersections [HUB], del Pezzo surfaces [ES, Corollary 6.5]. The case of K3 surfaces is treated in [AFO]. In this short note we show that the existence of Ulrich bundles for abelian surfaces follows easily from Serre’s construction: Theorem. Any abelian surface X ⊂ PN carries a rank 2 Ulrich bundle. Proof: We put dimH0(X,OX(1)) = n. Let C be a smooth curve in |OX(1) | ; we have OC(1) ∼ = ωC, and g(C) = n + 1. We choose a subset Z ⊂ C of n general points. Then Z has the Cayley-Bacharach property on X (see for instance [HL], Theorem 5.1.1): for every p ∈ Z, any section of H0(X,OX(1)) vanishing on Z r {p} vanishes on Z. Indeed, the image V of the restriction ma