Weak Brill–Noether on abelian surfaces

We study the cohomology of a general stable sheaf on an abelian surface. We say that a moduli space satisfies weak Brill–Noether if the general sheaf has at most one non-zero cohomology group. Let (X, H) be a polarized abelian surface and let v=(r,ξ,a) be a Mukai vector on X with v2⩾0, r>0 and ξ·H>0. We show that if ρ(X)=1 or ρ(X)=2 and X contains an elliptic curve, then all the moduli spaces MX,H(v) satisfy weak Brill–Noether. Conversely, if ρ(X)>2 or ρ(X)=2 and X does not contain an elliptic curve, we show that there are infinitely many moduli spaces MX,H(v) that fail weak Brill–Noether. As a consequence, we classify Chern classes of Ulrich bundles on abelian surfaces.