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 \({\textbf{v}}=(r, \xi , a)\) be a Mukai vector on X with \({\textbf{v}}^2 \geqslant 0\) , \(r>0\) and \(\xi \cdot H>0\) . We show that if \(\rho (X)=1\) or \(\rho (X)=2\) and X contains an elliptic curve, then all the moduli spaces \(M_{X,H}({\textbf{v}})\) satisfy weak Brill–Noether. Conversely, if \(\rho (X)>2\) or \(\rho (X)=2\) and X does not contain an elliptic curve, we show that there are infinitely many moduli spaces \(M_{X,H}({\textbf{v}})\) that fail weak Brill–Noether. As a consequence, we classify Chern classes of Ulrich bundles on abelian surfaces.
