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 $\mathbf{v}=(r,\xi,a)$ be a Mukai vector on $X$ with $\mathbf{v}^2\ge 0$,$r>0$, 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 $M_{X,H}(\mathbf{v})$ 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 $M_{X,H}(\mathbf{v})$ that fail weak Brill-Noether. As a consequence, we classify Chern classes of Ulrich bundles on abelian surfaces.