Complex curves in o-minimal geometry

There has recently been considerable progress relating o-minimality to complex analytic geometry. Yet almost nothing is known about coherent cohomology or the classification of vector bundles, even for curves. In ${\mathbb{R}}_{\mathrm{a}\mathrm{n}}$ and similar structures, we show that cohomology of noncompact curves is concentrated entirely at punctures. As an application, we compute the cohomology of the structure sheaf on the affine line and describe a connection to Diophantine approximation. Finally, we use similar techniques to characterize which definable Riemann surfaces have definable compactifications. The proofs are based on a careful analysis of boundary behavior for definable holomorphic functions.