Let $S$ be a Riemann surface of genus $g$, $p\in S$ and $ E$ be the holomorphic line bundle associated with the divisor $p$. This means that $E$ admits a section $\sigma$ with a simple zero at $p$ and non vanishing everywhere else. Does this imply that the $\bar{\partial}$ cohomology group $H_{\bar{\partial}}^1$ is trivial?
I think so, because then you can consider ${\sigma}^{-1}$, and this is a section with a simple pole at $p$. Do you agree with me?