Let $C$ be a smooth irreducible complex curve on a complex surface $S$, defined as the zero locus of a section of a complex line bundle $L$. Note the normal bundle is the restriction $N_{C/S}=L|_C$. I would expect vanishing of the sheaf cohomologies $h^1(L)=h^2(L)=0$, when some nice properties hold of $S$.
For example, if the anti-canonical divisor of $S$ is effective, then $h^2(L)=0$ since \begin{equation} H^2(L)\cong H^0(L^*\otimes K_S) \subseteq H^0(L^*)=0\,, \end{equation} where the first step used Serre duality, the second the fact the anti-canonical bundle $K_S^*$ corresponds to an effective divisor, and the third that $L^*$ does not correspond to an effective divisor.
Can one show that $h^1(L)=0$ under this assumption? If not, does it hold under stronger assumptions, e.g. when the anti-canonical divisor of $S$ is nef, or if $S$ is weak Fano?
No. Take $S=\mathbb{F}_2$, the Hirzebruch surface and let $C$ be the unique rational curve with $C^2=-2$. Then, $H^1(\mathcal{O}_S(C))\neq 0$, while $-K_S$ is effective and $S$ is weak Fano.