Let $R$ be a complete Henselian discrete valuation ring, $\pi:X \to \mathrm{Spec} (R)$ be a smooth, proper, integral, flat $\mathrm{Spec} (R)$-scheme of dimension $2$. Assume that the genus of the special fiber (which is a smooth curve) is at least $2$. Is there any condition on $X$ such that $H^1(\mathcal{O}\_X)=0$?



Cohomology of a fibered surface
Let $s,\eta $ be the closed and generic points of $\mathrm{Spec}(R)$. Since you assume $X$ smooth over $R$, we have $h^0(\mathscr{O}\_{X\_\eta })=h^0(\mathscr{O}\_{X\_s})=1$, hence $h^1(\mathscr{O}\_{X\_\eta })=h^1(\mathscr{O}\_{X\_s})\geq 2$. Thus $H^1(X,\mathscr{O}\_X)$ is a $R$-module of rank $\geq 2$, it cannot be 0. 


