Let $X$ be a smooth variety over a field $k=\bar k$, $\rm char$ $k=0$. Assume that $\mathcal{L}$ is a nef and big line bundle. Is there some bound saying that $$ h^1(\mathcal{L}^{\otimes n}) \le f(n) $$ where $f(n)$ is a function only depending on $n$? For example, a polynomial.

And how about the case when $\rm char$ $k>0$?