In complex geometry, various vanishing theorems for cohomology
groups of a hermitian line bundle E over a compact complex manifold X have been found. 


My question is


Is there some vanishing theorems over a **general noncompact** complex manifold exist? (Except shose on Stein manifolds)



Vanishing theorems on a non-compact manifold
A complex manifold of dimension $n$ is non-compact if and only if $H^n(X,{\mathcal F})=0$ for any coherent sheaf ${\mathcal F}$ on $X$. This is the only general vanishing result that I know of on non-compact manifolds. 


But other than that, there are some other vanishing theorems on non-compact manifolds. For instance, if $X$ is $q$-complete, then $H^r(X, {\mathcal F})=0,\forall r\geq q$.


Or if $X$ is weakly $1$-complete, and $L$ is a positive line bundle on $X$, then $H^{n,q}(X,L)=0$, $\forall q\geq 1$.


In general, for a non-compact manifold, you know nothing about the cohmology groups, they can even be non-Hausdorff. 


