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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)

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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.

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  • $\begingroup$ $H^{q, n}(X, L)$ or $H^{n, q}(X, L)$ in para3? $\endgroup$
    – jack lion
    Commented May 10, 2020 at 5:24
  • $\begingroup$ You're right, thank you. $\endgroup$
    – user48958
    Commented May 12, 2020 at 20:27

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