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This is a general question: As we know there are a lot of vanishing theorems like Fujita vanishing, kodaira Nakano vanishing, vanishing for big nef line bundle, Kollár vanishing, etc. Those just for projective guys.

My question is: are there some important vanishing problems for general complete algebraic manifold? Although we don't have nice line bundle like $\mathcal{O}_X(1)$, are there still some vanishing expectation for cohomology of coherent sheaves in some sense.

More specific question, if we have a proper surjective morphism $f:X\rightarrow Y$, $X$ complete, $Y$ projective, do we expect that $R^jf_*\mathcal{O}_X(K_X)=0$ for $j>dim X-dim Y$?

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I guess I could have just told you in person, but anyway, yes your specific question has a positive answer. To see this, use Chow's theorem and resolution of singularities to find a birational map $\pi:\tilde X\to X$ with $\tilde X$ smooth and projective. Now apply Kollár vanishing twice (or really Grauert-Riemenschneider) to get $R^if_*K_X= R^i(f\circ \pi)_* K_{\tilde X} = 0$ for $i>\dim X-\dim Y$.

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  • $\begingroup$ Thank you for your answer. I was trying to reproof Bertini, f:X-->Y from complete to projective, and show pull back of hyperplane divisor f^-1(D) is connected. At first I have to use Chow's lemma π:X~ --->X, g:X~ --->Y, to pullback D to the projective guy X~, and prove H^n-1(X~, K_x+ng*(D))=0 by second stage of the spectral sequence all vanish. then π maps g^-1(D) connected to connected divisor f^-1(D) in X. But now I can do the same thing on X directly because of above vanishing. Anyway thank you. $\endgroup$
    – Feng Hao
    Sep 27 '15 at 1:40

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