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This is a general question: As we know there are lot of vanishing theorem 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 vanishing problems for general complete algebraic manifold which is worthy to solve? 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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  • $\begingroup$ Trivial comment: if you have good control on the fibers of $f$ then you would hope that the dimension of the cohomology of $K_X$ might be constant, and then you can apply base change and Grothendieck vanishing. $\endgroup$ Sep 27, 2015 at 3:33
  • $\begingroup$ @AlexYoucis: I am not sure I understand your comment. when you say good control you mean like positive curvature ? why do I expect that cohomology of Kx to be constant? $\endgroup$
    – Feng Hao
    Sep 27, 2015 at 3:37
  • $\begingroup$ @AlexYoucis Actually Donu Arapura give an answer: mathoverflow.net/questions/219355/… $\endgroup$
    – Feng Hao
    Sep 27, 2015 at 3:39
  • $\begingroup$ Yes, Donu's answer makes anything I say seem somewhat silly :) Glad you got an answer! $\endgroup$ Sep 27, 2015 at 8:28

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This question was asked and answered on MathOverflow. I have replicated the accepted answer by Donu Arapura below.

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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