Can one prove vanishing of higher direct images fiber-wise? Let $\pi:X\to Y$ be a  proper map of algebraic varieties (over $\mathbb C$) which is a bi-rational equivalence.
are the following statements equivalent?


*

*The derived direct image of $O_X$ is $O_Y$.

*For any $y \in Y$ we have $H^*(O_{\pi^{-1}(y)})= \mathbb C$
Remarks:


*

*I do not assume that $X$ or $Y$ are smooth.

*The fiber is considered scheme theoretically

*By point one can mean a scheme theoretic point or a closed point or a geometric point. I think it is does not matter.

*I also have 2 variations of the question:
a) What  happens  we consider  conditions  (1) and (2) only on the level of the zero cohomology?
b) What  happens  we consider conditions (1) and (2) only on the level of higher cohomologies?

*Variation (b) make sense without the conditions on $\pi$. I need the answer only for the case I described (since I'm interested in rational singularities) but I'll be happy to know what happens in general
 A: Karl is correct. Here is an example which at least sheds some light, I hope. Embed the projective line with a large degree line bundle in $n$-space, so that the embedding is not linearly normal. Let $Y$ be the cone (polynomial ring in $n+1$ variables modulo the ideal of forms vanishing on the curve), so that $Y$ is not normal. Let $X$ be the blow up of the irrelevant maximal ideal. Then one can easily check that $X$ is smooth (a $\mathbb{P}^1$ bundle over $\mathbb{P}^1$), $\pi:X\to Y$ is birational with only one exceptional point, namely the vertex of $Y$ and the scheme theoretic inverse image of this point is just a $\mathbb{P}^1$. Thus condition b) is satisfied. But, $\pi_*O_X$ is the normalization of $O_Y$.
A: Hi Rami,
What about using the Grothendieck complex? Thus, there is a bounded complex $K$ of coherent locally free sheaves on Y, such that $K_y$, the application of "fiber at y" functor elementwise to $K$, computes cohomologies of fibers.Then backwards induction seems to give that if (2) holds, the direct image of $O_X$ is concentrated in degree 0.
Maybe I am wrong? I did not check carefully. Also, I did not check if we can extract information about image being exactly $O_Y$.
Sasha
