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