TITLE: Top cohomology of resolution of singularities
QUESTION [5 upvotes]: Let $X$ be a projective variety over $\mathbb C$ of dimension $n$.  Let $\tilde{X} \to X$ be a resolution of singularities.  Suppose that $H^n(\tilde{X}, \mathcal O_{\tilde{X}}) = 0$.  What can we say about $H^n(X, \mathcal O_X)$?  When is it $0$?

REPLY [2 votes]: A very easy counterexample: let $X$ be a nodal cubic in $\mathbb{P}^3$. Then the resolution of singularities of $X$ is $\bar X = \mathbb{P}^1$, so $H^1(\bar X, \mathcal{O}_{\bar X}) = 0$. On the other hand, the exact sequence of sheaves on $\mathbb{P}^2$
$$ 0\to \mathcal{O}(-3)\to \mathcal{O} \to \mathcal{O}_X \to 0 $$
shows that $H^1(X, \mathcal{O}_X)$ is one-dimensional.