Dualizing sheaf under birational morphism

Question

Let $f:X\to Y$ be a birational morphism of irreducible projective varieties over $\mathbb{C}$ (so $f$ is defined everywhere but invertible only on a open dense subset). If $X$ and $Y$ are smooth, then there is a natural isomorphism between the vector spaces $\Gamma(X,\omega_X)$ and $\Gamma(Y,\omega_Y)$ where $\omega_X$ and $\omega_Y$ are the dualizing sheaves of $X$ and $Y$ (as for example shown in Hartshorne's book). Now assume that only $X$ is smooth and $Y$ is not necessarily smooth (but say the singular locus has codimension at least two if necessary). Do we still have such a natural isomorphism?

Answer 1

This should not be true in general. Note that your condition on $f:X\to Y$ implies that it's actually a proper birational map from a smooth variety and thus a resolution of singularities of $Y$. If $Y$ is Cohen-Macaulay and has rational singularities, one has that $f_*\Omega_Y \to \Omega_X$ must be isomorphism of sheaves (actually, of dualizing complexes) which would imply the result you ask for on global sections. But there are certainly non-rational singularities.