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Let $Y$ be a normal projective variety and let $f:X\to Y$ be a desingularization. Define $\mathcal K_X=f_*\omega_X$, the Grauert--Riemenschneider canonical sheaf of $X$. It is independent of the resolution. Does this have some good properties?

In particular, are there mild assumptions that ensure it is Cohen--Macaulay?

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Let me expand Donu's answer a bit. Say that $Y$ is normal for simplicity. Then we always have:

$$f_* \omega_X \subseteq \omega_Y.$$

The sheaf $\omega_Y$ is Cohen-Macaulay if $Y$ is Cohen-Macaulay, but in general, it is not.

On the other hand, $\omega_Y$ is always S2 (see for instance Kollár-Mori). Furthermore, if $Y$ is normal, then $f_* \omega_X \subseteq \omega_Y$ is an isomorphism outside a set of codimension 2. Hence

Observation For normal $Y$, $f_* \omega_X$ will be S2 if and only if $f_* \omega_X = \omega_Y$. This equality is equivalent to rational singularities if $Y$ is Cohen-Macaulay.

In other words, the only reasonable way to guarantee that $f_* \omega_X$ is Cohen-Macaulay is to require that $Y$ has rational singularities (at least if $Y$ is normal).

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  • $\begingroup$ Just to extend on Karl's excellent answer: a slightly more general situation (that is, not just for $\omega_X$) is handled in Kollár's recent book (goo.gl/JE2CyM). Look at section 2.5.In particular, Theorem 2.74 gives exactly what Karl is saying with the substitution $\mathscr G=\omega_X$. $\endgroup$ Apr 14, 2015 at 15:48
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Yes, it behaves like the canonical sheaf for smooth varieties in many respects. For example, it satisfies Kodaira vanishing $H^i(Y, \mathcal{K}_Y\otimes L)=0$ when $L$ is ample when $i>0$. To prove this, use Grauert-Riemenschneider vanishing to rewrite this as $H^i(X,\omega_X\otimes f^*L)$, and then apply Kawamata-Viehweg to show that this is zero.

Regarding the last question, if $\mathcal{K}_X$ equals the (shifted) dualizing complex, then $X$ will be CM, and in fact it will have rational singularities. I'm not sure if this is the kind of answer you're looking for.

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  • $\begingroup$ Sorry, I meant that $\mathcal K_X$ is a CM sheaf. Is this often the case? $\endgroup$
    – gsvr
    Apr 4, 2015 at 13:07

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