# Vanishing theorems for pluri-canonical bundle

I would like to know if Grauer-Riemenschneider vanishing theorem is still true in the setting of pluri-canonical bundle, i.e. the power of canonical bundle.

Let me recall

Grauer-Riemenschneider vanishing theroem: Let $\pi: X\to Y$ be a surjective projective morphism of two varieties with $X$ smooth. Let $K_X$ be the canonical line bundle. Then $$R^i\pi_*(K_X)= 0, \qquad \text{ if } i> \dim Y-\dim X.$$

In the book by Lazarfeld's book, positivity in algebraic geometry, this theorem follow from Kawamata-Viehweg's vanishing theorem: Let $X$ be a projective smooth variety. Then for any nef and big line bundle $L$, $$H^i(X,K_X\otimes L)=0, \qquad, \text{ for any } i>0 .$$

My question is: if we replace $K_X$ by $K_X^{\otimes n}$, $n>0$ is Grauer-Riemenschneider vanishing theorem still holds?

As pointed out by Jason Starr that Kawamata-Viehweg would fail for pluri-canonical line bundle. Still it is quite possible to some generalization, for example take $L^{\otimes r_n}$, where $r_n$ depends on $n$.

• No, certainly not. Consider the case that $X$ is Fano, e.g., $\mathbb{P}^r$. Jun 26 '14 at 23:04
• Is there any chance that Grauer-Riemenschneider theorem can be generalized? It is what I really want. I read the book by Lazarsfeld, he explained that K-V theorem imply G-R theorem.
– JJH
Jun 26 '14 at 23:27

First of all the theorem you are citing is not Grauert-Riemenschneider vanishing, but Kollár's vanishing and I doubt that you can prove it as a consequence of Kawamata-Viehweg vanishing. GR vanishing is for the case $\dim Y=\dim X$.

Second, of course, you can do what you are asking with an $r_n$: Let $r_n$ be such that $\omega_X^{n-1}\otimes L^{r_n}$ is ample and apply the previous version.

As Jason explained, you will always run into trouble with Fano varieties.

Here is the issue: cohomology of the canonical bundle is very different from that of pluricanonical bundles. Some aspects of this is explained in the introduction of Kollár's paper on subadditivity of Kodaira dimension. One simple difference is that the canonical bundle appears in the Hodge decomposition while the pluricanonical ones don't. Even if this does not seem a big difference, it is.

If you want vanishing of an adjoint bundle, i.e., something like $\omega_X\otimes L$, then you generally need some positivity of $L$. Just think of the dual version of KV vanishing and write $L^{-1}$ as an adjoint bundle. So to have vanishing for pluricanonical bundles you need some positivity of them.

• In Theorem 3.1 of this note: homepages.math.uic.edu/~mpopa/lectures/istanbul.pdf , the author refer the general case as G-R vanishing theorem. Anyway I'm actually only interested in the case when dim X=dim Y, in which case it does follow from K-V theorem. In fact, I only care when $X\to Y$ is a resolution of singularity.
– JJH
Jun 27 '14 at 15:39
• And if you look at the proof in the cited paper, the author first comments that this is a special case of Kollár's vanishing and immediately goes to saying that he will only prove it for the case $\dim X=\dim Y$. I think he called it GR vanishing mistakenly. It's not a big deal to me, but Kollár's vanishing is a lot harder than GR vanishing, so he should get the credit for it. Jun 28 '14 at 3:38