Is there an analog of Kodaira vanishing for singular varieties - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T22:23:51Z http://mathoverflow.net/feeds/question/78132 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/78132/is-there-an-analog-of-kodaira-vanishing-for-singular-varieties Is there an analog of Kodaira vanishing for singular varieties Alexander Braverman 2011-10-14T13:18:19Z 2011-10-14T15:50:21Z <p>I would like to know what kind of analogs of Kodaira vanishing theorem are valid for singular varieties. For example, is the following true: let $X$ be a projective Gorenstein variety and let $\omega_X$ be its canonical bundle. Is it true that $H^i(L\otimes \omega_X)=0$ for $i>0$ for an ample line bundle $L$?</p> http://mathoverflow.net/questions/78132/is-there-an-analog-of-kodaira-vanishing-for-singular-varieties/78135#78135 Answer by J.C. Ottem for Is there an analog of Kodaira vanishing for singular varieties J.C. Ottem 2011-10-14T13:33:29Z 2011-10-14T13:38:40Z <p>No. The following counterexample is due to Sommese:</p> <p>Let $Y$ be the projective bundle $\pi:\mathbb{P}(O\oplus O(1)^{\oplus 3})\to \mathbb{P}^1$. Let $M$ be the tautological bundle on $Y$ and take a general member $X\in|M\otimes \pi^*O(-1)^{\oplus 4})|$. Then $X$ is a normal, projective, Gorenstein 3-fold. If $L$ is the line budle $M\otimes \pi^*O(1)$, one can also check that $H^1(X,O(K_X+L))=\mathbb{C}$.</p> <p>However, it is known that the Kodaira vanishing theorem holds if $X$ has log canonical singularities. There are also weaker versions in the theorem in the paper 'D. Arapura and D. B. Jaffe <a href="http://www.jstor.org/pss/2047052" rel="nofollow">On Kodaira Vanishing for Singular Varieties</a> Proc. A.M.S, 105, No. 4, pp. 911-916, 1989.'</p> http://mathoverflow.net/questions/78132/is-there-an-analog-of-kodaira-vanishing-for-singular-varieties/78142#78142 Answer by Karl Schwede for Is there an analog of Kodaira vanishing for singular varieties Karl Schwede 2011-10-14T15:50:21Z 2011-10-14T15:50:21Z <p>Indeed, as JC Ottem points out, Kodaira holds for log canonical (even semi-log canonical singularities). There's also a way to quickly deduce that Kodaira vanishing holds for Du Bois singularities (either from the Ambro-Fujino machinary or mimicking arguments of Kollar, let me know if you want details, perhaps I should put it on mathoverflow since it's not written down anywhere).</p> <p>However, I should probably point out that it's totally trivial to see that Kodaira vanishing holds for rational singularities. Here's the proof:</p> <p>Let $\pi : Y \to X$ be a resolution. Note $R \pi_* O_Y \cong O_X$ and so $R \pi O_Y(-\pi^* L) \cong O_X(-L)$ for any line bundle $L$. Fix $L$ to be ample. By a spectral sequence/composition of derived functors argument:</p> <p>$$H^i(X, O_X(-L)) = H^i(Y, O_Y(-\pi^* L)).$$</p> <p>But $\pi^* L$ is nef and big and the vanishing of the right hand side is just Kawamata-Viehweg vanishing and Serre duality. </p>