Do some of the local cohomology groups of the structure sheaf on the singular locus vanish? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T14:28:09Z http://mathoverflow.net/feeds/question/63257 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/63257/do-some-of-the-local-cohomology-groups-of-the-structure-sheaf-on-the-singular-loc Do some of the local cohomology groups of the structure sheaf on the singular locus vanish? Ben Webster 2011-04-28T05:39:26Z 2011-04-28T06:32:35Z <p>Ok, so this is where I reveal my ignorance as an algebraic geometer: I had previously asked about pushforward of line bundles from the smooth locus of a variety to the whole thing. I think I understand basically how that picture works now, but I have a variation thereof that I would like to ask about. </p> <p>Let $X$ is a normal variety (let's say irreducible quasi-projective of finite type over $\mathbb{C}$; you can even assume terminal and $\mathbb{Q}$-factorial if you like), $Y$ its smooth locus and $Z$ its singular locus. In fact, let's say the dimension of $Z$ is at most $\dim X - 4$ just for good measure.</p> <blockquote> <p>What can be said about the local cohomology $H^i_Z(\mathcal{O}_X)$ which fills in the exact sequence $$\cdots \to H^i_Z(\mathcal{O}_X)\to H^i(X;\mathcal{O}_X)\to H^i(Y;\mathcal{O}_Y)\to \cdots?$$</p> </blockquote> <p>What I'd love to say is that this is 0 in degrees $\leq 3$, but obviously, I'd accept other answers if they happen to be true. </p> http://mathoverflow.net/questions/63257/do-some-of-the-local-cohomology-groups-of-the-structure-sheaf-on-the-singular-loc/63261#63261 Answer by Sándor Kovács for Do some of the local cohomology groups of the structure sheaf on the singular locus vanish? Sándor Kovács 2011-04-28T06:08:13Z 2011-04-28T06:32:35Z <p>This is essentially answered in my answer to a question raised after my answer to an MO question <a href="http://mathoverflow.net/questions/45347/why-does-the-s2-property-of-a-ring-correspond-to-the-hartogs-phenomenon/45616#45616" rel="nofollow">here</a>.</p> <p>Basically the point is that what you want is true provided your variety is $S_4$. (See the above link for a proof). In particular, if your variety is Cohen-Macaulay (which terminal and even klt singularities are, but not log canonical) then you're in business.</p> <p>In general, this vanishing is essentially equivalent to that your variety be $S_4$. If you consider a $4$-dimensional variety $X$ with a single singular point $z\in X$ that is $S_2$, but not $S_4$, then $X$ is normal but $H^i_z(X,\mathscr O_X)\neq 0$.</p> <p>For an explicit example of such a singularity consider a cone over an abelian threefold. This is a non-klt log canonical singularity. The fact that this is such an example follows from the condition that tells you what $S_m$ a cone is based on the cohomology groups of the scheme it is a cone over. For a complete proof of that criterion in a rather general case see Lemma 3.1 of <a href="http://arxiv.org/abs/1005.5207" rel="nofollow">this paper</a> of Patakfalvi.</p> <p>You might also need the cohomological interpretation of depth, which is (one of) Grothendieck's vanishing theorem. See for example in <a href="http://books.google.com/books?id=ouCysVw20GAC&amp;lpg=PR12&amp;dq=bruns%2520herzog%2520grothendieck%2520vanishing&amp;pg=PA132#v=onepage&amp;q=grothendieck%2520vanishing&amp;f=false" rel="nofollow">Bruns-Hezog</a>.</p>