Lefschetz Hyper-plane theorem for singular projective varieties? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T10:54:19Z http://mathoverflow.net/feeds/question/57744 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/57744/lefschetz-hyper-plane-theorem-for-singular-projective-varieties Lefschetz Hyper-plane theorem for singular projective varieties? Mohammad F.Tehrani 2011-03-07T22:58:13Z 2011-05-13T00:36:36Z <p>Lefschetz hyper-plane theorem for smooth projective varieties, $X\subset \mathbb{P}^{n+1}$ says:</p> <p>For smooth hyperplane section $Y= X\cap H$, the restriction map</p> <p>$H^i(X) \rightarrow H^i(Y)$ is an isomorphism for $0\leq i \leq n$ and an injection for $i=n$. Similarly we get an statement for homologies.</p> <p>For a singular variety projective variety $X$, lets consider the singular homology (or singular cohomology)</p> <p>Here is the question: Is there a Lefschetz hyperplane theorem in for projective varieties with possible canonical singularities?</p> <p>What which I expect is some thing like this:</p> <p>$X$ as before and assume $X_{sing}$ (singular locus of $X$) has codimension at least $k$ in $X$; then for generic hyperplane section $Y$ we have an isomorphism:</p> <p>$H_i(X)\cong H_i(Y)$ for $i$ less then some function of $k$ and $n$ !!!</p> http://mathoverflow.net/questions/57744/lefschetz-hyper-plane-theorem-for-singular-projective-varieties/57771#57771 Answer by Sándor Kovács for Lefschetz Hyper-plane theorem for singular projective varieties? Sándor Kovács 2011-03-08T04:22:52Z 2011-03-08T04:22:52Z <p>There are actually several versions of the Lefschetz hyperplane theorem for singular varieties. The main point is that this is ultimately a Hodge theoretic statement, one proof is using the Kodaira-Akizuki-Nakano vanishing theorem to establish the analogous statement for the Hodge components of singular cohomology. </p> <p>Hartshorne proved that the usual statement holds if $X\setminus Y$ is smooth, in other words, if $Y$ contains the singular locus of $X$. See 4.3 of <a href="http://www.springerlink.com/content/x336284161121452/" rel="nofollow">this paper</a>. </p> <p>With the development of a sensible Hodge theory for singular varieties this has been further generalized. I am not sure whom to attribute the credit for this. You can find a version for the case when $X$ is a local complete intersection in III.3.12(iii) of <a href="http://www.springer.com/mathematics/algebra/book/978-3-540-50023-0" rel="nofollow">this volume</a>.</p> http://mathoverflow.net/questions/57744/lefschetz-hyper-plane-theorem-for-singular-projective-varieties/64816#64816 Answer by Dmitri for Lefschetz Hyper-plane theorem for singular projective varieties? Dmitri 2011-05-12T17:09:11Z 2011-05-12T17:09:11Z <p>I would like to provide two more references:</p> <p>First is "Positivity in algebraic geometry I" by Lazarsfeld, section 3.1 There is a nice counterexample there, showing that even if $X$ has an isolated singularity, we can have $H_1(X)\ne H_1(Y)$ for $X$ of dimension 3 and higher. Namely we should just take any smooth $X$ with $\pi_1(X)=0$ and identify two points $x,y$ on it. Then $\pi_1$ of the obtained variety will be $\mathbb Z$, while for generic $Y$ we have $\pi_1(Y)=0$. But surely this singularity is by no means canonical. </p> <p>Second reference (advised in the book of Lazarsfeld) is "On topology of algebraic varieties Fulton". If you google exactly this phrase (with " "), you will get the article. It treats in particular the case when $X$ is a local complete intersection on the complement to the hyperplane, mentioned by Sandor (first theorem in chapter 3).</p> http://mathoverflow.net/questions/57744/lefschetz-hyper-plane-theorem-for-singular-projective-varieties/64854#64854 Answer by algori for Lefschetz Hyper-plane theorem for singular projective varieties? algori 2011-05-12T23:36:16Z 2011-05-13T00:36:36Z <p>I think what you are looking for is the Lefschetz hyperplane theorem for singular varieties from Goresky and MacPherson's " Stratified Morse theory" (part II, section 1.2). The range in which there is an isomorphism depends on the number of equations needed to define $X$ locally. The theorem says (after some deciphering) that if this number is $\leq k$ for the points of $X$ outside the hyperplane, then the hyperplane section map is an isomorphism in degrees $&lt; N-k-1$ where $N$ is the dimension of the ambient projective space.</p> <p>Also, for the middle perversity intersection homology the Lefschetz theorem is stated almost exactly as for smooth varieties and ordinary homology: for a generic hyperplane the hyperplane section map in homology is an isomorphism in degrees $&lt;\dim X-1$ and is surjective in degree $\dim X-1$.</p>