Lefschetz hyper-plane theorem for singular projective varieties?

Question

Lefschetz hyper-plane theorem for smooth projective varieties, $X\subset \mathbb{P}^{n+1}$ says:

For smooth hyperplane section $Y= X\cap H$, the restriction map

$H^i(X) \rightarrow H^i(Y)$ is an isomorphism for $0\leq i \leq n$ and an injection for $i=n$. Similarly we get an statement for homologies.

For a singular variety projective variety $X$, lets consider the singular homology (or singular cohomology)

Here is the question: Is there a Lefschetz hyperplane theorem in for projective varieties with possible canonical singularities?

What which I expect is some thing like this:

$X$ as before and assume $X_{\mathrm{sing}}$ (singular locus of $X$) has codimension at least $k$ in $X$; then for generic hyperplane section $Y$ we have an isomorphism:

$H_i(X)\cong H_i(Y)$ for $i$ less then some function of $k$ and $n$...

Answer 1

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.

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 this paper.

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 this volume.

Answer 2

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 $< N-k-1$ where $N$ is the dimension of the ambient projective space.

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 $<\dim X-1$ and is surjective in degree $\dim X-1$.

Answer 3

I would like to provide two more references:

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.

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).