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 $X_{sing}$ (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 $n$ !!!
Best Answer
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.