TITLE: Lefschetz Hyper-plane theorem for singular projective varieties?
QUESTION [10 upvotes]: 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 $n$ !!!

REPLY [4 votes]: 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 $\dim X-1$.