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 $n$ !!!

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 $n$...