# Lefschetz hyper-plane theorem for singular projective varieties?

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 $$n$$...

• A slight typo: the $n$ in "$0\leq i\leq n$" is not the same as in "$\mathbb P^{n+1}.$" BTW, I wonder if one can deduce something from the version for intersection cohomology, combined with the constraint on codim of the singular locus. Mar 8, 2011 at 13:19
• In a related but different setup you should see Dan Halpern-Leistner's paper "The Lefschetz Hyperplane theorem for stacks". May 13, 2011 at 10:03
• There is also this version of Weak Lefschetz Hyperplane Theorem for normal projective varieties and smooth divisors. Jul 1, 2022 at 14:59

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.

• the version where the hyperplane contains the singular points is already in the standard reference of milnor, morse theory. p.41. at least over the complex numbers. Mar 8, 2011 at 4:50
• If you downvote (Roy, I know it wasn't you), would you mind telling me why? Mar 8, 2011 at 15:10
• so you mean there is nothing none yet when $Y$ is a generic hyperplane section and not necessarily one which has the singular locus? Mar 8, 2011 at 19:44
• can you say more details on the first paragraph of your answer. or can you mention one of those several versions which is most relevant to this case. The case I have in mind is when $X$ is a projective 4-fold with canonical singularities and $Y$ is a hyperplane section. You can even assume $X$ is toric. Mar 8, 2011 at 19:46
• Is the weak Lefschetz theorem for intersection cohomology of interest? igitur-archive.library.uu.nl/math/2007-0224-201418/… Mar 10, 2011 at 16:42

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 $\dim X-1$.

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).