Let $X$ be a projective, smooth, algebraic variety over a subfield of the complex numbers, and let $Y \hookrightarrow X$ be a smooth hyperplane section of $X$. The classical Lefschetz theorem claims that the morphism induced by restriction

$$ H^j(X) \longrightarrow H^j(Y) $$

is an isomorphism whenever $j\leq \dim X-2$. Here $H^j(\cdot)$ could be singular cohomology with rational coefficients or algebraic de Rham cohomology.

I'm interested in the following variant. Let $D$ be a normal crossings divisor on $X$ and $(E, \nabla)$ an integrable algebraic connection with regular singularities along $D$. Let $Y$ be a smooth hyperplane section of $X$ meeting properly all the intersections of the irreducible components of $D$. Denote by $H^j(X, E, \nabla)$ the hypercohomology of the de Rham complex of the connection, that is $$ H^j(X, E, \nabla):=\mathbb{H}^j(X, E \to E \otimes_{\mathcal{O}_X} \Omega^1_X(\log D) \to \cdots). $$

Question: Is is true that $$ H^j(X, E, \nabla) \longrightarrow H^j(Y, E_{| Y}, \nabla_{| Y}) $$ is an isomorphism for $j \leq \dim X -2$ ?


$E_{|Y}=E \otimes_{\mathcal{O}_X} \mathcal{O}_Y$

is the restriction of $E$ to $Y$ and $\nabla_{| Y}$ is the induced connection on $E_{| Y}$.

If this asking too much, are there some natural conditions on $(E, \nabla)$ for the question to have a positive answer?

Thanks for your help!

  • $\begingroup$ In the definition of your de Rham complex I assume you want to put $\Omega^i_X(D)$ instead of just $\Omega^1_X$. Assuming this is the case, then what you want probably follows from the homotopy version of the Lefschetz theorem for quasi-projective varieties (applied to $X - D$). $\endgroup$ – ulrich May 6 '13 at 10:37
  • $\begingroup$ By Cor. II.6.11 in Deligne's "Equations différentielles..." (LNM 163), the cohomology groups you want to compute are the cohomology groups of the corresponding local systems (for the ordinary topology). Now notice that the topological proof of Lefschetz's theorem is based on the result that a non-singular $k$-dim. affine variety has the homotopy type of $k$-dim. CW complex. So you are led to the question: does the cohomology of a local systems on a $k$-dim. CW complex vanish above $2k$ ? This is unlikely to be true in general (classifying spaces probably provide counterexamples). $\endgroup$ – Damian Rössler May 6 '13 at 10:38
  • $\begingroup$ ... what I wrote above does not provide a counterexample but it might suggest how to construct one (or provide a proof !). $\endgroup$ – Damian Rössler May 6 '13 at 10:39
  • 1
    $\begingroup$ @Damian: The relevant variety need not be affine because of the presence of $D$. In any case, for affine varieties it is standard that the cohomology with coefficients in any constructible sheaf vanishes in degrees greater than the dimension... $\endgroup$ – ulrich May 6 '13 at 12:46
  • $\begingroup$ @leffe93: See, for example, the paper by Hamm and Lê: Lefschetz theorems on quasi-projective varieties. Bulletin de la Société Mathématique de France, 113 (1985), p. 123-142 $\endgroup$ – ulrich May 6 '13 at 13:08

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.