0
$\begingroup$

What is the hodge structure given to the cohomology of the complement of a closed subset with respect to a smooth variety?

This is not quite what is wanted. In fact, The hodge structure is constructed as in Deligne's Hodge theory II where it is shown that it does not depend on the compactification but the question is: can we compare the Hodge structure of $ H^i(U, C) $ and $ H^i_c( U,C) $ asumming that $ U = X -Y $ where $ X$ is a smooth, compact ( say $P^4$) and $ Y $ is a hypersurface which admits only isolated double points?

What we can actually construct is the Gauss-Mannin connection on $Gr_F ^3 H^4(U,C)$ but we would like to relate it to the given by taking cohomology with compact supports on $ H^4_c( U,C) $. Of course, you can always think of an exact sequence involving $ H^i( X,C) $ and $ H^i_c( U, C) $ but remember $X$ is singular.

Improve this question
$\endgroup$
3
  • 4
    $\begingroup$ Well it's a mixed Hodge structure in general. As to how you construct it, take a look at the book by Peters and Steenbrink on MHS, or the original paper of Deligne Theorie de Hodge II, $\endgroup$
    – Donu Arapura
    Sep 27, 2015 at 18:31
  • 1
    $\begingroup$ $H^{i}_c(U,\mathbb{C})$ is dual to $H^{2n-i}(U,\mathbb{C})$ (as mixed Hodge structures), with $n=\dim U$. Apparently you are considering the case $i=n=4$ so this should answer your question. $\endgroup$
    – abx
    Sep 29, 2015 at 7:26
  • $\begingroup$ You mean exactly the mapping given by $\endgroup$
    – Qgeom
    Sep 30, 2015 at 19:19

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.