1
$\begingroup$

Let $X$ be a smooth projective variety over a field $k$ of characteristic zero and let $D$ be a simple normal crossing divisor on $X$, with irreducible components $D_i$.

Does there exist a nonzero global section $\omega \in H^0(X, \Omega^1_X(\log D))$ such that $\mathrm{Res}_{D_i}\omega \neq 0$ for every component $D_i$?

I think the case of curves should follow somehow from Riemann-Roch but I don't see how.

Thanks for your help

$\endgroup$
2
  • 1
    $\begingroup$ Take $X$ a curve and $D$ a single point... probably you should assume that $\mathrm{gr}^W_2 H^1(X \setminus D) \neq 0$. $\endgroup$ Nov 8, 2013 at 9:51
  • 1
    $\begingroup$ Since abx and Dan have shown that your initial guess is wrong, I recommend that you try some simple examples on your own before asking further questions about this. $\endgroup$ Nov 8, 2013 at 14:38

1 Answer 1

2
$\begingroup$

With no further hypothesis, the answer is no. Take $X$ with $H^0(X,\Omega ^1_X)=0$, and $D$ a smooth divisor. Then from the exact sequence $$0\rightarrow \Omega ^1_X\rightarrow \Omega ^1_X(\log D) {\buildrel {\mathrm{Res}}\over {\longrightarrow}} \mathcal{O}_D\rightarrow 0$$you get $H^0(X,\Omega ^1_X(\log D) )=0$ (the section $1$ of $\mathcal{O}_D$ goes to the class of $D$ in $H^1(X,\Omega ^1_X)$, which is nonzero).

$\endgroup$
6
  • $\begingroup$ Oh, thank you for this observation. What happens if we assume $H^0(X, \Omega^1_X) \neq 0$? $\endgroup$
    – resid
    Nov 8, 2013 at 12:27
  • $\begingroup$ That doesn't change the problem : the map $\mathrm{Res}: H^0(X,\Omega ^1_X(\log D))\rightarrow H^0(D,\mathcal{O}_D)$ is still the zero map, for the same reason. $\endgroup$
    – abx
    Nov 8, 2013 at 12:51
  • $\begingroup$ Are you saying that every global section of $\Omega^1_X(\log D)$ has zero residues at all $D_i$? $\endgroup$
    – resid
    Nov 8, 2013 at 13:16
  • $\begingroup$ In the case there is just one $D_i$, yes -- and therefore every such section comes from $H^0(X,\Omega ^1_X)$. $\endgroup$
    – abx
    Nov 8, 2013 at 13:29
  • $\begingroup$ What happens if there are several $D_i$? $\endgroup$
    – resid
    Nov 8, 2013 at 13:43

Your Answer

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

Not the answer you're looking for? Browse other questions tagged or ask your own question.