Let $X$ be a complex variety of dimension $n$ and $D$ a smooth hypersurface.

Let $\Omega_X(logD)^*$ be the holomorphic logarithmic De Rham complex: $\omega\in \Omega_X(logD)^k$ is a form of degree $k$ on $X$, holomorphic outside $D$ and with a logarithmic pole on $D$.

I want to prove that the complex $\Omega_X(logD)^*$ is exact for degrees $k\ge 2$: here is my proof

where $D$ is defined by $\{z=0\}$, $\alpha\in \Omega_X(logD)^k$, $k\ge 2$ can be written as $\alpha=\frac{dz\wedge \beta}{z}+\gamma$, with both $\beta$ and $\gamma$ holomorphic forms which do not contain $dz$.

$d\alpha=0$ implies $\frac{dz\wedge d\beta}{z}+d\gamma=0$ and $\beta$ and $\gamma$ are closed. So by the holomorphic version of the Poincarè lemma there are holomorphic forms $\beta'$ and $\gamma'$ such that $\beta=d\beta'$ and $\gamma=d\gamma'$.

So $\alpha=d(\frac{dz\wedge \beta'}{z}+\gamma')$

Am i right? Is that so simple?

  • 2
    $\begingroup$ To be clear, you are only proving local exactness. You're argument looks fine to me, although you might need a sign on the $\beta'$ in the last formula. $\endgroup$ Jul 4, 2013 at 16:04
  • $\begingroup$ by global exactness you mean exactness of global sections? of course not, i want to prove local exactness $\endgroup$ Jul 4, 2013 at 16:12

1 Answer 1


See Corollary 1.10 in Steenbrink "Limits of Hodge structures" (Invent. Math. 1976).

Nb. In your definition of the log de Rham complex you need to assume $\omega$ and $d\omega$ have simple poles along $D$ (take e.g. $k=0$, $\omega = 1/f$ where $f$ is a local equation for $D$).


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.