Holomorphic logarithmic De Rham complex

Question

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?

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 $D$).