Inclusion of logarithmic de-Rham complex into differentials

2012-09-27T00:59:14Z

Let $X$ be a complex manifold and $D$ a normal crossing divisor. Let $U = X - D$ and $j: U \rightarrow X$ the natural map. Voisen observes that there is a natural inclusion $$\Omega^k_X(\log D) \subset j_*\Omega_U^k$$ Why is this so? Certainly, by adjointness of $j^{-1}, j_*$ we get a natural map $\Omega^k_X(\log D) \rightarrow j_*\Omega_U^k$. It's not obvious to me that this map is injective at the level of stalks. Basically I have two questions:

1.) Is it infact obvious that the natural map produced by adjointness is injective at the level of stalks? (Does this follow from more general "sheaf theory" theorems"?)

2.) Are you able to see in a more obvious way that a map $\Omega^k_X(\log D) \rightarrow j_*\Omega_U^k$ exists, and is an inclusion of sheaves "geometrically", that is, without using adjointness of $j^{-1}, j_*$?

Answer by Sándor Kovács for Inclusion of logarithmic de-Rham complex into differentials

2012-09-27T03:13:24Z

Answers to the numbered questions:

Yes, it is in fact obvious that the natural map you describe is injective, because it is injective, in fact an isomorphism, on $U$ which is dense in $X$ and $\Omega_X^k(\log D)$ is locally free (it would be enough that it is torsion-free).
I would actually say that the adjointness you are using is both geometric and obvious. In other words, your map is simply the restriction of logarithmic differentials from an open set $V\subseteq X$ to $U\cap V$. (Note that by the definition of $U$, $U\cap V\neq\emptyset$): $$ \Gamma(V,\Omega_X^k(\log D))\to \Gamma(U\cap V, \Omega_X^k(\log D))\simeq \Gamma (U\cap V, \Omega_U^k)\simeq \Gamma (V, j_*\Omega_U^k). $$

Inclusion of logarithmic de-Rham complex into differentials - MathOverflow

Inclusion of logarithmic de-Rham complex into differentials

Answer by Sándor Kovács for Inclusion of logarithmic de-Rham complex into differentials