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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:
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 $j^{-1}, j_*$?