Let $V$ be a complex manifold and $D \subset V$ a smooth divisor.
Question 1 Is $H^i(V \setminus D, \mathbb{C}) \simeq \mathbb{H}^i ( V, \Omega^{\bullet}_V(\log D)) $ ?
Question 2(Edited) Ok, 1 is true. Is it possible to define naturally a homomorphism $H^2(V \setminus D, \mathbb{C}) \rightarrow H^1(V, \Omega_V^1 (\log D))$?
(In my case $V$ is of the form (3-dimensional $U \setminus p $ where $U$ is a $3$-dimensional smooth Stein complex manifold) space and $D$ is of the form $\Delta \point. setminus p$ where $\Delta \subset U$ is a divosor with an isolated singularity at $p$. Then $H^1(V, \Omega_V^1 (\log D))$ is the set of 1st order deformations of the pair $(U, \Delta)$. Since $\Delta$ has only isolated singularities, this is finite dimensional.)
I think it is true when $V$ is compact. How about non-compact case?
