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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?

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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 $H^2(V \setminus D, \mathbb{C}) \rightarrow H^1(V, \Omega_V^1 (\log D))$?

(And is $H^1( V, \Omega_V^1( \log D))$ a direct summand of $\mathbb{H}^2 (V, \Omega^{\bullet}_V(\log D))$?)

In my case $V$ is of the form (3-dimensional Stein complex manifold) \ point. I think it is true when $V$ is compact. How about non-compact case?

3 deleted 240 characters in body; added 4 characters in body; added 8 characters in body

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 Assume that (Edited) Ok, 1 is true. Consider the following homomorphisms; Is it possible to define naturally a homomorphism $H^1( V, \Omega_V^1( \log D)) \rightarrow \mathbb{H}^2 (V, \Omega^{\bullet}_V(\log D)) H^2(V \rightarrow H^1( Vsetminus D, \Omega_V^1( \log D))$

induced by the homomorphisms of complexes $\Omega_V^1( \log D)[-1] \rightarrow \Omega_V^{\bullet}( \log D) mathbb{C}) \rightarrow \Omega_V^1( H^1(V, \log D)[-1]$. Omega_V^1 (Is this consideration well?)

Then \log D))$? (And is$H^1( V, \Omega_V^1( \log D))$a direct summand of$\mathbb{H}^2 (V, \Omega^{\bullet}_V(\log D)) $? ?) In my case$V$is of the form (3-dimensional Stein complex manifold) \ point. I think it is true when$V\$ is compact. How about non-compact case?

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