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For a non-compact Riemann surface $X$ there is an isomorphism: $$\Omega(X)/\mathrm d \mathcal O(X)\simeq H^1(X,\mathbb C)$$ where $\Omega$ is the sheaf of holomorphic forms on $X$. The group on the left can be understood as the "holomorphic de Rham" cohomology group $H^1_\mathrm{dR,hol}(X)$. This fact can be generalized to Stein manifolds, but for simplicity I consider this particular case. My questions are:

  1. Can this isomorphism be described as follows: $\mathcal I:H^1_\mathrm{dR,hol}(X) \rightarrow H^1_{\mathrm{sing}}(X,\mathbb C)$ $$\mathcal I[\omega][\gamma] = \int_\gamma \omega,$$ in other words as the analog of de Rham isomorphism, but with holomorphic $1$-forms?
  2. Is it true that given a smooth complex-valued closed $1$-form on $X$, we can find a holomorphic $1$-form which is cohomologous to it?

EDIT: In order to clarify the context and give some reference I'll write the proof I know, which uses arguments of sheaf cohomology. First of all, for non-compact Riemann surfaces we have $$H^1(X,\mathcal O) = 0,\qquad \qquad(1)$$ which is a non-trivial fact (see Forster, Lectures on Riemann Surfaces, Theorem 26.1). Now we argue like in Forster, Theorem 15.13: consider the exact sequence $0 \rightarrow \mathbb C \rightarrow \mathcal O \xrightarrow{\mathrm d} \Omega \rightarrow 0$, it induces a long exact sequence in cohomology, where we find $$\mathcal O(X) \xrightarrow{\mathrm{d}} \Omega(X) \rightarrow H^1(X,\mathbb C) \rightarrow H^1(X,\mathcal O) = 0 \quad\text{ (because of $(1)$)}$$ and so $\Omega(X)/\mathrm d \mathcal O(X)\simeq H^1(X,\mathbb C)$. This argument, for the general case of Stein manifolds, can be found in Serre, Quelques problemes globaux relatifs aux varietes de Stein.

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  • $\begingroup$ Tell me if I understood : $\Omega$ is the vector space of holomorphic differentials on $X$, $d\mathcal{O} = \{ \omega \in \Omega, \omega = f'(z)dz, f \text{ holomorphic}\}$. Let $\Gamma$ be the group of closed loops modulo homotopy, and $$J(\omega)(\gamma) =I(\gamma)(\omega) = \int_\gamma \omega$$ then $I(\gamma) \in \Omega'$ (it is a linear map $\Omega \to \mathbb{C}$) and $I$ is an injective group morphism $\Gamma \to \Omega'$, while $J(\omega)$ is a group morphism $\Gamma \to \mathbb{C}$ and $J$ is a group isomorphism $\Omega/d \mathcal{O} \to \Gamma'$. $\endgroup$
    – reuns
    Commented Jun 18, 2017 at 22:36
  • $\begingroup$ It seems your question 2. is about harmonic $1$-forms which are not always holomorphic $\endgroup$
    – reuns
    Commented Jun 18, 2017 at 23:13
  • $\begingroup$ @user1952009, thank you for your answer. Could you please explain in some more detail? $\endgroup$
    – user336494
    Commented Jun 19, 2017 at 10:49
  • $\begingroup$ I (as many of us) am not confortable with the notations. Why not write explicitly the complex analytic and group objects you are considering ? $\endgroup$
    – reuns
    Commented Jun 19, 2017 at 20:40

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