Suppose $X$ is a compact Riemann surface of genus $g$, $\mathcal{O}$ represents the sheaf of holomorphic functions and $\Omega$ the sheaf of holomorphic 1-forms

I want to show the dimension of $H^1(X, \mathbb{C})$ equals $2g$ by using the short exact sequence $0 \to \mathbb{C} \to \mathcal{O} \xrightarrow{d} \Omega \to 0$.

My approach so far is to use the long sequence in cech cohomology of the given short exact sequence $$0 \to \mathbb{C} \xrightarrow{i} \check{H}^0(X,\mathcal{O}) \xrightarrow{d} \check{H}^0(X,\Omega) \xrightarrow{\Delta} \check{H}^1(X, \mathbb{C}) \xrightarrow{i^1} \check{H}^1(X, \mathcal{O}) \xrightarrow{d^1} H^1(X, \Omega)$$ where $\Delta$ is the connecting homomorphism and the maps $i^1$ and $d^1$ are the induced maps from the respective sheaf maps.

By the Serre-Duality Theorem, I know that dim$\check{H}^0(X,\Omega) = g$ and dim$\check{H}^1(X, \Omega) = 1$. After repeated applications of the Rank-Nullity theorem, I can show that dim$($Ker$(i^1)) = g$. But I am having trouble showing that dim$($Im$(i^1)) = g$ as well. It seems like I'm missing something really really straightforward.

The books I am using are Rick Miranda's Algebraic Curves and Riemann Surfaces and Otto Forster's Lectures on Riemann Surfaces.

Would greatly appreciate any help or hints! Thank you so much!

  • $\begingroup$ It's not difficult to compute the $d^{1}$ explicitly on cocycles. $d^{1} (f_{ij}) = (df_{ij})$. So you just have to compute the kernel of this map. $\endgroup$ – user40276 Sep 17 '15 at 6:07
  • $\begingroup$ Remember that you can integrate. $\endgroup$ – user40276 Sep 17 '15 at 6:15

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