Proof of Serre duality for $D=0$

Question

I have been working through a proof of Serre duality, which proceeds by induction on the divisor $D$, but I am having trouble with the base-case. How can I prove that on a riemann surface X, $H^0(X, \Omega) = H^1(X, \mathscr{O})^*$, where $\Omega$ is the sheaf of holomorphic 1-forms, $\mathscr{O}$ the sheaf of holomorphic functions. I know I have the linear map $H^0(X, \Omega) \times H^1(X, \mathscr{O}) \to H^1(X, \Omega) \to \mathbb{C}$, with the second arrow being the residue map. How do I show that this induces an isomorphism of the duals? I am hoping this is much simpler than the full proof of serre-duality, in which it is quite tedious to prove this induces an isomorphism.

Answer 1

Here is an explicit definition of the pairing in the case of the projective line. For $\mathcal{M}$ a coherent sheaf, an element of $H^1(\mathcal{M})$ represents an extension$$0 \to \mathcal{M} \to \text{?} \to \mathcal{O} \to 0.$$If we are given a homomorphism $\mathcal{M} \to \mathcal{O}(-2)$, this induces an extension$$0 \to \mathcal{O}(-2) \to \text{?} \to \mathcal{O} \to 0.$$Now, there is one sort of universal such extension with $\text{?} = \mathcal{O}(-1) + \mathcal{O}(-1)$, and any such extension is a scalar multiple of the universal one, so we get a scalar at the end. This defines the pairing.

Understanding the universal extension is key. One way of defining it via the principal parts sequence for $\mathcal{O}(1)$ and then twisted by $\mathcal{O}(-1)$. It also relates closely to the extension connecting the tangent bundle of affine space with that of projective space, which is really where all this comes from most explicitly.