In this post, $X$ will mean a compact Riemann surface (without boundary).

Let us start by considering a form of the Mittag-Leffler problem: Given a point $p \in X$, can we find a global meromorphic function $f \in \mathcal{M}(X)$ such that $(f)_{\infty} = p$ and $\text{res}_p(f) = \lambda$? i.e. such that the only poles of $f$ is a simple pole at $p$ with residue $\lambda$ in some coordinates. Since this would give a biholomorphic function $f : X \to \mathbb{P}^1$, this is only possible if the genus of $X$ is zero, but let us see how this obstruction manifests itself.

We will consider a covering $\mathfrak{U} = U_1, U_2$ of $X$ where $U_1$ is a coordinate patch centered at $p$ with (holomorphic) coordinate $z$, while $U_2$ is just $X \setminus p$. We can reformulate the question as follows: does there exist $f \in \mathcal{M}(X)$ such that $f_1 \in \frac{\lambda}{z} + \mathcal{O}(U_1)$ and $f_2 \in \mathcal{O}(U_2)$? NB: Here $f_i$ means the restriction $f|_{U_i}$.

Before going further, note that the residue of a meromorphic function is only invariantly defined as a tangent vector: if $f \in \mathcal{M}(X)$ has a simple pole at $p$, then we define, for $\alpha_p \in T^*_pX$,

$\langle \tilde{\alpha}, \text{res}_p(f) \rangle = \text{res}_p(f\alpha)$

where $\tilde{\alpha}$ is any extension of $\alpha_p$, and where

$\text{res}_p(\alpha) = \frac{1}{2\pi i}\int_{\gamma}\alpha$

is a well-defined residue for any meromorphic 1-form $\alpha$ defined around $p$, where $\gamma$ is a small loop around $p$. Another way to see this is to take $z$ a holomorphic coordinate around $p$ and $f$ a meromorphic function with a simple pole at $p$, say $f(z) = \frac{\lambda}{z} + h(z)$ for $h$ holomorphic. Then if $\tilde{z} = c_1z + c_2z^2 + \cdots$ is another choice of coordinates, then $\frac{1}{z} = \frac{c_1}{\tilde{z}} + \frac{c_2z}{\tilde{z}} + \cdots$ so

$f(\tilde{z}) = \frac{c_1\lambda}{\tilde{z}} + g(\tilde{z})$

for some holomorphic $g$. In other words,

$(\text{res}_pf)_{\tilde{z}} = \left(\frac{d\tilde{z}}{dz}|_p\right)(\text{res}_pf)_{z}$

i.e.

$\text{res}_p(f) = \lambda\frac{\partial}{\partial z}$

is well-defined as an element of $T_pX$.

First approach: Our first approach to solving the Mittag-Leffler problem is with Cech cohomology: What we want is to find a global meromorphic function $f$ such that on $U_1 \cap U_2 = U_1 \setminus p$ we have $f = \frac{\lambda}{z}$. Thus we see $f \in C^1(\mathfrak{U},\mathcal{O})$, as a Cech 1-cochain in the sheaf of holomorphic functions and ask that $f = f_2 - f_1$ on $U_1 \setminus p$ for $f_i \in \mathcal{O}(U_i)$, i.e. that $f$ defines a coboundary in $H^1(X,\mathcal{O})$. Indeed, $f_2$ would then be holomorphic on $X \setminus p$ with $f_2 = \frac{\lambda}{z} + f_1$ near $p$.

Another way to realize this is via the following short exact sequence of sheaves:

$0 \to \mathcal{O} \to \mathcal{O}(p) \overset{\text{res}_p}\to T_pX \to 0$

where $\mathcal{O}(p)$ is the sheaf of meromorphic functions having at worst a simple pole at $p$ and where we consider $T_pX$ as the associated skyscraper sheaf. From the induced long exact sequence we get

$H^0(X,\mathcal{O}(p)) \to T_pX \overset{\Delta}\to H^1(X,\mathcal{O})$

and in this formulation, finding a meromorphic function with at worst a simple pole at $p$ with residue $v = \lambda \frac{\partial}{\partial z}$ is possible if and only if $\Delta(v) = 0 \in H^1(X,\mathcal{O})$. But unwinding the definition of $\Delta$, we see that $\Delta(v) = f$ iff $\exists f_i \in \mathcal{O}(p)(U_i)$ such that $\text{res}_p(f_1) = v$ and $f_1 - f_2 = f$ in $U_1 \setminus p$. Thus we can take $f_1 = \frac{\lambda}{z}$ and $f_2 = 0$, and

$\Delta(\lambda\frac{\partial}{\partial z}) = [\frac{\lambda}{z}] \in H^1(X,\mathcal{O})$.

We once again come to the conclusion that the problem is solvable exactly when $\frac{\lambda}{z}$ is a coboundary.

Second approach: Consider a smooth cut-off function $\beta$ supported in $U_1$ and identically equal to 1 near $p$. Then $\beta \frac{\lambda}{z} \in C^{\infty}(U_2)$ and the probem reduces to finding $g \in C^{\infty}(X)$ such that $\overline{\partial}(\beta \frac{\lambda}{z}) = \overline{\partial}g$. Note that $\overline{\partial}\beta = 0$ near $p$ so we can consider

$\Psi = \overline{\partial}(\beta)\frac{\lambda}{z}$

as an element of $A^{0,1}(X)$, i.e. a $(0,1)$-form on $X$. Reformulating this, we can solve the problem exactly when $[\Psi] = 0 \in H^{0,1}_{\overline{\partial}}(X)$.

To relate this to our first approach, we note $A^{p,q}$ the sheaf of $(p,q)$-forms and use the short exact sequence of sheaves

$0 \to \mathcal{O} \to A^{0,0} \overset{\overline{\partial}}\to A^{0,1} \to 0$

which gives the exact sequence

$H^0(X,A^{0,0}) \to H^0(X,A^{0,1}) \overset{\Delta'}\to H^1(X,\mathcal{O}) \to 0$.

In other words, this $\Delta'$ gives an isomorphism

$H^{0,1}_{\overline{\partial}}(X) = \text{coker} (\overline{\partial}:A^{0,0}(X) \to A^{0,1}(X)) \cong H^1(X,\mathcal{O}).$

Unwinding the definition of $\Delta'$, we see that $\Delta'(\overline{\partial}(f_i)) = f$ iff $f = f_2 - f_1$ in $U_1 \cap U_2$, for $f_i \in \mathcal{O}(U_i)$.

This isomorphism $H^{0,1}_{\overline{\partial}}(X) \cong H^1(X,\mathcal{O})$ is the simplest instance of the so-called Dolbeault isomorphism which is a similar isomorphism $H^{p,q}_{\overline{\partial}}(X,E) \cong H^q(X,\Omega^p(E))$ valid for arbitrary compact complex manifolds and complex vector bundle $E$, where $\Omega^p(E)$ is the sheaf of holomorphic p-forms with values in $E$.We can recap our discussion with the following:

Proposition: The obstruction to solving the Mittag-Leffler problem coming from our second approach, $[\Psi] \in H^{0,1}_{\overline{\partial}}(X)$ which is represented by the 1-cocycle $\overline{\partial}(\beta)\frac{\lambda}{z} \in A^{0,1}(X)$, corresponds under Dolbeault’s isomorphism to $[\frac{\lambda}{z}] \in H^1(X,\mathcal{O})$, the obstruction from our first approach.