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The Mittag-Leffler problem and the Dolbeault isomorphism.

June 10, 2014

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.

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