# The Mittag-Leffler problem and the Dolbeault isomorphism.

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

Let us start by considering a form of the **Mittag-Leffler problem**: Given a point , can we find a global meromorphic function such that and ? i.e. such that the only poles of is a simple pole at with residue in some coordinates. Since this would give a biholomorphic function , this is only possible if the genus of is zero, but let us see how this obstruction manifests itself.

We will consider a covering of where is a coordinate patch centered at with (holomorphic) coordinate , while is just . We can reformulate the question as follows: does there exist such that and ? NB: Here means the restriction .

Before going further, note that the **residue** of a meromorphic function is only invariantly defined as a **tangent vector**: if has a simple pole at , then we define, for ,

where is any extension of , and where

is a well-defined residue for any meromorphic 1-form defined around , where is a small loop around . Another way to see this is to take a holomorphic coordinate around and a meromorphic function with a simple pole at , say for holomorphic. Then if is another choice of coordinates, then so

for some holomorphic . In other words,

i.e.

is well-defined as an element of .

**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 such that on we have . Thus we see , as a **Cech 1-cochain** in the sheaf of holomorphic functions and ask that on for , i.e. that defines a **co****boundary** in . Indeed, would then be holomorphic on with near .

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

where is the sheaf of meromorphic functions having at worst a simple pole at and where we consider as the associated skyscraper sheaf. From the induced long exact sequence we get

and in this formulation, finding a meromorphic function with at worst a simple pole at with residue is possible if and only if . But unwinding the definition of , we see that iff such that and in . Thus we can take and , and

.

We once again come to the conclusion that the problem is solvable exactly when is a coboundary.

**Second approach:** Consider a smooth cut-off function supported in and identically equal to 1 near . Then and the probem reduces to finding such that . Note that near so we can consider

as an element of , i.e. a -form on . Reformulating this, we can solve the problem exactly when .

To relate this to our first approach, we note the sheaf of -forms and use the short exact sequence of sheaves

which gives the exact sequence

.

In other words, this gives an isomorphism

Unwinding the definition of , we see that iff in , for .

This isomorphism is the simplest instance of the so-called **Dolbeault isomorphism** which is a similar isomorphism valid for arbitrary compact complex manifolds and complex vector bundle , where is the sheaf of holomorphic p-forms with values in .We can recap our discussion with the following:

Proposition:The obstruction to solving the Mittag-Leffler problem coming from our second approach, which is represented by the 1-cocycle , corresponds under Dolbeault’s isomorphism to , the obstruction from our first approach.

## Trackbacks & Pingbacks