The Residue of Meromorphic Connections

/ Alexander

1. Introduction

A connection with some good properties will become a very precious thing on a holomorphic vector bundle over a complex manifold, for example, Atiyah-Weil theorem states that:

Atiyah-Weil

A holomorphic vector bundle E over a compact Riemann surface admits a holomorphic connection, if and only if the degree of E^{\oplus k} for any positive integer k is 0.

So, if a connection \nabla is admitted on some vector bundle, \nabla will must contain something that can reflects the property of the underlying vector bundle. In this blog, I will give an example in a simple case: the meromorphic connection on a line bundle over Riemann sphere \mathbb{P}^1.

(The main proof will be followed my post on the Math Stack Exchange)

2. Definitions and Main Results

Let D=k_1a_1+\cdots+k_ma_m be an effective divisor on \mathbb{P}^1, where k_i>0 for each i. Suppose L is a holomorphic line bundle on \mathbb{P}^1.

Recall that a meromorphic connection on L with poles on the divisor D is a sheaf morphism:

\nabla: L\longrightarrow L\otimes\Omega^1_D\left(\mathbb{P}^1\right)

where L is the sheaf of sections of L, and \Omega^1_D(\mathbb{P}^1) is the sheaf of meromorphic 1-forms with poles on D.

Hence locally, \nabla is a meromorphic 1-form, more precisely, if we choose a local trivialization of L near a pole a_i, namely

\phi_i: L|_{D_i}\longrightarrow D_i\times \mathbb{C}

where D_i is a coordinate neighborhood at a_i which sends a_i to zero, then we can express \nabla in terms of Laurent expansion:

\begin{aligned}\nabla=d-\left(\frac{\xi_k}{z^k}+\cdots+\frac{\xi_1}{z}+\cdots\right)dz\end{aligned}

We call the coefficient \xi_1 the residue of \nabla at the pole a_i, denoted by

\mathrm{Res}_{a_i}(\nabla):=\xi_1

Well, it is not hard to see the term \xi_1 is independent of the choice of the local trivialization, because using different local trivialization yields differed by a gauge transformation, that will not change the term \xi, thus this definition is well-defined.

We define the residue of the meromorphic connection to be the sum of all residues near each pole:

\begin{aligned}\mathrm{Res}(\nabla):=\sum_{i=1}^{m}\mathrm{Res}_{a_i}(\nabla)\end{aligned}

The main theorem asserts as follows:

Theorem: The residue of a meromorphic connection \nabla on a line bundle L over \mathbb{P}^1 equals to the negative degree of L:

\mathrm{Res}(\nabla)=-\deg L

Proof. Let

\begin{aligned}U_1=\left\{[x,y]\in\mathbb{P}^1:x\neq 0\right\}\quad U_2=\left\{[x,y]\in\mathbb{P}^1:y\neq 0\right\}\end{aligned}

be the standard covering on \mathbb{P}^1 . The transition function of L is given by

\begin{aligned}g_{12}[x,y]=\left(\frac{x}{y}\right)^{\deg L}: U_1\cap U_2\longrightarrow\mathbb{C}^*\end{aligned}

WLOG, we can assume that all poles are contained in U_1, i.e, a_1,...,a_m\in U_1.

Now, we choose a nowhere vanishing section of L on U_1, namely \sigma_1\in\Gamma(U_1;L), and \sigma_2 a section of L on U_2 determined by

\sigma_2=g_{12}\sigma_1: U_2\longrightarrow L\qquad\qquad (\mathbf{*})

hence no hard to see \sigma_2 is also nowhere vanishing on U_2, hence we can find two meromorphic 1-forms \eta_1,\eta_2\in\Omega^1_D such that

\nabla\sigma_1=\eta_1\otimes\sigma_1\quad\nabla\sigma_2=\eta_2\otimes\sigma_2

Hence by (\mathbf{*}), we have

\begin{aligned}\frac{dg_{12}}{g_{12}}=\eta_2-\eta_1\end{aligned}

As we are assuming U_1 containing all poles of \nabla, hence \eta_2 is in fact a holomorphic 1-form, and \eta_1 has the same residue as \nabla, i.e, \mathrm{Res}\eta_1=\mathrm{Res}\nabla !!!!

Now, we use the coordinate z=x/y on U_1\cap U_2, and consider the equality

\begin{aligned}\eta_1=\eta_2-\frac{\deg L}{z}\end{aligned}

by taking residue on both sides yields:

\mathrm{Res}(\nabla)=\mathrm{Res}(\eta_1)=-\deg L\qquad\blacksquare

3. Remarks

This result remains true for high rank vector bundles. Also, there is a generalization to the high dimensional compact complex manifolds, which asserts that the first Chern class c_1(E) of the vector bundle E equals to the corresponding Chern class of the negative residue divisor \mathrm{Res}(\nabla):

c_1(E)=-c(\mathrm{Res}(\nabla))\in H^2(X;\mathbb{C})

This formula is called the residue formula for the meromorphic connections. But in this case we need to impose that the connection is flat. This result was due to JORGE VITÓRIO PEREIRA.

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