# Contents

## Idea

The theory of meromorphic connections is a modern viewpoint on local behaviour of a class of systems of ODE-s with meromorphic coefficients in a complex domain.

## Definition

### 1-Dimensional case

Consider the field $\mathcal{K}$ of meromorphic functions in a neighborhood of $0\in \mathbb{C}$ with possible pole at $0$ and a finite dimensional $\mathcal{K}$-module $M$. A meromorphic connection at $x=0$ is a $\mathbb{C}$-linear operator $\nabla:M\to M$ satisfying

$\nabla(fu) = \frac{df}{dx}u + f\nabla(u),\,\,\,\,f\in\mathcal{K}, u\in M$

In fact, it is customary, in modern literature to consider just a germ: the connections on two different neighborhoods agreeing on the intersection are identified. This way $\mathcal{K}$ is isomorphic to the field of formal Laurant power series $\mathbb{C}[ [u] ][u^{-1}]$.

There is a natural tensor product on the category of $\mathcal{K}$-modules with meromorphic connections. Namely $(M,\nabla_M)\otimes (N\nabla_N) = (M\otimes N,\nabla)$ where

$\nabla(u\otimes v) = \nabla_M (u)\otimes v + u\otimes\nabla_N(v)$

There is also an inner hom, namely $HOM((M,\nabla_M),(P,\nabla_P))$ is $Hom_{\mathcal{K}}(M,P)$ with a meromorphic connection

$\nabla(\phi)(u) = \nabla_P (\phi(u)) - \phi(\nabla_M (u)),\,\,\,\,u\in M, \phi:M\to N.$
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• Pierre Deligne, Équations différentielles à points singuliers réguliers, Lect. Notes in Math. 163, Springer-Verlag (1970)