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

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

$\nabla(\phi)(u) = \nabla_P (\phi(u)) - \phi(\nabla_M (u)),\,\,\,\,u\in M, \phi:M\to N.$

References

chapter 5, Theory of meromorphic connections, from R. Hotta, K. Takeuchi, T. Tanisaki, D-modules, perverse sheaves, and representation theory, Progress in Mathematics 236, Birkhäuser

C. Sabbah, Isomonodromic deformations and Frobenius manifolds, Springer 2007, doi, errata

P. Maisonobe, C. Sabbah, D-modules cohérents et holonomes, Hermann, Paris 1993.

D. Babbitt, V.S. Varadarajan, Deformations of nilpotent matrices over rings and reduction of analytic families of meromorphic differential equations, Mem. Amer. Math. Soc. 55 (325), iv+147, 1985; Local moduli for meromorphic differential equations, Astérisque 169-170 (1989), 1–217.

V.S. Varadarajan, Linear meromorphic differential equation: a modern point of view, Bull. AMS 33, n. 1, 1996, pdf, citeseer:pdf.

Pierre Deligne, Équations différentielles à points singuliers réguliers, Lect. Notes in Math. 163, Springer-Verlag (1970)

Revised on March 29, 2010 14:10:51
by Zoran Škoda
(161.53.130.104)