nLab
meromorphic connection

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

$$\nabla (\mathrm{fu})=\frac{\mathrm{df}}{\mathrm{dx}}u+f\nabla (u),f\in 𝒦,u\in M$$\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 $𝒦$ is isomorphic to the field of formal Laurant power series $ℂ[[u]][u^{-1}]$.

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

$$\nabla (u\otimes v)={\nabla }_M(u)\otimes v+u\otimes {\nabla }_N(v)$$\nabla(u\otimes v) = \nabla_M (u)\otimes v + u\otimes\nabla_N(v)

There is also an inner hom, namely $\mathrm{HOM}((M,{\nabla }_M),(P,{\nabla }_P))$ is ${\mathrm{Hom}}_{𝒦}(M,P)$ with a meromorphic connection

$$\nabla (\varphi )(u)={\nabla }_P(\varphi (u))-\varphi ({\nabla }_M(u)),u\in M,\varphi :M\to N.$$\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.

• L.Katzarkov, M.Kontsevich, T.Pantev, Hodge theoretic aspects of mirror symmetry, arxiv/0806.0107

• 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)