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.
Consider the field of meromorphic functions in a neighborhood of with possible pole at and a finite dimensional -module . A meromorphic connection at is a -linear operator 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 is isomorphic to the field of formal Laurent series .
There is a natural tensor product on the category of -modules with meromorphic connections. Namely where
There is also an inner hom, namely is with a meromorphic connection
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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)
A discussion of meromorphic connections on the complex projective line:
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