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meromorphic connection

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 𝒦\mathcal{K} of meromorphic functions in a neighborhood of 00\in \mathbb{C} with possible pole at 00 and a finite dimensional 𝒦\mathcal{K}-module MM. A meromorphic connection at x=0x=0 is a \mathbb{C}-linear operator :MM\nabla:M\to M satisfying

(fu)=dfdxu+f(u),f𝒦,uM \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\mathcal{K} is isomorphic to the field of formal Laurant power series [[u]][u 1]\mathbb{C}[ [u] ][u^{-1}].

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

(uv)= M(u)v+u N(v) \nabla(u\otimes v) = \nabla_M (u)\otimes v + u\otimes\nabla_N(v)

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

(ϕ)(u)= P(ϕ(u))ϕ( M(u)),uM,ϕ:MN. \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)

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