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\begin{document}

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\section*{meromorphic connection}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{1dimensional_case}{1-Dimensional case}\dotfill \pageref*{1dimensional_case} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The theory of \textbf{meromorphic connections} is a modern viewpoint on local behaviour of a class of [[ordinary differential equation|systems of ODE]]-s with meromorphic coefficients in a complex domain.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{1dimensional_case}{}\subsubsection*{{1-Dimensional case}}\label{1dimensional_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 \textbf{meromorphic} connection at $x=0$ is a $\mathbb{C}$-linear operator $\nabla:M\to M$ satisfying

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

\begin{displaymath}
\nabla(u\otimes v) = \nabla_M (u)\otimes v + u\otimes\nabla_N(v)
\end{displaymath}
There is also an inner hom, namely $HOM((M,\nabla_M),(P,\nabla_P))$ is $Hom_{\mathcal{K}}(M,P)$ with a meromorphic connection

\begin{displaymath}
\nabla(\phi)(u) = \nabla_P (\phi(u)) - \phi(\nabla_M (u)),\,\,\,\,u\in M, \phi:M\to N.
\end{displaymath}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item chapter 5, \emph{Theory of meromorphic connections}, from R. Hotta, K. Takeuchi, T. Tanisaki, \emph{D-modules, perverse sheaves, and representation theory}, Progress in Mathematics \textbf{236}, Birkh\"a{}user


\item [[Claude Sabbah|C. Sabbah]], \emph{Isomonodromic deformations and Frobenius manifolds}, Springer 2007, \href{http://dx.doi.org/10.1007/978-1-84800-054-4}{doi}, \href{http://www.math.polytechnique.fr/cmat/sabbah/coursP6errata.pdf}{errata}


\item P. Maisonobe, C. Sabbah, \emph{D-modules coh\'e{}rents et holonomes}, Hermann, Paris 1993.


\item L.Katzarkov, [[Maxim Kontsevich|M.Kontsevich]], T.Pantev, \emph{Hodge theoretic aspects of mirror symmetry}, \href{http://arxiv.org/abs/0806.0107}{arxiv/0806.0107}


\item D. Babbitt, V.S. Varadarajan, \emph{Deformations of nilpotent matrices over rings and reduction of analytic families of meromorphic differential equations}, Mem. Amer. Math. Soc. \textbf{55} (325), iv+147, 1985; \emph{Local moduli for meromorphic differential equations}, Ast\'e{}risque 169-170 (1989), 1--217.


\item V.S. Varadarajan, \emph{Linear meromorphic differential equation: a modern point of view}, Bull. AMS \textbf{33}, n. 1, 1996, \href{http://www.ams.org/bull/1996-33-01/S0273-0979-96-00624-6/S0273-0979-96-00624-6.pdf}{pdf}, \href{http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.104.8107&rep=rep1&type=pdf}{citeseer:pdf}.


\item [[Pierre Deligne]], \emph{\'E{}quations diff\'e{}rentielles \`a{} points singuliers r\'e{}guliers}, Lect. Notes in Math. \textbf{163}, Springer-Verlag (1970)



\end{itemize}


\end{document}