Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Moduli of parabolic connections on curves and the Riemann-Hilbert correspondence


Author: Michi-aki Inaba
Journal: J. Algebraic Geom. 22 (2013), 407-480
DOI: https://doi.org/10.1090/S1056-3911-2013-00621-9
Published electronically: February 14, 2013
MathSciNet review: 3048542
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Abstract: Let $ (C,\mathbf {t})$ ( $ \mathbf {t}=(t_1,\ldots ,t_n)$) be an $ n$-pointed smooth projective curve of genus $ g$ and take an element $ \boldsymbol {\lambda }=(\lambda ^{(i)}_j)\in \mathbf {C}^{nr}$ such that $ -\sum _{i,j}\lambda ^{(i)}_j=d\in \mathbf {Z}$. For a weight $ \boldsymbol {\alpha }$, let $ M_C^{\boldsymbol {\alpha }}(\mathbf {t},\boldsymbol {\lambda })$ be the moduli space of $ \boldsymbol {\alpha }$-stable $ (\mathbf {t},\boldsymbol {\lambda })$-parabolic connections on $ C$ and let $ RP_r(C,\mathbf {t})_{\mathbf {a}}$ be the moduli space of representations of the fundamental group $ \pi _1(C\setminus \{t_1,\ldots ,t_n\},*)$ with the local monodromy data $ \mathbf {a}$ for a certain $ \mathbf {a}\in \mathbf {C}^{nr}$. Then we prove that the morphism $ \mathbf {RH}:M_C^{\boldsymbol {\alpha }}(\mathbf {t},\boldsymbol {\lambda })\rightarrow RP_r(C,\mathbf {t})_{\mathbf {a}}$ determined by the Riemann-Hilbert correspondence is a proper surjective bimeromorphic morphism. As a corollary, we prove the geometric Painlevé property of the isomonodromic deformation defined on the moduli space of parabolic connections.


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Michi-aki Inaba
Affiliation: Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
Email: inaba@math.kyoto-u.ac.jp

DOI: https://doi.org/10.1090/S1056-3911-2013-00621-9
Received by editor(s): August 4, 2009
Received by editor(s) in revised form: November 25, 2010, and July 30, 2012
Published electronically: February 14, 2013
Article copyright: © Copyright 2013 University Press, Inc.

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