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

Inaba, Michi-aki

doi:10.1090/S1056-3911-2013-00621-9

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 | References | Additional Information

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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Additional Information

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

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.