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

Abstract

Let ( C , t ) (C,\mathbf {t}) ( t = ( t 1 , … , t n ) \mathbf {t}=(t_1,\ldots ,t_n) ) be an n n -pointed smooth projective curve of genus g g and take an element λ = ( λ j ( i ) ) ∈ C n r \boldsymbol {\lambda }=(\lambda ^{(i)}_j)\in \mathbf {C}^{nr} such that − ∑ i , j λ j ( i ) = d ∈ Z -\sum _{i,j}\lambda ^{(i)}_j=d\in \mathbf {Z} . For a weight α \boldsymbol {\alpha } , let M C α ( t , λ ) M_C^{\boldsymbol {\alpha }}(\mathbf {t},\boldsymbol {\lambda }) be the moduli space of α \boldsymbol {\alpha } -stable ( t , λ ) (\mathbf {t},\boldsymbol {\lambda }) -parabolic connections on C C and let R P r ( C , t ) a RP_r(C,\mathbf {t})_{\mathbf {a}} be the moduli space of representations of the fundamental group π 1 ( C ∖ { t 1 , … , t n } , ∗ ) \pi _1(C\setminus \{t_1,\ldots ,t_n\},*) with the local monodromy data a \mathbf {a} for a certain a ∈ C n r \mathbf {a}\in \mathbf {C}^{nr} . Then we prove that the morphism R H : M C α ( t , λ ) → R P r ( C , t ) a \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.