Michi-Aki Inaba

We introduce the notion of factorized ramified structure on a generic ramified irregular singular connection on a smooth projective curve. By using the deformation theory of connections with factorized ramified structure, we construct a canonical 2-form on the moduli space of ramified connections. Since the factorized ramified structure provides a duality on the tangent space of the moduli space, the 2-form becomes nondegenerate. We prove that the 2-form on the moduli space of ramified connections is d -closed via constructing an unfolding of the moduli space. Based on the Stokes data, we introduce the notion of local generalized isomonodromic deformation for generic unramified irregular singular connections on a unit disk. Applying the Jimbo-Miwa-Ueno theory to generic unramified connections, the local generalized isomonodromic deformationis equivalent to the extendability of the family of connections to an integrable connection. We give the same statement for ramified connections. Based on this principle of Jimbo-Miwa-Ueno theory, we construct a global generalized isomonodromic deformation on the moduli space of generic ramified connections by constructing a horizontal lift of a universal family of connections. As a consequence of the global generalized isomonodromic deformation, we can lift the relative symplectic form on the moduli space to a total closed form, which is called a generalized isomonodromic 2-form.

We introduce an unfolded moduli space of connections, which is an algebraic relative moduli space of connections on complex smooth projective curves, whose generic fiber is a moduli space of regular singular connections and whose special fiber is a moduli space of unramified irregular singular connections. On the moduli space of unramified irregular singular connections, there is a subbundle of the tangent bundle defining the generalized isomonodromic deformation produced by the Jimbo-Miwa-Ueno theory. On an analytic open subset of the unfolded moduli space of connections, we construct a non-canonical lift of this subbundle, which we call an unfolding of the unramified irregular singular generalized isomonodromic deformation. Our construction of an unfolding of the unramified irregular singular generalized isomonodromic deformation is not compatible with the asymptotic property in the unfolding theory established by Hurtubise, Lambert and Rousseau which gives unfolded Stokes matrices for an unfolded linear differential equation in a general framework.

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We give an algebraic construction of the moduli space of irregular singular connections of generic ramified type on a smooth projective curve. We prove that the moduli space is smooth and give its dimension. Under the assumption that the exponent of ramified type is generic, we give an algebraic symplectic form on the moduli space.

We define a moduli space of stable regular singular parabolic connections of spectral type on smooth projective curves and show the smoothness of the moduli space and give a relative symplectic structure on the moduli space. Moreover, we define the isomonodromic deformation on this moduli space and prove the geometric Painlev\'e property of the isomonodromic deformation.

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.