Michi-aki Inaba

Moduli of Stable Parabolic Connections, Riemann–Hilbert Correspondence and Geometry of Painlevé Equation of Type VI, Part I

In this paper, we will give a complete geometric background for the geometry of Painleve VI and Garnier equations. By geometric invariant theory, we will construct a smooth fine moduli space M ¸ n (t,- ,L ) of stable parabolic connections on P 1 with log- arithmic poles at D(t )= t1+ ··· +tn as well as its natural compactification. Moreover the moduli space R(Pn,t)a of Jordan equivalence classes of SL2(C)-representations of the fundamental group π1(P 1 \ D(t), ∗) are defined as the categorical quotient. We define the Riemann-Hilbert correspondence RH : M ¸ n (t,- ,L ) −→ R(Pn,t)a and prove that RH is a bimeromorphic proper surjective analytic map. Painlevea nd Garnier equations can be derived from the isomonodromic flows and Painlevep rop- erty of these equations are easily derived from the properties of RH .W e also prove

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

Let (C, t) (t = (t1, . . . , tn)) be an n-pointed smooth projective curve of genus g and take an element λ = (λ (i) j ) ∈ Cnr such that − ∑ i,j λ (i) j = d ∈ Z. For a weight α, let Mα C (t,λ) be the moduli space of α-stable (t,λ)-parabolic connections on C and let RPr(C, t)a be the moduli space of representations of the fundamental group π1(C \{t1, . . . , tn}, ∗) with the local monodromy data a for a certain a ∈ Cnr. Then we prove that the morphism RH : Mα C (t,λ) → RPr(C, t)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.

MODULI OF UNRAMIFIED IRREGULAR SINGULAR PARABOLIC CONNECTIONS ON A SMOOTH PROJECTIVE CURVE.

In this paper we construct a coarse moduli scheme of stable unramified irregular singular parabolic connections on a smooth projective curve and prove that the constructed moduli space is smooth and has a symplectic structure. Moreover we will construct the moduli space of generalized monodromy data coming from topological monodromies, formal monodromies, links and Stokes data associated to the generic irregular connections. We will prove that for a generic choice of generalized local exponents, the generalized Riemann- Hilbert correspondence from the moduli space of the connections to the moduli space of the associated generalized monodromy data gives an analytic isomorphism. This shows that differential systems arising from (generalized) isomonodromic deformations of corresponding unramified irregular singular parabolic connections admit geometric Painleve property as in the regular singular cases proved generally in (8).

Bäcklund transformations of the sixth Painlevé equation in terms of Riemann-Hilbert correspondence

It is well known that the sixth Painleve equation PVI admits a group of Backlund transformations which is isomorphic to the affine Weyl group of type D (1) . Although var- ious aspects of this unexpectedly large symmetry have been discussed by many authors, there still remains a basic problem yet to be considered, that is, the problem of charac- terizing the Backlund transformations in terms of Riemann-Hilbert correspondence. In this direction, we show that the Backlund transformations are just the pull-back of very simple transformations on the moduli of monodromy representations by the Riemann- Hilbert correspondence. This result gives a natural and clear picture of the Backlund

On the moduli of stable sheaves on a reducible projective scheme and examples on a reducible quadric surface

We study the moduli space of stable sheaves on a reducible projective scheme by use of a suitable stratification of the moduli space. Each stratum is the moduli space of “triples”, which is the main object investigated in this paper. As an application, we can see that the relative moduli space of rank two stable sheaves on quadric surfaces gives a nontrivial example of the relative moduli space which is not flat over the base space.