In this paper, we will give a complete geometric background for the geometry of Painlevé $VI$ and Garnier equations. By geometric invariant theory, we will construct a smooth coarse moduli space $M_n^{\balpha}(\bt, \blambda, L) $ of stable parabolic connection on $\BP^1$ with logarithmic poles at $D(\bt) = t_1 + ... + t_n$ as well as its natural compactification. Moreover the moduli space $\cR(\cP_{n, \bt})_{\ba}$ of Jordan equivalence classes of $SL_2(\C)$-representations of the fundamental group $π_1(\BP^1 \setminus D(\bt),\ast)$ are defined as the categorical quotient. We define the Riemann-Hilbert correspondence $\RH: M_n^{\balpha}(\bt, \blambda, L) \lra \cR(\cP_{n, \bt})_{\ba}$ and prove that $\RH$ is a bimeromorphic proper surjective analytic map. Painlevé and Garnier equations can be derived from the isomonodromic flows and Painlevé property of these equations are easily derived from the properties of $\RH$. We also prove that the smooth parts of both moduli spaces have natural symplectic structures and $\RH$ is a symplectic resolution of singularities of $\cR(\cP_{n, \bt})_{\ba}$, from which one can give geometric backgrounds for other interesting phenomena, like Hamiltonian structures, Bäcklund transformations, special solutions of these equations.
In this paper, we show that the family of moduli spaces of $\balpha'$-stable $(\bt, \blambda)$-parabolic $ϕ$-connections of rank 2 over $\BP^1$ with 4-regular singular points and the fixed determinant bundle of degree -1 is isomorphic to the family of Okamoto--Painlevé pairs introduced by Okamoto \cite{O1} and \cite{STT02}. We also discuss about the generalization of our theory to the case where the rank of the connections and genus of the base curve are arbitrary. Defining isomonodromic flows on the family of moduli space of stable parabolic connections via the Riemann-Hilbert correspondences…
This paper applies methods of Van der Put and Van derPut-Saito to the fourth Painlevé equation. One obtains a Riemann--Hilbert correspondence between moduli spaces of rank two connections on $\mathbb{P}^1$ and moduli spaces for the monodromy data. The moduli spaces for these connections are identified with Okamoto--Painlevé varieties and the Painlevé property follows. For an explicit computation of the full group of Bäcklund transformations, rank three connections on $\mathbb{P}^1$ are introduced, inspired by the symmetric form for ${\rm PIV}$ as was studied by M. Noumi and Y. Yamada.
It is well known that the sixth Painlevé equation $\PVI$ admits a group of Bäcklund transformations which is isomorphic to the affine Weyl group of type $\mathrm{D}_4^{(1)}$. Although various 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 characterizing the Bäcklund transformations in terms of Riemann-Hilbert correspondence. In this direction, we show that the Bäcklund transformations are just the pull-back of very simple transformations on the moduli of monodromy representations…
In this paper we introduce the concept of decorated character variety for the Riemann surfaces arising in the theory of the Painlevé differential equations. Since all Painlevé differential equations (apart from the sixth one) exhibit Stokes phenomenon, it is natural to consider Riemann spheres with holes and bordered cusps on such holes. The decorated character is defined as complexification of the bordered cusped Teichmüller space introduced in arXiv:1509.07044. We show that the decorated character variety of a Riemann sphere with s holes and n>1 bordered cusps is a Poisson manifold of dimens…
The Riemann-Hilbert approach for the equations ${\rm PIII(D_6)}$ and ${\rm PIII(D_7)}$ is studied in detail, involving moduli spaces for connections and monodromy data, Okamoto-Painlevé varieties, the Painlevé property, special solutions and explicit Bäcklund transformations.
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 Painlevé property of the isomonodromic deformation.
The sixth Painlevé equation is hiding extremely rich geometric structures behind its outward appearance. This article tries to give as a total picture as possible of its dynamical natures, based on the Riemann-Hilbert approach recently developed by the authors, using various techniques from algebraic geometry. A good part of the contents is extended to Garnier systems, while this article is restricted to the original sixth Painlevé equation.
Every finite branch solutions to the sixth Painleve equation around a fixed singular point is an algebraic branch solution. In particular a global solution is an algebraic solution if and only if it is finitely many-valued globally. The proof of this result relies on algebraic geometry of Painleve VI, Riemann-Hilbert correspondence, geometry and dynamics on cubic surfaces, resolutions of Kleinian singularities, and power geometry of algebraic differential equations. In the course of the proof we are also able to classify all finite branch solutions up to Backlund transformations.
In this paper, we study rank 2 (quasi) parabolic bundles over the Riemann sphere with an effective divisor and these moduli spaces. First we consider a criterium when a parabolic bundle admits a unramified irregular singular parabolic connection. Second, to give a good compactification of the moduli space of semistable parabolic bundles, we introduce a generalization of parabolic bundles, which is called refined parabolic bundles. Third, we discuss a stability condition of refined parabolic bundles and define elementary transformations of the refined parabolic bundles. Finally, we describe the…
We construct explicit geometric models for moduli spaces of semi-stable strongly parabolic Higgs bundles over the Riemann sphere, in the case of rank two, four marked points, arbitrary degree, and arbitrary weights. The mechanism of construction relies on elementary geometric and combinatorial techniques, based on a detailed study of orbit stability of (in general non-reductive) bundle automorphism groups on certain carefully crafted spaces. The aforementioned techniques are not exclusive to the case we examine, and this work elucidates a general approach to construct arbitrary moduli spaces o…
We will describe a method for constructing explicit algebraic solutions to the sixth Painleve equation, generalising that of Dubrovin-Mazzocco. There are basically two steps: First we explain how to construct finite braid group orbits of triples of elements of SL_2(C) out of triples of generators of three-dimensional complex reflection groups. (This involves the Fourier-Laplace transform for certain irregular connections.) Then we adapt a result of Jimbo to produce the Painleve VI solutions. (In particular this solves a Riemann-Hilbert problem explicitly.) Each step will be illustrated using t…
In this paper, we give a systematic construction of ten isomonodromic families of connections of rank two on $\mathbb{P}^1$ inducing Painlevé equations. The classification of ten families is given by considering the Riemann-Hilbert morphism from a moduli space of connections with certain type of regular and irregular singularities to a corresponding catetorical moduli space of analytic data (i.e., ordinary monodromy, Stokes matrices and links), which is called the monodromy space. Explicit equations of the monodromy spaces for ten families are calculated. Moreover, we obtain natural explicit f…
To a finite quiver equipped with a positive integer on each of its vertices, we associate a holomorphic symplectic manifold having some parameters. This coincides with Nakajima's quiver variety with no stability parameter/framing if the integers attached on the vertices are all equal to one. The construction of reflection functors for quiver varieties are generalized to our case, in which these relate to simple reflections in the Weyl group of some symmetrizable, possibly non-symmetric Kac-Moody algebra. The moduli spaces of meromorphic connections on the rank 2 trivial bundle over the Riemann…
Let $(X,\,D)$ be an $m$-pointed compact Riemann surface of genus at least $2$. For each $x \,\in\, D$, fix full flag and concentrated weight system $α$. Let $P \mathcal{M}_ξ$ denote the moduli space of semi-stable parabolic vector bundles of rank $r$ and determinant $ξ$ over $X$ with weight system $α$, where $r$ is a prime number and $ξ$ is a holomorphic line bundle over $X$ of degree $d$ which is not a multiple of $r$. We compute the Chen-Ruan cohomology of the orbifold for the action on $P \mathcal{M}_ξ$ of the group of $r$-torsion points in ${\rm Pic}^0(X)$.
The cohomology ring of the moduli space of stable holomorphic vector bundles of rank n and degree d over a Riemann surface of genus g>1 has a standard set of generators when n and d are coprime. When n=2 the relations between these generators are well understood, and in particular a conjecture of Mumford, that a certain set of relations is a complete set, is known to be true. In this article generalisations are given of Mumford's relations to the cases when n>2 and also when the bundles are parabolic bundles, and these are shown to form complete sets of relations.
Let $(C,\bt)$ ($\bt=(t_1,...,t_n)$) be an $n$-pointed smooth projective curve of genus $g$ and take an element $\blambda=(λ^{(i)}_j)\in\C^{nr}$ such that $-\sum_{i,j}λ^{(i)}_j=d\in\mathbf{Z}$. For a weight $\balpha$, let $M_C^{\balpha}(\bt,\blambda)$ be the moduli space of $\balpha$-stable $(\bt,\blambda)$-parabolic connections on $C$ and let $RP_r(C,\bt)_{\ba}$ be the moduli space of representations of the fundamental group $π_1(C\setminus\{t_1,...,t_n\},*)$ with the local monodromy data $\ba$ for a certain $\ba\in\C^{nr}$. Then we prove that the morphism $\RH:M_C^{\balpha}(\bt,\blambda)\righ…
A local Riemann-Hilbert correspondence for tame meromorphic connections on a curve compatible with a parahoric level structure will be established. Special cases include logarithmic connections on G-bundles and on parabolic G-bundles, where G is a complex reductive group. The corresponding Betti data involves pairs (M,P) consisting of the local monodromy M in G and a (weighted) parabolic subgroup P of G such that M is in P, as in the multiplicative Brieskorn-Grothendieck-Springer resolution (extended to the parabolic case). The natural quasi-Hamiltonian structures that arise on such spaces of …
For an elliptic surface $q:Y \to Σ$, with prescribed singular fibres, Stefan Bauer proved directly via algebraic geometry that the stable bundles over $Y$, whose chern classes are pull backs from $Σ$, correspond to the stable (V-)bundles over $Σ$. We show, via a short proof in differential geometry, a generalisation to stable parabolic bundles. This uses extensions of Donaldson's deep result, giving the existence of Hermitian-Yang-Mills (or anti-self-dual) connections on stable parabolic bundles. In our cases these connections are flat and hence, correspond to representations of certain fundam…
We prove the SYZ (Strominger-Yau-Zaslow) duality for the moduli space of full flag parabolic Higgs bundles over a compact Riemann surface.The SYZ duality was proved for moduli spaces of Higgs vector bundles over a compact Riemann surface by Hausel and Thaddeus.
Mehta and Seshadri have proved that the set of equivalence classes of irreducible unitary representations of the fundamental group of a punctured compact Riemann surface, can be identified with equivalence classes of stable parabolic bundles of parabolic degree zero on the compact Riemann surface. In this note, we discuss the Mehta-Seshadri correspondence over an irreducible projective curve with at most nodes as singularities.
A Lamé connection is a logarithmic $\mathrm{sl}(2,\mathbb C)$-connection $(E,\nabla)$ over an elliptic curve $X:\{y^2=x(x-1)(x-t)\}$, $t\not=0,1$, having a single pole at infinity. When this connection is irreducible, we show that it is invariant by the standart involution and can be pushed down as a logarithmic $\mathrm{sl}(2,\mathbb C)$-connection over $\mathbb P^1$ with poles at $0$, $1$, $t$ and $\infty$. Therefore, the isomonodromic deformation $(E_t,\nabla_t)$ of an irreducible Lamé connection, when the elliptic curve $X_t$ varry in the Legendre family, is parametrized by a solution $q(t…
Let $X$ be a compact Riemann surface of genus $g \geq 3$ and $S$ a finite subset of $X$. Let $ξ$ be fixed a holomorphic line bundle over $X$ of degree $d$. Let $\mathcal{M}_{pc}(r, d, α)$ (respectively, $\mathcal{M}_{pc}(r, α, ξ)$ ) denote the moduli space of parabolic connections of rank $r$, degree $d$ and full flag rational generic weight system $α$, (respectively, with the fixed determinant $ξ$) singular over the parabolic points $S \subset X$. Let $\mathcal{M}'_{pc}(r, d, α)$ (respectively, $\mathcal{M}'_{pc}(r, α, ξ)$) be the Zariski dense open subset of $\mathcal{M}_{pc}(r, d, α)$ (resp…
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 modu…
It is proved that the Painlevé VI equation $(PVI_{\al,\be,\ga,\de})$ for the special values of constants $(\al=\frac{ν^2}{4},\be=-\frac{ν^2}{4}, \ga=\frac{ν^2}{4},\de=\f1{2}-\frac{ν^2}{4})$ is a reduced hamiltonian system. Its phase space is the set of flat $SL(2,C)$ connections over elliptic curves with a marked point and time of the system is given by the elliptic module. This equation can be derived via reduction procedure from the free infinite hamiltonian system. The phase space of later is the affine space of smooth connections and the "times are the Beltrami differentials. This approach…
