Description of generalized isomonodromic deformations of rank two linear differential equations using apparent singularities

The dimension of the space of Garnier equations with fixed locus of apparent singularities

We show that the conditions imposed on a second order linear differential equation with rational coefficients on the complex line by requiring it to have regular singularities with fixed exponents at the points of a finite set $P$ and apparent singularities at a finite set $Q$ (disjoint from $P$) determine a linear system of maximal rank. In addition, we show that certain auxiliary parameters can also be fixed. This enables us to conclude that the family of such differential equations is of the expected dimension and to define a birational map between an open subset of the moduli space of loga…

从 $\lambda$-连接到具有显奇点的 $PSL_2(\mathbb{C})$-opers

From $λ$-connections to $PSL_2(\mathbb{C})$-opers with apparent singularities

本文在亏格大于1的黎曼曲面上，从SL2(C) λ-连接和子线丛构造具有显奇点的opers，证明所得有理映射是泊松映射，并探讨其与wobbly丛及模空间中拉格朗日子流形的联系。

Algebraic and hamiltonian approaches to isostokes deformations

We study a generalization of the isomonodromic deformation to the case of connections with irregular singularities. We call this generalization Isostokes Deformation. A new deformation parameter arises: one can deform the formal normal forms of connections at irregular points. We study this part of the deformation, giving an algebraic description. Then we show how to use loop groups and hypercohomology to write explicit hamiltonians. We work on an arbitrary complete algebraic curve, the structure group is an arbitrary semisimiple group.

The locus of the representation of logarithmic connections by Fuchsian equations

The generic element of the moduli space of logarithmic connections with parabolic points on holomorphic vector bundle over the Riemann sphere can be represented by a Fuchsian equation with some singularities and some apparent singularities. We analyze the case of rank $3$ vector bundle which leads to third order Fuchsian equation. We find coordinates on an open subset of the moduli space and we construct a non-trivial part of the moduli space by blowing up along a variety in a special case.

Foliations on the moduli space of rank two connections on the projective line minus four points

We look at natural foliations on the Painlevé VI moduli space of regular connections of rank 2 on $\pp ^1 -{t_1,t_2,t_3,t_4}$. These foliations are fibrations, and are interpreted in terms of the nonabelian Hodge filtration, giving a proof of the nonabelian Hodge foliation conjecture in this case. Two basic kinds of fibrations arise: from apparent singularities, and from quasiparabolic bundles. We show that these are transverse. Okamoto's additional symmetry, which may be seen as Katz's middle convolution, exchanges the quasiparabolic and apparent-singularity foliations.

Lagrangian fibration in duality on moduli space of rank two logarithmic connections over the projective line

We study the moduli space of logarithmic connections of rank 2 on the Riemann sphere minus n points with fixed spectral data. There are two natural Lagrangian maps: one towards apparent singularities of the associated fuchsian scalar equation, and another one towards moduli of parabolic bundles. We show that these are transversal and dual to each other. In case n=5, we recover the beautiful geometry of Del Pezzo surfaces of degree 4.

Explicit description of jumping phenomena on moduli spaces of parabolic connections and Hilbert schemes of points on surfaces

In this paper, we investigate the apparent singularities and the dual parameters of rank 2 parabolic connections on $\mathbb{P}^1$ and rank 2 (parabolic) Higgs bundle on $\mathbb{P}^1$. Then we obtain explicit descriptions of Zariski open sets of the moduli space of the parabolic connections and the moduli space of the Higgs bundles. For $n=5$, we can give global descriptions of the moduli spaces in detail.

Deformations of Fuchsian equations and logarithmic connections

We give a geometric proof to the classical fact that the dimension of the deformations of a given generic Fuchsian equation without changing the semi-simple conjugacy class of its local monodromies (``number of accessory parameters'') is equal to half the dimension of the moduli space of deformations of the associated local system. We do this by constructing a weight 1 Hodge structure on the infinitesimal deformations of logarithmic connections, such that deformations as an equation correspond to the $(1,0)$-part. This answers a question of Nicholas Katz, who noticed the dimension doubling men…

秩为2的对数联络模空间的双有理描述

Birational description of moduli spaces of rank 2 logarithmic connections

本文通过分析抛物丛和表观奇点，证明射影空间与抛物丛模空间乘积的 Zariski 开子集可为秩2抛物对数联络模空间提供 Darboux 坐标，从而给出其双有理描述。

具有三个极点的射影直线上秩三对数联络的模空间

Moduli space of rank three logarithmic connections on the projective line with three poles

本文刻画了射影直线上带三个极点的秩三抛物对数联络模空间，证明其与 Sakai 分类中的 $A^{(1)*}_2$-曲面族同构，并借此揭示了表观奇点与底层抛物丛的关系。

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 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…