Painlevé equations and the middle convolution

On the $\ell$-adic Fourier transform and the determinant of the middle convolution

We study the relation of the middle convolution to the $\ell$-adic Fourier transformation in the étale context. Using Katz' work and Laumon's theory of local Fourier transformations we obtain a detailed description of the local monodromy and the determinant of Katz' middle convolution functor $\MC_χ$ in the tame case. The theory of local $ε$-constants then implies that the property of an étale sheaf of having an at most quadratic determinant is often preserved under $\MC_χ$ if $χ$ is quadratic.

李代数表示中的中卷积

Middle convolution for Lie algebra representations

本文提出李代数的中卷积函子，将其视为Long-Moody函子的推广，并展示其在Fuchsian系统和logarithmic connections中的应用，最终建立holonomy李代数与局部系统间的Riemann-Hilbert对应。

Behaviour of some Hodge invariants by middle convolution

Following an article of Dettweiler and Sabbah, this article studies the behaviour of various Hodge invariants by middle additive convolution with a Kummer module. The main result gives the behaviour of the nearby cycle local Hodge numerical data at infinity. We also give expressions for Hodge numbers and degrees of some Hodge bundles without making the hypothesis of scalar monodromy at infinity, which generalizes the resultsof Dettweiler and Sabbah.

On the middle convolution

We present a cohomological interpretation of the middle convolution functor MC and find an explicit Riemann-Hilbert correspondence for MC_λ. This leads to an algorithm for the construction of Fuchsian systems which correspond to irreducible rigid local systems under the Riemann-Hilbert correspondence. Also, the effect of MC_λon the p-curvatures ist determined and new examples of differential systems are found which satisfy the Grothendieck-Katz p-curvature conjecture.

Katz's middle convolution algorithm

This is an expository account of Katz's middle convolution operation on local systems over ${\bf P}^1-\{q\_1,..., q\_n\}$. We describe the Betti and de Rham versions, and point out that they give isomorphisms between different moduli spaces of local systems, following Völklein, Dettweiler-Reiter, Haraoka-Yokoyama. Kostov's program for applying the Katz algorithm is to say that in the range where middle convolution no longer reduces the rank, one should give a direct construction of local systems. This has been done by Kostov and Crawley-Boevey. We describe here an alternative construction usin…

关于具有好直线的超平面排列的Riemann-Hilbert问题

Middle convolution for affine hyperplane arrangements and Aomoto realizability

研究超平面排列补集上的Riemann-Hilbert问题变体，通过推广Katz的中部卷积为函子，证明其保持该问题的可解性。

通过p进霍奇理论与朗兰兹对应构造4个穿孔的复射影直线上的$\text{GL}_2$型阿贝尔簇族及其在Painleve VI方程代数解中的应用

Constructing families of abelian varieties of $\text{GL}_2$-type over $4$-punctured complex projective line via $p$-adic Hodge theory and Langlands correspondence and application to algebraic solutions of Painleve VI equation

本研究借助p进霍奇理论与朗兰兹对应，构造了4穿孔射影直线上无穷多族带特定坏约化的$\text{GL}_2$型阿贝尔簇，结合相关猜想证明其可由椭圆曲线泛族挠截面参数化，且可导出Painleve VI方程的代数解。

Arithmetic of degenerating principal variations of Hodge structure: examples arising from mirror symmetry and middle convolution

We collect evidence in support of a conjecture of Griffiths, Green and Kerr on the arithmetic of extension classes of limiting mixed Hodge structures arising from semistable degenerations over a number field. After briefly summarizing how a result of Iritani implies this conjecture for a collection of hypergeometric Calabi-Yau threefold examples studied by Doran and Morgan, the authors investigate a sequence of (non-hypergeometric) examples in dimensions 1 through 6 arising from Katz's theory of the middle convolution. A crucial role is played by the Mumford-Tate group (of type G2) of the fami…

Halphen's transform and middle convolution

We show that the Halphen transform of a Lamé equation can be written as the symmetric square of the Lamé equation followed by an Euler transform. We use this to compute a list of Lamé equations with arithmetic Fuchsian monodromy group. It contains all those Lamé equations where the quaternion algebra $A$ over $k$ associated to the arithmetic Fuchsian group is a quaternion algebra $A$ over $\mathbb{Q}$. Further we classify all geometric braid group orbits in $\rm{SL}_2(\ZZ)^4$ with the possible exception of three orbits.

算术型完整全纯D-模与中间卷积

Holonomic $\mathcal{D}$-modules of arithmetic type and middle convolution

本文建立将G-联络扩展为完整全纯D-模的框架并证明其导出范畴具有格罗滕迪克六函子形式体系，由此得到中间卷积关于全局逆半径的稳定性，进而证明刚性Fuchs系统上多种算术性质等价，肯定回答了André与Baldassarri的猜想。

Hodge theory of the middle convolution

We compute the behaviour of Hodge data by tensor product with a unitary rank-one local system and middle convolution by a Kummer unitary rank-one local system for an irreducible variation of polarized complex Hodge structure on a punctured complex affine line. We give applications of these formulas to local systems with G_2-monodromy.

Generalized Kuga-Satake theory and rigid local systems, I: the middle convolution

This note explains an approach to producing examples of 'generalized Kuga-Satake theory' based on establishing special cases of Simpson's conjecture that rigid local systems are motivic. This strategy is then carried out, using work of Bogner and Reiter, in a low-dimensional example, producing the first non-trivial examples of generalized Kuga-Satake theory outside of the Tannakian category of motives generated by abelian varieties. For a more robust collection of examples, see the sequel "Generalized Kuga-Satake theory and rigid local systems, II: rigid Hecke eigensheaves."

On globally nilpotent differential equations

In a previous work of the authors, a middle convolution operation on the category of Fuchsian differential systems was introduced. In this note we show that the middle convolution of Fuchsian systems preserves the property of global nilpotence. This leads to a globally nilpotent Fuchsian system of rank two which does not belong to the known classes of globally nilpotent rank two systems. Moreover, we give a globally nilpotent Fuchsian system of rank seven whose differential Galois group is isomorphic to the exceptional simple algebraic group of type $G_2.$

枚举 $SL_2(\C)$ -特征簇上的有限辫群轨道

Enumerating Finite Braid Group Orbits on $SL_2(\C)$-Character Varieties

研究辫群在 $n$ 次穿孔球面的 $SL_2(\C)$ -特征簇上的有限轨道，基于中间卷积和有限复反射群的结果，运用卡茨中间卷积分类有限轨道，给出非本原群相关理论结果、本原群计算结果及相关计算机代码。

关于特殊线性群的伽罗瓦实现

On Galois realizations of special linear groups

本文通过研究中间卷积构造的平展层的行列式，给出在有理数域上将特殊线性群正则实现为伽罗瓦群的方法。

特权局部系统的Grothendieck-Katz猜想

The Grothendieck-Katz Conjecture for privileged local systems

本文定义了光滑拟射影复簇上具有固定边界拟单值单性的特权局部系统，推广了Katz在射影直线上的物理刚性概念，并给出避免中间卷积的新证明，从而在高维上验证了p-曲率猜想。