Gauß-Manin determinants for rank one irregular connections on curves

Chern classes of Gauß-Manin bundles of weight 1 vanish

We show that the Chern character of a variation of polarized Hodge structures of weight one with nilpotent residues at $\infty$ dies up to torsion in the Chow ring, except in codimension 0.

Characteristic classes of flat bundles and determinant of the Gauss-Manin connection

The purpose of this note is to give a survey on recent progress on characteristic classes of flat bundles, and how they behave in a family.

Exponents of some one-dimensional Gauss-Manin cohomologies

In this paper we provide a purely algebraic characterization of the exponents of one-dimensional direct images of a structure sheaf by a rational function, related to the vanishing of the cohomologies of a certain Koszul complex associated with such a morphism. This can be extended to a more general family of Gauss-Manin systems. As an application, we calculate a set of possible exponents of the Gauss-Manin cohomology of some arrangements of hyperplanes with multiplicities, relevant to Dwork families and mirror symmetry.

Gauß-Manin connection and Tannaka duality

In analogy with the étale fundamental groups, we express the Gauß-Manin connection for $H^1$ in Tannaka terms. One difficulty is that unlike for fundamental groups, the Tannaka group scheme of relative connections, and the groupoid schemes of absolute connections and of connections on the base do not form an exact sequence. We added a theorem due to Pierre Deligne which improves the main result of the article. It allows to compare the Tannaka group scheme of relative connections with the kernel of the map from the groupoid scheme of absolute connections to the one of the base.

塔恩卡伊对偶与曲线族的高斯-曼金连接

Tannakian duality and Gauss-Manin connections for a family of curves

本文研究了光滑曲线族的微分基本群方案与高斯-曼金连接之间的关系，证明了在高斯-曼金连接中几何相对基本群的上同调映射是同构的，从而将高斯-曼金连接解释为微分基本群的上同调。

Gauss-Manin connections for arrangements, III Formal connections

We study the Gauss-Manin connection for the moduli space of an arrangement of complex hyperplanes in the cohomology of a complex rank one local system. We define formal Gauss-Manin connection matrices in the Aomoto complex and prove that, for all arrangements and all local systems, these formal connection matrices specialize to Gauss-Manin connection matrices.

Gauss-Manin connections for arrangements, II Nonresonant weights

We study the Gauss-Manin connection for the moduli space of an arrangement of complex hyperplanes in the cohomology of a nonresonant complex rank one local system. Aomoto and Kita determined this Gauss-Manin connection for arrangements in general position. We use their results and an algorithm constructed in this paper to determine this Gauss-Manin connection for all arrangements.

Gauss-Manin联络与Lie-Rinehart上同调

Gauss-Manin connections and Lie-Rinehart cohomology

本文为Lie-Rinehart代数上同调构造了广义Gauss-Manin联络，并以此构建了平坦联络K群间的Gysin映射，同时计算了超曲面奇点上的相关实例。

Gauss-Manin Connections for Arrangements

We construct a formal connection on the Aomoto complex of an arrangement of hyperplanes, and use it to study the Gauss-Manin connection for the moduli space of the arrangement in the cohomology of a complex rank one local system. We prove that the eigenvalues of the Gauss-Manin connection are integral linear combinations of the weights which define the local system.

Gauss-Manin stratification and stratified fundamental group schemes

We defined the Gauß-Manin stratification of a stratified bundle with respect to a smooth morphism and use it to study the homotopy sequence of stratified fundamental group schemes.

滤过与上同调 II：高斯-马宁联络

Filtrations and cohomology II: the Gauss-Manin connection

该研究用导出方法分析多类同调/上同调中的高斯-马宁联络，给出零幂不变性等问题的新方法，推广Bhatt特征零结果至混合特征，揭示与哈特肖恩复形的关联并解释特征零棱柱上同调特征。

Gauss-Manin connections for arrangements, IV Nonresonant eigenvalues

An arrangement is a finite set of hyperplanes in a finite dimensional complex affine space. A complex rank one local system on the arrangement complement is determined by a set of complex weights for the hyperplanes. We study the Gauss-Manin connection for the moduli space of arrangements of fixed combinatorial type in the cohomology of the complement with coefficients in the local system determined by the weights. For nonresonant weights, we solve the eigenvalue problem for the endomorphisms arising in the 1-form associated to the Gauss-Manin connection.

Calculation of mixed Hodge structures, Gauss-Manin connections and Picard-Fuchs equations

In this article we introduce algorithms which compute iterations of Gauss-Manin connections, Picard-Fuchs equations of Abelian integrals and mixed Hodge structure of affine varieties of dimension $n$ in terms of differential forms. In the case $n=1$ such computations have many applications in differential equations and counting their limit cycles. For $n>3$, these computations give us an explicit definition of Hodge cycles.

A note on the Gauss-Manin connection for abelian schemes

We study differential forms on the universal vector extension $A^\natural$ of an abelian scheme $A$ in characteristic zero, and derive a new construction of the $D$-group scheme structure on $A^\natural$. This gives, in particular, a rather simple description of the Gauss--Manin connection on the de Rham cohomology of $A$ in terms of global algebraic differential forms on $A^\natural$. The key ingredient is the computation of the coherent cohomology of $A^{\natural}$, due to Coleman and Laumon.

Connexion de Gauss-Manin associée à la déformationverselle de la singularité et zérosde l'intégrale hyperelliptique

One studies the system of differential equations satisfied by the hyperelliptic integral associated to a vanishing cycle defined for the versal deformation of the $A-μ$ singularity. As an application, the estimates on the multiplicity of zeros of the hyperelliptic integral are given as a linear function of the degree of present polynomials.

Gauss-Manin connections for p-adic families of nearly overconvergent modular forms

We interpolate the Gauss-Manin connection in p-adic families of nearly overconvergent modular forms. This gives a family of Maass-Shimura type differential operators from the space of nearly overconvergent modular forms of type r to the space of nearly overconvergent modular forms of type r + 1 with p-adic weight shifted by 2. Our construction is purely geometric, using Andreatta-Iovita-Stevens and Pilloni's geometric construction of eigencurves, and should thus generalize to higher rank groups.

Deep Learning Gauss-Manin Connections

The Gauss-Manin connection of a family of hypersurfaces governs the change of the period matrix along the family. This connection can be complicated even when the equations defining the family look simple. When this is the case, it is computationally expensive to compute the period matrices of varieties in the family via homotopy continuation. We train neural networks that can quickly and reliably guess the complexity of the Gauss-Manin connection of a pencil of hypersurfaces. As an application, we compute the periods of 96% of smooth quartic surfaces in projective 3-space whose defining equat…

Gauss-Manin Connections for Boundary Singularities and Isochore Deformations

We study here the relative cohomology and the Gauss-Manin connections associated to an isolated singularity of a function on a manifold with boundary, i.e. with a fixed hyperplane section. We prove several relative analogs of classical theorems obtained mainly by E. Brieskorn and B. Malgrange, concerning the properties of the Gauss-Manin connection as well as its relations with the Picard-Lefschetz monodromy and the asymptotics of integrals of holomorphic forms along the vanishing cycles. Finally, we give an application in isochore deformation theory, i.e. the deformation theory of boundary si…

Gauss-Manin connection in disguise: Quasi Jacobi forms of index zero

We consider the moduli space of abelian varieties with two marked points and a frame of the relative de Rham cohomolgy with boundary at these points compatible with its mixed Hodge structure. Such a moduli space gives a natural algebro-geometric framework for higher genus quasi Jacobi forms of index zero and their differential equations which are given as vector fields. In the case of elliptic curves we compute explicitly the Gauss-Manin connection and such vector fields.

The $p$-curvature conjecture for the non-abelian Gauss-Manin connection

Originally conjectured unpublished by Grothendieck, then formulated precisely by Katz, the $p$-curvature conjecture is a local-global principle for algebraic differential equations. It is at present open, though various cases are known. Katz subsequently proved this conjecture in a wide range of cases, for differential equations corresponding to the Gauss-Manin connection on algebraic de Rham cohomology. This dissertation addresses the non-abelian analogue of Katz' theorem, in the sense of Simpson's non-abelian Hodge theory, surveyed by Simpson and later developed in characteristic $p$ by Ogus…

A Boundedness theorem for nearby slopes of holonomic D-modules

Using twisted nearby cycles, we define a new notion of slopes for complex holonomic D-modules. We prove a boundedness result for these slopes, study their functoriality and use them to characterize regularity. For a family of (possibly irregular) algebraic connections E\_t parametrized by a smooth curve, we deduce under natural conditions an explicit bound for the usual slopes of the differential equation satisfied by the family of irregular periods of the E\_t. This generalizes the regularity of the Gauss-Manin connection proved by Katz and Deligne. Finally, we address some questions about an…

Gauss-Manin connection in disguise: Genus two curves

We describe an algebra of meromorphic functions on the Siegel domain of genus two which contains Siegel modular forms for an arithmetic index six subgroup of the symplectic group and it is closed under three canonical derivations of the Siegel domain. The main ingredients of our study are the moduli of enhanced genus two curves, Gauss-Manin connection and the modular vector fields living on such moduli spaces.

Gauss--Manin 联络延拓到全纯余容许 D-cap-模

Gauss--Manin connections extend to holonomic coadmissible D-cap-modules

本文证明光滑刚性解析簇上 Gauss--Manin 联络沿 Zariski 开浸入的推出是余容许且全纯的，推广了经典正则奇点定理，并借助 p-adic 方法与对数联络延拓性建立结论。

伪装的高斯-曼宁联络：全纯反常方程与高亏格实格罗莫夫-维滕不变量

Gauss-Manin Connection in Disguise: Holomorphic Anomaly Equations and Higher Genus Real Gromov-Witten Invariants

该研究构建扩展全纯反常方程的代数框架，以五次三维簇为模型求解方程并确定亏格5内的歧义性，得到整数型实格罗莫夫-维滕不变量预测值，还扩展框架计算双三次曲面亏格9内的对应预测值。

Contruction of quasi-invariant holomorphic parameters for the Gauss-Manin connection of a holomorphic map to a curve (second version)

In this paper we consider holomorphic families of frescos (i.e. filtered differential equations with a regular singularity) and we construct a locally versal holomorphic family for every fixed Bernstein polynomial. We construct also several holomorphic parameters (a holomorphic parameter is a function defined on a set of isomorphism classes of frescos) which are quasi-invariant by changes of variable. This is motivated by the fact that a fresco is associated to a relative de Rham cohomology class on a one parameter degeneration of compact complex manifolds, up to a change of variable in the pa…

Gauss-Manin connection in disguise: Calabi-Yau threefolds

We describe a Lie Algebra on the moduli space of Calabi-Yau threefolds enhanced with differential forms and its relation to the Bershadsky-Cecotti-Ooguri-Vafa holomorphic anomaly equation. In particular, we describe algebraic topological string partition functions $F_g^{alg}, g\geq 1$, which encode the polynomial structure of holomorphic and non-holomorphic topological string partition functions. Our approach is based on Grothendieck's algebraic de Rham cohomology and on the algebraic Gauss-Manin connection. In this way, we recover a result of Yamaguchi-Yau and Alim-Länge in an algebraic conte…

Gauss-Manin Connection in Disguise: Dwork Family

We study the moduli space $\textsf{T}$ of the Calabi-Yau $n$-folds arising from the Dwork family and enhanced with bases of the $n$-th de Rham cohomology with constant cup product and compatible with Hodge filtration. We also describe a unique vector field $\textsf{R}$ in $\textsf{T}$ which contracted with the Gauss-Manin connection gives an upper triangular matrix with some non-constant entries which are natural generalizations of Yukawa couplings. For $n=1,2$ we compute explicit expressions of $\textsf{R}$ and give a solution of $\textsf{R}$ in terms of quasi-modular forms. The moduli space …

The Gauss-Manin connection on the periodic cyclic homology

It is expected that the periodic cyclic homology of a DG algebra over the field of complex numbers (and, more generally, the periodic cyclic homology of a DG category) carries a lot of additional structure similar to the mixed Hodge structure on the de Rham cohomology of algebraic varieties. Whereas a construction of such a structure seems to be out of reach at the moment its counterpart in finite characteristic is much better understood thanks to recent groundbreaking works of Kaledin. In particular, it is proven by Kaledin that under some assumptions on a DG algebra $A$ over a perfect field …

Free products, cyclic homology, and the Gauss-Manin connection

We present a new approach to cyclic homology that does not involve the Connes differential and is based on a `noncommutative equivariant de Rham complex' of an associative algebra. The differential in that complex is a sum of the Karoubi-de Rham differential, which replaces the Connes differential, and another operation analogous to contraction with a vector field. As a byproduct, we give a simple explicit construction of the Gauss-Manin connection, introduced earlier by E. Getzler, on the relative cyclic homology of a flat family of associative algebras over a central base ring. We introduce …

手性Cartan微积分

Chiral Cartan calculus

本文研究手性de Rham复形上的Gauss-Manin联络，探讨其数学结构与性质。

A Riemann-Roch theorem for flat bundles, with values in the algebraic Chern-Simons theory

Let $f: X \to S$ be flat morphism over an algebraically closed field $k$ with a relative normal crossings divisor $Y\subset X$, $(E, \nabla)$ be a bundle with a connection with log poles along $Y$ and curvature with values in $f^*Ω^2_{k(S)}$. Then the Gauß-Manin sheaf $R^if_*(Ω^*_{X/S}({\rm log} Y)\otimes E)$ carries a Gauß-Manin connection $GM^i(\nabla)$. We establish a Riemann-Roch formula relating the algebraic Chern-Simons invariants of $\nabla$, $GM^i(\nabla)$ and the top Chern class of $Ω^1_{X/S}({\rm log}Y)$.

伪装中的Gauss-Manin联络：开Gromov-Witten不变量

Gauss-Manin connection in disguise: Open Gromov-Witten invariants

本文遵循H. Movasati的思想，构造了一个拟仿射空间来参数化镜像对称中的全纯圆盘及相关对象，并为相对de Rham上同调配备与混合Hodge结构相容的标架，同时计算了相应的模向量场。

Gauss-Manin connection in disguise: Noether-Lefschetz and Hodge loci

We give a classification of components of the Hodge locus in any parameter space of smooth projective varieties. This is done using determinantal varieties constructed from the infinitesimal variation of Hodge structures (IVHS) of the underlying family. As a corollary we prove that the minimum codimension for the components of the Hodge locus in the parameter space of $m$-dimensional hypersurfaces of degree $d$ with $d\geq 2+\frac{4}{m}$ and in a Zariski neighborhood of the point representing the Fermat variety, is obtained by the locus of hypersurfaces passing through an $\frac{m}{2}$-dimensi…

论Gauss-Manin联络与实奇点

On the Gauss-Manin Connection and Real Singularities

本研究针对实奇点构建对应微分方程组，证明其可确定Milnor纤维同调群，计算普通二次奇点情形并给出恢复任意孤立奇点最高同调群的条件，还基于奇点morsification构造了公钥加密方案。

Contruction of holomorphic parameters invariant by change of variable in the Gauss-Manin connection of an holomorphic map to a disc

When we consider a proper holomorphic map \ $\tilde{f}: X \to C$ \ of a complex manifold \ $X$ \ on a smooth complex curve \ $C$ \ with a critical value at a point \ $0$ \ in \ $C$, the choice of a local coordinate near this point allows to dispose of an holomorphic function \ $f$. Then we may construct, using this function, an (a,b)-modules structure on the cohomology sheaves of the formal completion (in \ $f$) \ of the complex of sheaves \ $(Ker\, df^{\bullet},d^{\bullet})$. These (a,b)-modules represent a filtered version of the Gauss-Manin connection of \ $f$. The most simple example of th…

Residues of connections and the Chevalley-Weil formula for curves

Given a finite group of automorphisms of a compact Riemann surface, the Chevalley-Weil formula computes the character valued Euler characteristic of an equivariant line bundle. The goal of this article is to give a proof by computing using residues of a Gauss-Manin connection.

Conformal blocks and the cohomology of configuration spaces of curves

We realize any space of conformal blocks attached to a punctured curve inside the cohomology of a configuration space of that curve and compare the WZW connection with the Gauss-Manin connection.

Unfolding of Orbifold LG B-Models: A Case Study

In this note we explore the variation of Hodge structures associated to the orbifold Landau-Ginzburg B-model whose superpotential has two variables. We extend the Getzler-Gauss-Manin connection to Hochschild chains twisted by group action. As an application, we provide explicit computations for the Getzler-Gauss-Manin connection on the universal (noncommutative) unfolding of $\mathbb{Z}_2$-orbifolding of A-type singularities. The result verifies an example of deformed version of Mckay correspondence.

Arrangements and Frobenius like structures

We consider a family of generic weighted arrangements of $n$ hyperplanes in $\C^k$ and show that the Gauss-Manin connection for the associated hypergeometric integrals, the contravariant form on the space of singular vectors, and the algebra of functions on the critical set of the master function define a Frobenius like structure on the base of the family. As a result of this construction we show that the matrix elements of the linear operators of the Gauss-Manin connection are given by the 2k+1-st derivatives of a single function on the base of the family, the function called the potential of…

Topological computation of the Stokes matrices of the weighted projective line $\mathbb{P}(1,3)$

The localized Fourier-Laplace transform of the Gauß-Manin system of $f\colon \mathbb{G}_m \to \mathbb{A}^1,\ x \mapsto x + x^{-3}$ is a $\mathcal{D}_{\mathbb{G}_m}$-module, having a regular singularity at $0$ and an irregular one at $\infty$. By mirror symmetry, it is closely related to the quantum connection of the weighted projective line $\mathbb{P}(1,3)$. Following results of A. D'Agnolo, M. Hien, G. Morando and C. Sabbah from 2017, we compute its Stokes multipliers at $\infty$ by purely topological methods. We compare it to the Gram matrix of the Euler-Poincaré pairing on $D^b(\text{Coh}(…

An algorithm of computing cohomology intersection number of hypergeometric integrals

We show that the cohomology intersection number of a twisted Gauss-Manin connection with regularization condition is a rational function. As an application, we obtain a new quadratic relation associated to period integrals of a certain family of K3 surfaces.