Hélène Esnault, Eckart Viehweg
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.
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We give an explicit formula for the determinant of the Gauß-Manin connection for an irregular connection on a Zariski open set of the projective line $¶^1_K$ over a function field $K$ over a field $k$ of characteristic zero.
Gauß-Manin determinant connections associated to irregular connections on a curve are studied. The determinant of the Fourier transform of an irregular connection is calculated. The determinant of cohomology of the standard rank 2 Kloosterman sheaf is computed modulo 2 torsion. Periods associated to irregular connections are studied in the very basic $\exp(f)$ case, and analogies with the Gauß-Manin determinant are discussed.
Unlocking the Hodge Conjecture: A Spectral Fingerprint Approach via Gauss-Manin Derivatives
本研究提出基于埃尔米特谱指纹的符号解析框架，通过全纯形式及其Gauss-Manin导数的投影构造恒零指纹泛函，证明投影导数张成正交补，结合算术几何结果在框架内解决霍奇猜想。
Non-vanishing of homotopy groups of Manin--Schechtman arrangements
本文研究Manin-Schechtman排列的补集，证明其高维同调群非零，从而推翻其为K(π,1)空间的假设。
Pour tout schéma simplicial complexe $X_{\bullet}$ il existe une application canonique $\nabla:H^{\ast}(X_{\bullet})\longrightarrow Ω^1_{{\mathbb C}/{\mathbb Q}}\otimes H^{\ast}(X_{\bullet})$, appelée la connexion de Gauß-Manin. Nous montrons qu'il existe une unique connexion fonctorielle sur toute structure de Hodge-Tate mixte ayant certaines propriétés de la connexion de Gauß-Manin. Cette connexion n'est pas intégrable en général, et alors son intégrabilité est une condition non triviale pour qu'une structure de Hodge soit géométrique. Dans des cas particuliers, je donne des formules explici…
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.
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.
Pour tout $t\in\mN$ nous définissons un certain entier positif $\N_t$ et nous conjecturons: si $H$ est un fibré de Gauss-Manin d'une fibration semi-stable alors la $t$-ème classe de Chern de $H$ est annulée par $\N_t$. Nous démontrons diverses conséquences de cette conjecture. For any $t\in\mN$, we define a certain positive integer $\N_t$ and we conjecture: if $H$ is a Gauss-Manin bundle of a semi-stable fibration then the $t$-th Chern class of $H$ is kiled by $\N_t$. We prove various consequences of this conjecture.
The Gauss-Manin system of an ICIS
本文研究了孤立完全交集奇点（ICIS）的Gauss-Manin系统及其Hodge filtration，揭示了其与广义Brieskorn格的关系，并将饱和Brieskorn格上的residue与Torrelli的b-function联系起来。
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.
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.
see the old abstract and the comments here.
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.
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.
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.
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.
We show that hypergeometric differential equations, unitary and Gauss-Manin connections give rise to de Rham cohomology sheaves which do not admit a Bloch-Ogus resolution. The latter is in contrast to Panin's theorem asserting that corresponding étale cohomology sheaves do fulfill Bloch-Ogus theory.
Our results concern analytic functions on the open unit $p$-adic poly-disc in $\mathbb{C}^n_p$ centered at the multiplicative unit and we prove that such functions only vanish at finitely many $n$-tuples of roots of unity $(ζ_1-1,\ldots,ζ_n-1)$ unless they vanish along a translate of the formal multiplicative group. For polynomial functions, this follows from the multiplicative Manin-Mumford conjecture. However we allow for a much wider class of analytic functions; in particular we establish a rigidity result for formal tori. Moreover, our methods apply to Lubin-Tate formal groups beyond just …
We associate to any convenient nondegenerate Laurent polynomial on the complex torus (C^*)^n a canonical Frobenius-Saito structure on the base space of its universal unfolding. According to the method of K. Saito (primitive forms) and of M. Saito (good basis of the Gauss-Manin system), the main problem, which is solved in this article, is the analysis of the Gauss-Manin system of the Laurent polynomial (or its universal unfolding) and of the corresponding Hodge theory.
Gauss-Manin System and the Virtual Structure Constants
本文应用Givental微分方程解决射影超曲面有理曲线计数问题，证明了前文关于量子上同调结构常数的猜想，并阐明了Calabi-Yau及一般型超曲面中虚拟结构常数与该微分方程的对应关系。
We give an explicit description of the canonical Frobenius structure attached (by the results of the first part of this article) to the polynomial f(u_0,...,u_n)=w_0u_0+...+w_nu_n restricted to the torus u_0^{w_0}...u_n^{w_n}=1, for any family of positive integers w_0,...,w_n such that gcd(w_0,...,w_n)=1.
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 …
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 $Ω^1_{X/S}({\rm log}Y)$.
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…
We explain how to construct a Frobenius structure on the parameter space of the universal unfolding of a Laurent polynomial using a result of C. Hertling and Y. Manin. This new approach greatly simplifies the (classic) one used in the paper "Gauss-Manin systems, Brieskorn lattices and Frobenius structures (I)", written with C. Sabbah.
The crystalline Chern classes of the value of a locally free crystal vanish on a smooth variety defined over a perfect field. Out of this we conclude new cases of de Jong's conjecture relating the geometric étale fundamental group of a smooth projective variety defined over a perfect field and the triviality of its category of isocrystals. We also discuss the case of the Gauß-Manin convergent isocrystal.
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。
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.
In this note, we investigate the Chern classes of flat bundles in the arithmetic Deligne Cohomology, introduced by Green-Griffiths, Asakura-Saito. We show nontriviality of the Chern classes in some cases and the proof also indicates that generic flat bundles (in its moduli space) can be expected to have nontrivial classes, which is in contrast with some Gauss-Manin systems.
If $β\in\CC^d$ is integral but not a strongly resonant parameter for the homogeneous matrix $A\in\ZZ^{d\times n}$ with $\ZZ A=\ZZ^d$, then the associated GKZ-system carries a naturally defined mixed Hodge module structure. We study here in the normal case the corresponding weight filtration by computing the intersection complexes with respective multiplicities on the associated graded parts. We do this by computing the weight filtration of a Gauss-Manin system with respect to a locally closed embedding of a torus inside an affine space. We then produce a result, based on a Fourier-Sato transfo…
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…
Newton Polyhedrons and Hodge Numbers of Non-degenerate Laurent Polynomials
研究无穷远牛顿多面体为单纯形的非退化便利洛朗多项式，利用堆叠扇上的逐锥多项式函数环刻画消失圈的极化混合霍奇结构，计算其霍奇菱形，并以初等方法解决相应Birkhoff问题。
This article gives a survey of recent results on a generalization of the notion of a Hodge structure. The main example is related to the Fourier-Laplace transform of a variation of polarizable Hodge structure on the punctured affine line, like the Gauss-Manin systems of a proper or tame algebraic function on a smooth quasi-projective variety. Variations of non-commutative Hodge structures often occur on the tangent bundle of Frobenius manifolds, giving rise to a tt* geometry.
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