Gauss-Manin Systems Research Articles

On Multivariate Picard–Fuchs Systems and Equations

In this paper, we studied the Picard–Fuchs systems and equations which appear in the theory of Gauss–Manin systems and connections associated with deformations of isolated singularities. Among other things, we describe some interesting properties of such systems and relationships between them. Then we show how to calculate the fundamental solutions to the Gauss–Manin system for Aμ-singularities and to the corresponding generalized Legendre equations in terms of the multidimensional Horn’s hypergeometric functions. In conclusion, some important questions concerning basic properties of the local and global Picard–Fuchs systems of Pfaffian type, involving integrability conditions and commuting relations, are discussed in some detail.

IRREGULAR HODGE FILTRATION OF SOME CONFLUENT HYPERGEOMETRIC SYSTEMS

AbstractWe determine the irregular Hodge filtration, as introduced by Sabbah, for the purely irregular hypergeometric${\mathcal{D}}$-modules. We obtain, in particular, a formula for the irregular Hodge numbers of these systems. We use the reduction of hypergeometric systems from GKZ-systems as well as comparison results to Gauss–Manin systems of Laurent polynomials via Fourier–Laplace and Radon transformations.

A-hypergeometric modules and Gauss–Manin systems

Let A be a d×n integer matrix. Gel′fand et al. proved that most A-hypergeometric systems have an interpretation as a Fourier–Laplace transform of a direct image. The set of parameters for which this happens was later identified by Schulze and Walther as the set of not strongly resonant parameters of A. A similar statement relating A-hypergeometric systems to exceptional direct images was proved by Reichelt. In this article, we consider a hybrid approach involving neighborhoods U of the torus of A and consider compositions of direct and exceptional direct images. Our main results characterize for which parameters the associated A-hypergeometric system is the inverse Fourier–Laplace transform of such a “mixed Gauss–Manin” system.In order to describe which U work for such a parameter, we introduce the notions of fiber support and cofiber support of a D-module.If the semigroup ring C[NA] is normal, we show that every A-hypergeometric system is “mixed Gauss–Manin”. We also give an explicit description of the neighborhoods U which work for each parameter in terms of primitive integral support functions.

Exponents of Some One-Dimensional Gauss–Manin Systems

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.

Exponentially Twisted Cyclic Homology

By a theorem of Bernhard Keller the de Rham cohomology of a smooth variety is isomorphic to the periodic cyclic homology of the differential graded category of perfect complexes on the variety. Both the de Rham cohomology and the cyclic homology can be twisted by the exponential of a regular function on the variety. We explain that the isomorphism holds true in the twisted setting and draw some conclusions on derived invariance of the algebraic Gauss-Manin systems associated with regular functions.

ON THE IRREGULAR HODGE FILTRATION OF EXPONENTIALLY TWISTED MIXED HODGE MODULES

Given a mixed Hodge module$\mathcal{N}$and a meromorphic function$f$on a complex manifold, we associate to these data a filtration (the irregular Hodge filtration) on the exponentially twisted holonomic module$\mathcal{N}\otimes \mathcal{E}^{f}$, which extends the construction of Esnaultet al.($E_{1}$-degeneration of the irregular Hodge filtration (with an appendix by Saito),J. reine angew. Math.(2015), doi:10.1515/crelle-2014-0118). We show the strictness of the push-forward filtered${\mathcal{D}}$-module through any projective morphism${\it\pi}:X\rightarrow Y$, by using the theory of mixed twistor${\mathcal{D}}$-modules of Mochizuki. We consider the example of the rescaling of a regular function$f$, which leads to an expression of the irregular Hodge filtration of the Laplace transform of the Gauss–Manin systems of$f$in terms of the Harder–Narasimhan filtration of the Kontsevich bundles associated with$f$.

Brieskorn modules and Gauss-Manin systems for non-isolated hypersurface singularities

We study the Brieskorn modules associated to a germ of a holomorphic function with non-isolated singularities and show that the Brieskorn module has naturally the structure of a module over the ring of microdifferential operators of non-positive degree, and that the kernel of the morphism to the Gauss–Manin system coincides with the torsion part for the action of t and also with that for the action of the inverse of the Gauss–Manin connection. This torsion part is not finitely generated in general, and a sufficient condition for the finiteness is given here. A Thom–Sebastiani-type theorem for the sheaf of Brieskorn modules is also proved when one of two functions has an isolated singularity.

Irregularity of an analogue of the Gauss-Manin systems

Dans la theorie des D-modules, on definit les systemes de Gauss-Manin par l'image directe par un morphisme du faisceau structural O. Un resultat essentiel est leur regularite. On s'interesse a l'irregularite d'un analogue des systemes de Gauss-Manin. Il s'agit de l'image directe f+(Oe g ) par un polyome f d'un D-module tordu par une exponentielle d'un second polynome g, ou f et g sont des polynomes a deux variables. Les analogues des systemes de Gauss-Manin peuvent avoir des singularites irregulieres. On exprimera alors un invariant attache a l'irregularite en c ∈P 1 de ces systemes a l'aide de la geometrie de l'application (f, g).

Picard–Fuchs Equations and Gauss–Manin Systems with a View Towards the Riemann–Hilbert Problem

We study Abelian integrals associated with a tame polynomial function and their Picard---Fuchs equations using the theory of algebraic Gauss---Manin systems. Especially, we look for a basis of the Petrov module, in which the Picard---Fuchs equations become as simple as possible. As an application, we discuss the related Riemann---Hilbert problem and prove that it has a positive answer under some conditions. In this case, we compute the Jordan normal form of the residue matrices of the corresponding Fuchsian system in terms of local data.

Gauss-Manin systems, Brieskorn lattices and Frobenius structures (I)

We associate to any convenient nondegenerate Laurent polynomial f 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 f (or its universal unfolding) and of the corresponding Hodge theory.

Algebraic Gauss-Manin systems and Brieskorn modules

We study the algebraic Gauss-Manin system and the algebraic Brieskorn module associated to a polynomial mapping with isolated singularities. Since the algebraic Gauss-Manin system does not contain any information on the cohomology of singular fibers, we first construct a nonquasi-coherent sheaf which gives the cohomology of every fiber. Then we study the algebraic Brieskorn module, and show that its position in the algebraic Gauss-Manin system is determined by a natural map to quotients of local analytic Gauss-Manin systems, and its pole part by the vanishing cycles at infinity, comparing it with the Deligne extension. This implies for example a formula for the determinant of periods. In the two-dimensional case we can describe the global structure of the algebraic Gauss-Manin system rather explicitly.

Hyperbolic Cauchy problem and Leray's residue formula

We give an algebraic description of wave fronts that appear in strictly hyperbolic Cauchy problem. Concrete form of definig function of wave front issued from initial algebraic variety is obtained by the aid of the Gauss-Manin systems satisfied by Gel'fand -Leray integrals.

On geometry of fronts in wave propagations

We give a geometric descriptions of (wave) fronts in wave propagation processes. Concrete form of defining function of wave front issued from initial algebraic variety is obtained by the aid of Gauss-Manin systems associated with certain complete intersection singularities. In the case of propagations on the plane, we get restrictions on types of possible cusps that can appear on the wave front. 0. Introduction. In this note, we intend to develop a preparatory theory toward asymptotic analysis around singular loci that appear in wave propagation process with initial data whose C∞ singularities are located on certain smooth algebraic variety. For this purpose, we study the geometry of the (wave) front, i.e. the singular support S.S. of the solutions to the Cauchy problem where it loses C∞ smoothness. We give a concrete expression of defining function of the wave front in terms of the initial data (Theorem 10). We impose conditions on the initial data so that they are defined by a quasihomogeneous polynomial with isolated singularities. The key trick is to understand the (wave) front as a discriminantal set for a deformation of complete intersection singularity. This point of view has been exposed in [20]. As is well known, the fundamental solutions of hyperbolic operators are expressed by means of certain kinds of Gel′fand-Leray integrals. Thus the geometry of singular support (i.e. front) of the solutions is reduced to the study of singular support of these types of integrals. Homological and cohomological approach to the analysis of sharp-diffuse type of these integrals has been initiated by [3]. V. A. Vassiliev [21] uses F. Pham’s approach [15] analyzing homology classes to describe the ramification character of certain kinds of Gel′fand-Leray integrals, consequently to clarify the sharp-diffuse type (see Section 4 for 1991 Mathematics Subject Classification: Primary 35L25, 58G17, 33C75; Secondary 32S40, 78A05, 33C20. The paper is in final form and no version of it will be published elsewhere.

Duality for hypergeometric functions and invariant Gauss-Manin systems

We present some basic identities for hypergeometric functions associatedwith the integrals of Euler type. We give a geometrical proof for formulaesuch as the identity between the single and double integrals expressingAppell's hypergeometric series F1 (a, b, b' c; x, y).

Computing Gauss–Manin Systems for Complete Intersection Singularities Sμ

Abstract The Gauss–Manin systems with coefficients having logarithmic poles along the discriminant sets of the principal deformations of complete intersection quasihomogeneous singularities Sμ are calculated. Their solutions in the form of generalized hypergeometric functions are presented.

Computing Gauss-Manin systems for complete intersection singularitiesS μ

The Gauss-Manin systems with coecients having log- arithmic poles along the discriminant sets of the principal deforma- tions of complete intersection quasihomogeneous singularities Sn are calculated. Their solutions in the form of generalized hypergeometric functions are presented.

Hodge filtrations on Gauss-Manin systems, II

Hodge filtrations on Gauss-Manin systems, II