Gauss–Man i n Conne ction s for Arrangements, I E igenvalues

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.     

Irregularity and the period determinant for irregular singular connections on surfaces

In [6], we constructed a period pairing for flat irregular singular conncetions on surfaces. We now extend these constructions to a perfect period pairing between the irregularity complex of the connection and the complex of relative rapid decay chains. As a consequence, the period determinant of the connection decomposes as a product of a topological period determinant on the open surface and a determinant coming from the irregularity of the connection. Additionally, we deduce a method to compute the irregularity sheaf up to local isomorphism in topological terms (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The associated map of the nonabelian Gauss-Manin connection

The Gauss-Manin connection for nonabelian cohomology spaces is the isomonodromy flow. We write down explicitly the vector fields of the isomonodromy flow and calculate its induced vector fields on the associated graded space of the nonabelian Hogde filtration. The result turns out to be intimately related to the quadratic part of the Hitchin map.     

On the relative connections

Abstract

We investigate relative connections on a sheaf of modules. A sufficient condition is given for the existence of a relative holomorphic connection on a holomorphic vector bundle over a complex analytic family. We show that the relative Chern classes of a holomorphic vector bundle admitting relative holomorphic connection vanish, if each of the fiber of the complex analytic family is compact and Kähler.     

Universal extension crystals of 1-motives and applications

We use the crystalline nature of the universal extension of a 1-motive M to define a canonical Gauss-Manin connection on the de Rham realization of M. As an application we provide a construction of the so-called Manin map from a motivic point of view.     

Valuations with preassigned proximity relations

We characterize the infinite upper triangular matrices (which we call formal proximity matrices) that can arise as proximity matrices associated with zero-dimensional valuations dominating regular noetherian local rings. In particular, for every regular noetherian local ring R of the appropriate dimension, we give a sufficient condition for such a formal proximity matrix to be the proximity matrix associated with a real rank one valuation dominating R. Furthermore, we prove that in the special case of rational function fields, each formal proximity matrix arises as the proximity matrix of a valuation whose value group is computable from the formal proximity matrix. We also give an example to show that this is false for more general fields. Finally in the case of characteristic zero, our constructions can be seen as a particular case of a structure theorem for zero-dimensional valuations dominating equicharacteristic regular noetherian local rings.     

Center conditions: rigidity of logarithmic differential equations

In this paper, we prove that any degree d deformation of a generic logarithmic polynomial differential equation with a persistent center must be logarithmic again. This is a generalization of Ilyashenko's result on Hamiltonian differential equations. The main tools are Picard-Lefschetz theory of a polynomial with complex coefficients in two variables, specially the Gusein-Zade/A'Campo's theorem on calculating the Dynkin diagram of the polynomial, and the action of Gauss-Manin connection on the so-called Brieskorn lattice/Petrov module of the polynomial. We will also generalize J.P. Francoise recursion formula and (∗) condition for a polynomial which is a product of lines in a general position. Some applications on the cyclicity of cycles and the Bautin ideals will be given.     

Periods for irregular singular connections on surfaces

Given an integrable connection on a smooth quasi-projective algebraic surface U over a subfield k of the complex numbers, we define rapid decay homology groups with respect to the associated analytic connection which pair with the algebraic de Rham cohomology in terms of period integrals. These homology groups generalize the analogous groups in the same situation over curves defined by Bloch and Esnault. In dimension two, however, new features appear in this context which we explain in detail. Assuming a conjecture of Sabbah on the formal classification of meromorphic connections on surfaces (known to be true if the rank is lower than or equal to 5), we prove perfectness of the period pairing in dimension two.     

The uniqueness of tangent cones for Yang–Mills connections with isolated singularities

We proved a uniqueness theorem of tangent connections for a Yang–Mills connection with an isolated singularity with a quadratic growth of the curvature at the singularity. We also obtained control over the rate of the asymptotic convergence of the connection to the tangent connection if furthermore the connection is stationary or the tangent connection is integrable, with a stronger result in the latter case. There are parallel results for the cones at infinity of a Yang–Mills connection on an asymptotically flat manifold. We also gave an application of our methods to the Yang–Mills flow and proved that the Yang–Mills flow exists for all time and has asymptotic limit if the initial value is close to a smooth local minimizer of the Yang–Mills functional.     

Calculation of local Fourier transforms for formal connections

We calculate the local Fourier transforms for formal connections. In particular, we verify some formulas analogous to a conjecture of Laumon and Malgrange for-adic local Fourier transforms.     

Hypergeometric period for a tame polynomial

We analyze the Gauss-Manin system of differential equations (and its Fourier transform) attached to regular functions satisfying a tameness assumption on a smooth affine variety over C (e.g. tame polynomials on Cⁿ⁺¹). We give a solution to the Birkhoff problem for this system and prove Hodge-type results analogous to those existing for germs of isolated hypersurface singularities. We deduce a formula for the determinant of the “Aomoto complex”.     

A classification of locally homogeneous affine connections on compact surfaces

We classify the affine connections on compact orientable surfaces for which the pseudogroup of local isometries acts transitively. We prove that such a connection is either torsion-free and flat, the Levi–Civita connection of a Riemannian metric of constant curvature or the quotient of a translation-invariant connection in the plane. This refines previous results by Opozda.     

Type-II matrices in weighted Bose–Mesner algebras of ranks 2 and 3

Type-II matrices are nonzero complex matrices that were introduced in connection with spin models for link invariants. Type-II matrices have been found in connection with symmetric designs, sets of equiangular lines, strongly regular graphs, and some distance regular graphs. We investigate weighted complete and strongly regular graphs, and show that type-II matrices arise in this setting as well.     

Analytic methods in quantum computing

Here we consider a model of quantum computation, based on the monodromy representation of a Fuchsian system. The rôle of local and entangling operators in monodromic quantum computing is played by monodromy matrices of connections with logarithmic singularities acting on the fiber of a holomorphic vector bundle as on the space of qubits. The leading theme is the problem of constructing a set of universal gates as monodromy operators induced from a connection with logarithmic singularity. In the formal scheme developed by us, already known models — topological and holonomic — can be incorporated.     

Computing ECT-B-splines recursively

ECT-spline curves for sequences of multiple knots are generated from different local ECT-systems via connection matrices. Under appropriate assumptions there is a basis of the space of ECT-splines consisting of functions having minimal compact supports, normalized to form a nonnegative partition of unity. The basic functions can be defined by generalized divided differences [24]. This definition reduces to the classical one in case of a Schoenberg space. Under suitable assumptions it leads to a recursive method for computing the ECT-B-splines that reduces to the de Boor–Mansion–Cox recursion in case of ordinary polynomial splines and to Lyche's recursion in case of Tchebycheff splines. For sequences of simple knots and connection matrices that are nonsingular, lower triangular and totally positive the spline weights are identified as Neville–Aitken weights of certain generalized interpolation problems. For multiple knots they are limits of Neville–Aitken weights. In many cases the spline weights can be computed easily by recurrence. Our approach covers the case of Bézier-ECT-splines as well. They are defined by different local ECT-systems on knot intervals of a finite partition of a compact interval [a,b] connected at inner knots all of multiplicities zero by full connection matrices A ^[i] that are nonsingular, lower triangular and totally positive. In case of ordinary polynomials of order n they reduce to the classical Bézier polynomials. We also present a recursive algorithm of de Boor type computing ECT-spline curves pointwise. Examples of polynomial and rational B-splines constructed from given knot sequences and given connection matrices are added. For some of them we give explicit formulas of the spline weights, for others we display the B-splines or the B-spline curves. *Supported in part by INTAS 03-51-6637.     

基于复数理论的同异型联系数及其应用

虽然联系数中的i与复数中的i有不同的含义,但所定义的同异型联系数与复数在形式上完全相同.为此,依据复数理论给出了同异型联系数的三角函数与指数函数两种表述形式及其互相转换,举例说明其应用,从而为发展联系数理论提供了新的途径.     

Spectra of Cayley graphs of complex reflection groups

Renteln proved that the eigenvalues of the distance matrix of a Cayley graph of a real reflection group with respect to the set of all reflections are integral and provided a combinatorial formula for some such spectra. We prove the eigenvalues of the distance, adjacency, and codimension matrices of Cayley graphs of complex reflection groups with connection sets consisting of all reflections are integral and provide a combinatorial formula for the codimension spectra for a family of monomial complex reflection groups.     

Some Gauge-natural operators on linear connections

We determine explicitly all geometrical operators transforming a linear connection on a vector bundle :EM and a classical linear connection on the base manifoldM into a classical linear connection on the total spaceE.     

Relative connections on principal bundles and relative equivariant structures

We investigate relative holomorphic connections on a principal bundle over a family of compact complex manifolds. A sufficient condition is given for the existence of a relative holomorphic connection on a holomorphic principal bundle over a complex analytic family. We also introduce the notion of relative equivariant bundles and establish its relation with relative holomorphic connections on principal bundles.