On the purely irregular fundamental group

Based on the (not yet fully understood analogy) between irregular connections and wild ramification, we define a purely irregular fundamental group for complex algebraic varieties and prove some results about this fundamental group which are analogous to the p‐adic étale fundamental group of algebraic varieties over fields of characteristic p (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Gauß-Manin determinants for rank 1 irregular connections on curves

Let be a smooth open curve over a field , where k is an algebraically closed field of characteristic 0. Let be a (possibly irregular) absolutely integrable connection on a line bundle L. A formula is given for the determinant of de Rham cohomology with its Gau?-Manin connection . The formula is expressed as a norm from the curve of a cocycle with values in a complex defining algebraic differential characters [7], and this cocycle is shown to exist for connections of arbitrary rank. Received: 13 September 1999 / Published online: 17 August 2001     

Rational homotopy theory for non-simply connected spaces

We construct an algebraic rational homotopy theory for all connected CW spaces (with arbitrary fundamental group) whose universal cover is rationally of finite type. This construction extends the classical theory in the simply connected case and has two basic properties: (1) it induces a natural equivalence of the corresponding homotopy category to the homotopy category of spaces whose universal cover is rational and of finite type and (2) in the algebraic category, homotopy equivalences are isomorphisms. This algebraisation introduces a new homotopy invariant: a rational vector bundle with a distinguished class of linear connections.

     

Ramification of valuations

In this paper we develop a very explicit theory of ramification of general valuations in algebraic function fields. In characteristic zero and arbitrary dimension, we obtain the strongest possible generalization of the classical ramification theory of local Dedekind domains. We further develop a ramification theory of algebraic functions fields of dimension two in positive characteristic. We prove that local monomialization and simultaneous resolution hold under very mild assumptions, and give pathological examples.     

Some properties of Malgrange isomonodromic deformations of linear 2 × 2 systems

We study movable singularities of the Malgrange isomonodromic deformation of a linear differential 2 × 2 system with two irregular singularities of Poincaré rank 1 and with an arbitrary number of Fuchsian singular points.     

Algebraic approximation of attractors of dynamical systems on manifolds

We consider an algebraic approximation of attractors of dynamical systems defined on a Euclidean space, a flat cylinder, and a projective space. We present the Foias-Temam method for the approximation of attractors of systems with continuous time and apply it to the investigation of Lorenz and Rössler systems. A modification of this method for systems with discrete time is also described. We consider elements of the generalization of the method to the case of an arbitrary Riemannian analytic manifold.     

Nöther-type theorem of piecewise algebraic curves on triangulation

The piecewise algebraic curve is a kind generalization of the classical algebraic curve. Nöther-type theorem of piecewise algebraic curves on the cross-cut partition is very important to construct the Lagrange interpolation sets for a bivariate spline space. In this paper, using the properties of bivariate splines, the Nöther-type theorem of piecewise algebraic curves on the arbitrary triangulation is presented.     

Representation of algebraic numbers by periodic branching continued fractions

The notion of a periodic branching continued fraction is a natural generalization of the notion of a periodic continued fraction. It is shown that for an arbitrary positive algebraic number one can construct a periodic branching continued fraction with natural elements convergent to this number.     

N(o)ther-type theorem of piecewise algebraic curves on triangulation

The piecewise algebraic curve is a kind generalization of the classical algebraic curve.N(o)ther-type theorem of piecewise algebraic curves on the cross-cut partition is very important to construct the Lagrange interpolation sets for a bivariate spline space. In this paper, using the properties of bivariate splines, the N(o)ther-type theorem of piecewise algebraic curves on the arbitrary triangulation is presented.     

Moduli spaces of meromorphic connections and quiver varieties

We describe the moduli spaces of meromorphic connections on trivial holomorphic vector bundles over the Riemann sphere with at most one (unramified) irregular singularity and arbitrary number of simple poles as Nakajima's quiver varieties. This result enables us to solve partially the additive irregular Deligne–Simpson problem.     

From Klein to Painleve Via Fourier, Laplace and Jimbo

We describe a method for constructing explicit algebraic solutionsto the sixth Painlevé equation, generalising that ofDubrovin and Mazzocco. There are basically two steps. Firstwe explain how to construct finite braid group orbits of triplesof elements of SL₂(C) out of triples of generators of three-dimensionalcomplex reflection groups. (This involves the Fourier–Laplacetransform for certain irregular connections.) Then we adapta result of Jimbo to produce the Painlevé VI solutions.(In particular, this solves a Riemann–Hilbert problemexplicitly.) Each step is illustrated using the complex reflection groupassociated to Klein's simple group of order 168. This leadsto a new algebraic solution with seven branches. We also provethat, unlike the algebraic solutions of Dubrovin and Mazzoccoand Hitchin, this solution is not equivalent to any solutioncoming from a finite subgroup of SL₂(C). The results of this paper also yield a simple proof of a recenttheorem of Inaba, Iwasaki and Saito on the action of Okamoto'saffine D₄ symmetry group as well as the correct connection formulaefor generic Painlevé VI equations. 2000 Mathematics SubjectClassification 34M55, 34M40, 20F55.     

Syzygies of Multi-Variable Higher Spin Dirac Operators on $$\mathbb{R}^{3}$$

This paper is a short report on the generalization of some results of our previous paper [12] to the case of spin j/2 Dirac operators in real dimension three for arbitrary odd integer j. We use an explicit formula for the local expression of such operators to study their algebraic properties, construct the compatibility conditions of the overdetermined system associated to the operator in several spatial variables, and we prove that its associated algebraic complex, dual do the BGG sequence coming from representation theory, has substantially the same pattern as the Cauchy-Fueter complex. The author is a member of the Eduard Čech Center and his research is supported by the relative grants.     

On the period length of the generalized Lagrange algorithm

The generalized Lagrange algorithm is a number geometric generalization of Lagrange's continued fraction method for computing fundamental unit and class number of real quadratic number fields. This algorithm yields a system of fundamental units and the class number of an arbitrary algebraic number field by means of computing cycles of reduced ideals. In this paper we prove that the cardinality of a cycle of reduced ideals in an ideal class of an order of an algebraic number field is O(R), where R is the regulator of this order, and where the O-constant only depends on the degree of the field. We also give a lower bound on this cardinality.     

Trace Functionals on Noncommutative Deformations of Moduli Spaces of Flat Connections

We describe an efficient construction of a canonical noncommutative deformation of the algebraic functions on the moduli spaces of flat connections on a Riemann surface. The resulting algebra is a variant of the quantum moduli algebra introduced by Alekseev, Grosse, and Schomerus and Buffenoir and Roche. We construct a natural trace functional on this algebra and show that it is related to the canonical trace in the formal index theory of Fedosov and Nest and Tsygan via Verlinde's formula.     

On Weyl structures

We characterize the relation between the geometrical properties of Weyl manifolds and the algebraic properties of the Weyl algebras (§1) and the deformation algebras associated to two conformal Weyl connections (§2). The last section is devoted to the study of the Weyl-Lyra algebras associated to a conformal Weyl connection and a conformal semisymmetric connection.     

Closed Conjugacy Classes,Closed Orbits and Structure of Algebraic Groups

In this paper we prove that if an affine algebraic group (in characteristic zero) has all its conjugacy classes closed, then it is nilpotent. A classical result (called sometimes the Kostant-Rosenlicht Theorem) guarantees that if an affine algebraic group G is unipotent, then all its orbits on affine varieties are closed. We prove the converse of that theorem in arbitrary characteristics.     

Symmetric tensor fields of bounded deformation

We introduce and study spaces of symmetric tensor fields of bounded deformation for tensors of arbitrary order, i.e., where the symmetrized derivative is still a Radon measure. A Sobolev–Korn type estimate, a boundary trace theorem and continuous as well as compact embedding properties into Lebesgue spaces are obtained, showing that these spaces can be regarded as a natural generalization of the spaces of bounded deformation to higher-order symmetric tensors.     

Patchworking singular algebraic curves II

In this paper we present various particular versions of the general patch-working procedurefor the construction of reduced algebraic curves with prescribed singularities, on algebraic surfaces. Among the main examples are a deformation of reducible algebraic curves on reducible algebraic surfaces in the presence of non-transverse intersections of a curve with the singular locus of a surface, and a deformation of curves with multiple components. As an application we deduce, a significant asymptotical improvement for the sufficient existence criterion of algebraic curves with arbitrary prescribed singlarities in given linear systems on smooth projective algebraic surfaces. Both authors were partially supported by the Herman Minkowsky-Minerva Center for Geometry at Tel Aviv University, and by grant no. G-616-15.6/99 from the German-Israeli Foundation for Research and Development. The first author was also supported by the Bessel Research Award from the Alexander von Humboldt Foundation.     

Complete reducibility and Steinberg endomorphisms

Let G be a connected reductive algebraic group defined over an algebraically closed field of positive characteristic. We study a generalization of the notion of G-complete reducibility in the context of Steinberg endomorphisms of G. Our main theorem extends a special case of a rationality result in this setting.     

Cluster Algebras of Finite Type via Coxeter Elements and Principal Minors

We give a uniform geometric realization for the cluster algebra of an arbitrary finite type with principal coefficients at an arbitrary acyclic seed. This algebra is realized as the coordinate ring of a certain reduced double Bruhat cell in the simply connected semisimple algebraic group of the same Cartan–Killing type. In this realization, the cluster variables appear as certain (generalized) principal minors.