Existence of Moduli for Bi-Lipschitz Equivalence of Analytic Functions

We show that the bi-Lipschitz equivalence of analytic function germs (², 0)(, 0) admits continuous moduli. More precisely, we propose an invariant of the bi-Lipschitz equivalence of such germs that varies continuously in many analytic families f _t : (², 0)(, 0). For a single germ f the invariant of f is given in terms of the leading coefficients of the asymptotic expansions of f along the branches of generic polar curve of f.     

Surjectivity of quotient maps for algebraic (ℂ,+)-actions and polynomial maps with contractible fibers

In this paper we establish two results concerning algebraic (,+)-actions on ⁿ . First, let be an algebraic (,+)-action on ³. By a result of Miyanishi, its ring of invariants is isomorphic to [t ₁,t ₂]. Iff ₁,f ₂ generate this ring, the quotient map of is the mapF:³²,x(f ₁(x), f₂(x)). By using some topological arguments we prove thatF is always surjective. Secon, we are interested in dominant polynomial mapsF: ^{n n-1} whose connected components of their generic fibers are contractible. For such maps, we prove the existence of an algebraic (,+)-action on ⁿ for whichF is invariant. Moreover we give some conditions so thatF*([t ₁,...,t _n-1 ]) is the ring of invariants of .Dedicated to all my friends and my family     

On a global descent method for polynomials

Summary Given a complex polynomialp we determine a functionf _p : such that |p(f _p (z))||p(z)|,z withk<1. This result is used to introduce a global root-finding algorithm for polynomials.     

Über einen Satz von Shl. Sternberg in der Theorie der analytischen Iterationen

Let X=X ₁,...,X _n be the ring of formal power series inn indeterminates over . LetF:XAX+B(X)=(F ⁽¹⁾(X),...,F ⁽ⁿ⁾(X))(X) ⁿ denote an automorphism of X and let ₁,..., _n be the eigenvalues of the linear partA ofF. We will say thatF has an analytic iteration (a. i.) if there exists a family (F _t (itX)) _t of automorphisms such thatF _t(X) has coefficients analytic int and such thatF ₀=X,F ₁=F,F _t+t=F_tF_t for allt,t. Let now a set=(ln₁,...,ln _n ) of determinations of the logarithms be given. We ask if there exists an a. i. ofF such that the eigenvalues of the linear partA(t) ofF _t(X) are . We will give necessary and sufficient conditions forF to have such an a. i., namely thatF is conjugate to a semicanonical formN=T ^–1FT such that inN ^(k)(X) there appear at most monomialsX ₁ ¹ ...X _n ⁿ . This generalizes a result of Shl.Sternberg.

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Herrn Prof. Dr. E. Hlawka zum 60. Geburtstag gewidmet     

Spectral functions of zeros in the Besselq-functions

Zeta functions _v(z; q)= _n=1 [j_vn(q)]^–z and partition functions Z_v(t; q)=_n exp[–tj _vn ² (q)] related to the zeros j_vn(q) of the Bessel q-functions J_v(x; q) and J _v ⁽²⁾ (x; q) are studied and explicit formulas for _v(2n; q) at n=±1, ±2, ... are obtained. The poles of _v(z; q) in the complex plane and the corresponding residues are found. Asymptotics of the partition functions Z_v(t; q) at t 0 are investigated.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 3, pp. 397–414, June, 1996.     

On the Ordered K0‐Group of Universal, Free Product C*‐Algebras

The ordered K₀-group of the universal, unital free product C^*-algebra M_k()*M_l()is calculated in the case where k is prime and not a divisor in l. It is shown that the positive cone of K₀(M_k()*M_l())is as small as possible in this case. The article also contains results (full and partial) on the ordered K₀0-group of more general universal, unital free product C^* algebras.     

Charakterisierungen einiger Funktionenalgebren

One main result of this article is a characterization of all(G ₁) as topological algebras withG ₁ open in. For this and for similar results of Arens [4], Carpenter [6] and Brooks [5] Runge's approximation theorem is an important tool. It is extended to a characterization of all (G), whereG is a polynomially convex, open subset of a ^m .There is stated a similar characterization of allC(G) withG open in a ^m ,which is based on approximation by polynomials in theZ _j . and . A second main result is a characterization of allC(M),whereM is a paracompact manifold of even dimension and which proceeds from ideas in the article [3]. MoreoverC(P ₁()) is characterized as a top. algebra. All these characterizations base upon the theory of the Gelfand representation of seminormed-algebras.     

A spectral mapping theorem for scalar-type spectral operators in locally convex spaces

LetT be a continuous scalar-type spectral operator defined on a quasi-complete locally convex spaceX, that is,T=fdP whereP is an equicontinuous spectral measure inX andf is aP-integrable function. It is shown that (T) is precisely the closedP-essential range of the functionf or equivalently, that (T) is equal to the support of the (unique) equicontinuous spectral measureQ ^* defined on the Borel sets of the extended complex plane ^* such thatQ ^*({})=0 andT=zdQ _*(z). This result is then used to prove a spectral mapping theorem; namely, thatg((T))=(g(T)) for anyQ ^*-integrable functiong: ^{* *} which is continuous on (T). This is an improvement on previous results of this type since it covers the case wheng((T))/{} is an unbounded set in a phenomenon which occurs often for continuous operatorsT defined in non-normable spacesX.     

Quotient singularities which are complete intersections

Let G be a finite subgroup of GL_n() acting naturally on an affine space ⁿ. In this note we will determine G such that the quotient variety ⁿ/G is a complete intersection. For n=2 and 3, such a group G was classified in [13, 24, 32].     

Nicht endlich erzeugte Primideale in Steinschen Algebren

In this note we exhibit a closed prime idealF in the ring Ó(³) of all holomorphic functions on ³ which is not finitely generated.F is the ideal of a certain irreducible curve Y³, obtained as the image of a proper holomorphic map f³.

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Herrn Karl Stein gewidmet     

Some remarks on the construction of quantum symmetric spaces

We present a general survey of some recent developments regarding the construction of compact quantum symmetric spaces and the analysis of their zonal spherical functions in terms of q-orthogonal polynomials. In particular, we define a one-parameter family of two-sided coideals in U _q(g(n, )) and express the zonal spherical functions on the corresponding quantum projective spaces as Askey-Wilson polynomials containing two continuous and one discrete parameter.The author acknowledges financial support by the Japan Society for the Promotion of Science (JSPS) and the Netherlands Organization for Scientific Research (NWO).     

On the Fredholm theory of Wiener-Hopf equations and the coupling method

The Fredholm properties of the Wiener-Hopf operator onL _p(⁺,^m) are investigated using the coupling method for solving operator equations. The theory applies to equations whose kernel is an element ofL ₁(,^mxm). As usual the determinant of the symbol is assumed to have no zeros on the real line. The method of analysis is independent of the realization theory for symbols that are analytic in a strip containing the real axis although in some sense closely related to it. The connection between the two methods is briefly analysed in the paper.     

3-Mannigfaltigkeiten im ℙ5 und ihre zugehörigen Stabilen Garben

In this paper we begin to study 3-folds in a projective space of dimension 5. Using results from [9] we give a classification of all 3-folds in ⁵ , up to degree 6. There are only 3 different types of 3-folds in ⁵ of degree 6 which are not complete intersections. These manifolds can be represented as zero schemes of sections in certain (extremal) semistable reflexive sheaves of rank 2 on ⁵ . Finally we obtain examples of stable reflexive sheaves on ⁵ with homologieal dimension 1, which do not belong to the extremal sheaves [10].

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Dies ist ein Teil meiner Habilitationsschrift     

Quasi-ALE Metrics with Holonomy SU(m) and Sp(m)

Let G be a finite subgroup of U(m),and X a resolution of ^m /G. We define aspecial class of Kähler metrics g on Xcalled Quasi Asymptotically Locally Euclidean (QALE) metrics. Thesesatisfy a complicated asymptotic condition, implying that gis asymptotic to the Euclidean metric on ^m /G away fromits singular set. When ^m /Ghas an isolated singularity,QALE metrics are just ALE metrics. Our main result is an existencetheorem for Ricci-flat QALE Kähler metrics: if G is afinite subgroup of SU(m) and X a crepant resolution of ^m /G, then there is a unique Ricci-flat QALE Kähler metric on X in each Kähler class.This is proved using a version of the Calabi conjecture for QALEmanifolds. We also determine the holonomy group of the metrics in termsof G.     

KO?i groups of G3(?n), n odd

We use the work of Karoubi and Mudrinski on the real Grothendieck's groups of certain complex projective bundles to show that the torsion of the KO ^–i groups of G ₃( ⁿ ), n odd, are related to the known torsion of the KO ^–i groups of G₂( ⁿ ).     

Willmore Surfaces of ?4 and the Whitney Sphere

We make a contribution to the study of Willmore surfaces infour-dimensional Euclidean space ⁴ by making useof the identification of ⁴ with two-dimensionalcomplex Euclidean space ². We prove that theWhitney sphere is the only Willmore Lagrangian surface of genus zero in⁴ and establish some existence and uniquenessresults about Willmore Lagrangian tori in ^{4 2}.     

Sur la rigidité de certains groupes fondamentaux,l’arithméticité des réseaux hyperboliques complexes,et les “faux plans projectifs”

The motivation of this work comes from the study of lattices in real simple Lie groups. The famous Marguliss superrigidity theorem claims that finite dimensional reductive representations of any lattice of a real simple Lie group of real rank 2 are superrigid. As a corollary such a lattice is arithmetic. These results extend to the real rank one case for lattices in Sp(n,1) and F₄^(-20) by the work of Corlette and Gromov-Schoen. On the other hand Mostow and Deligne-Mostow exhibited arithmetic lattices with non-superrigid representations as well as non-arithmetic lattices in the unitary group PU(2,1). A natural question is then to find simple sufficient conditions for superrigidity or arithmeticity of lattices in PU(2,1). Rogawski conjectured the following: let be a torsion-free cocompact lattice in PU(2,1) such that the hyperbolic quotient M=\B² verifies the cohomogical conditions b₁(M)=0 and H^1,1(M,)H²(M,). Then is arithmetic. In this paper we consider a smooth complex projective surface M verifying the above cohomological assumptions and study Zariski-dense representations of the fundamental group ₁(M) in a simple k-group H of k-rank 2 (where k denotes a local field). Our main result states that there are strong restrictions on such representations, especially when k is non-archimedean (Theorem 5). We then consider some cocompact lattices in PU(2,1) of special geometric interest: recall that a fake P² is a smooth complex surface (distinct from P²) having the same Betti numbers as P². Fake P² exist by a result of Mumford and are complex hyperbolic quotients \H² by Yaus proof of the Calabi conjecture. They obviously verify the hypotheses of Rogawskis conjecture. In this case we prove that every Zariski-dense representation of in PGL(3) is superrigid in the sense of Margulis (Theorem 3). As a corollary every fake P² is an arithmetic quotient of the ball B².

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The degrees of polynomial self-mappings of ℂ2

Two results on the degrees of polynomial mappings ²² are obtained.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 527–534, April, 1998.     

Sur la compacité des ensembles de Julia des polynômes p-adiques

Let R be a rational function, of degree 2, with complex coefficients. Then the Julia set of R is a closed subset of ¹(), and therefore compact. If one replace by the field _p (completion of an algebraic closure of the field _p of the p-adic numbers), then one can define also a Julia set for a rational function with p-adic coefficients. But as _p is not locally compact, the Julia set may or may not be compact. In this paper, we study the compactness of the Julia set of p-adic polynomials. Mathematics Subject Classification (2000):11S99, 37B99.     

Asymptotically Locally Euclidean Metrics with Holonomy SU(m)

Let G be a finite subgroup of U(m) such that ^m /G has an isolated singularity at 0. Let X be a resolution of ^m /G, andg a Kähler metric on X. We callg Asymptotically Locally Euclidean (ALE) if it isasymptotic in a certain way to the Euclidean metric on ^m /G. In this paper we study Ricci-flat ALE Kähler metrics on X. We show that if G SU(m) and X is a crepant resolution of ^m /G, then there is a unique Ricci-flat ALE Kähler metric in each Kählerclass. This is proved using a version of the Calabi conjecture for ALEmanifolds. We also show the metrics have holonomy SU(m).