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.     

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 totally real embeddings of three-manifolds

LetV be a complex hypersurface in an open subset of ³, and letM be a smooth compact real hypersurface inV. Using a theorem of Gromov we prove that there exist small C¹ perturbations ofM in ³ such that is a totally real submanifold of ³. As a consequence we show that certain quotients of the three-sphere admit totally real embeddings into ³. In some special cases including the real projective three-space we find explicit totally real embeddings into ³. Our construction is similar to that of Ahern and Rudin who found a totally real embedding of the three-sphere into ³.Research supported by a fellowship from the Alfred P. Sloan foundation     

K-theory of fine topological algebras, Chern character, and assembly

TheK-theory of the group algebra [] for a countable, discrete group is defined in terms of the simplicial ring of smooth simplices on [], where [] is given the fine topology with respect to its finite-dimensional, linear subspaces. The assembly map for this theory :K _* B K _* [] is studied and shown to be a rational injection. The proof uses the Connes-Karoubi Chern character fromK-theory of Banach algebras to cyclic homology, here generalized to any fine topological algebra, and proved to be multiplicative.     

Translations and Associated Dirichlet Polyhedra in Complex Hyperbolic Space

We study a nonidentity transvection (i.e. (strictly) hyperbolic isometry) or nonidentity Heisenberg translation f of complex hyperbolic space H ⁿ and a Dirichlet polyhedron P of the cyclic group f. We have four main results: (a) If z & in H ⁿ and the axis of a nonidentity transvection are not complex collinear, then, roughly speaking, any two distinct 'naturally arising' geodesics passing through z are not complex collinear. (b) If g is also a transvection or Heisenberg translation of H ⁿ and z & in H ⁿ such that f(z)=g(z) and f ^–1(z)=g ^–1(z), then f=g. (c) We classify all this kind of polyhedra up to congruence in H ⁿ. (d) We obtain an equivalent condition for P to be cospinal (which means that the complex spines of the two sides of P coincide) in terms of the distance of the spines of the two sides of P.     

Invariant subrings of ℂ[X,Y,Z] which are complete intersections

It is known that for every finite subgroup G of SL(2,), the invariant subring [X,Y]^G is a hyper-surface. In this note we treat finite subgroups of SL(3,) and give complete classification of the finite subgroups of SL(3,) whose invariant subrings are complete intersections.     

On the Monodromy of a Multivalued Function along Its Ramification Locus

We consider multivalued analytic functions in ⁿ) whose set of singular points occupies a very small part of ⁿ). Under a mapping of a topological space Y into ⁿ), such a function f can induce a multivalued function on Y. This is possible even if the image of Y entirely lies in the ramification set of f. We estimate the monodromy group of the induced function via the monodromy group of f.     

Holomorphe Funktionen auf Gebieten über Banach-Räumen zu vorgegebenen Konvergenzradien

Given a function: ₊ on a domain spread over an infinite dimensional complex Banach space E with a Schauder basis such that -log is plurisubharmonic and d (d denotes the boundary distance on ) one can find a holomorphic function f: with _f, where _f is the radius of convergence of f. If, in addition, is locally Lipschitz continuous with constant 1, f can be chosen so that (3M)^–1 _f, where M is the basis constant of E. In the particular case of E= ₁ there are holomorphic functions f on with= _f.     

Ordinary and strong density continuity of complex analytic functions

In the paper we prove that the complex analytic functions are (ordinarily) density continuous. This stays in contrast with the fact that even such a simple function asG:²²,G(x,y)=(x,y ³ ), is not density continuous [1]. We will also characterize those analytic functions which are strongly density continuous at the given pointa . From this we conclude that a complex analytic functionf is strongly density continuous if and only iff(z)=a+bz, wherea, b andb is either real or imaginary.     

A and the vanishing topology of discriminants

Summary Suppose thatf: ⁿ , 0 ^p , 0 is finitely -determined withnp. We define a Milnor fiber for the discriminant off; it is the discriminant of a stabilization off. We prove that this discriminant Milnor fiber has the homotopy type of a wedge of spheres of dimensionp–1, whose number we denote byµ (f). One of the main theorems of the paper is a = type result: if (n, p) is in the range of nice dimensions in the sense of Mather, then -codium,with equality iff is weighted homogeneous. Outside the nice dimensions we obtain analogous formulae with correction terms measuring the presence of unstable but topologically stable germs in the stabilization. These results are further extended to nonlinear sections of free divisors.Oblatum 15-VIII-1990Partially supported by a grant from the National Science Foundation and a Fullbright Fellowship     

Embedding a non-embeddable stable plane

In [4], K. Strambach describes a 2-dimensional stable plane admitting =SL₂ as a group of automorphisms such that there exists no -equivarient embedding into a 2-dimensional projective plane. R. Löwen [3] has given a 4-dimensional analogue , admitting =SL₂. He posed the question whether there are embeddings of Strambach's plane into . We show that such embeddings exist, in fact we determine all -equivariant embeddings of 2-dimensional stable planes admitting as atransitive group of automorphisms.     

Common cyclic entire functions for partial differential operators

Let H(^N) denote the Fréchet space of all entire functions of N variables (N1). The purpose of this paper is to prove the existence of a dense set of functions f in H(^N) such that if L is any nonscalar linear differential operator with constant coefficients, then the set {p(L)fp(·) is a polynomial} is dense in H(^N).Research supported in part by an NSF grant     

Polynomially convex orbits of compact lie groups

LetV be a finite dimensional complex linear space and letG be a compact subgroup of GL(V). We prove that an orbitG, V, is polynomially convex if and only ifG is closed andG is the real form ofG . For every orbitG which is not polynomially convex we construct an analytic annulus or strip inG with the boundary inG. It is also proved that the group of holomorphic automorphisms ofG which commute withG acts transitively on the set of polynomially convexG-orbits. Further, an analog of the Kempf-Ness criterion is obtained and homogeneous spaces of compact Lie groups which admit only polynomially convex equivariant embeddings are characterized.Supported by Federal program Integratsiya, no. 586.Supported by INTAS grant 97/10170.     

Direct and inverse spectral problems for generalized strings

Let the functionQ be holomorphic in he upper half plane ⁺ and such that ImQ(z 0 and ImzQ(z) 0 ifz ⁺. A basic result of M.G. Krein states that these functionsQ are the principal Titchmarsh-Weyl coefficiens of a (regular or singular) stringS[L,m] with a (non-decreasing) mass distribution functionm on some interval [0,L) with a free left endpoint 0. This string corresponds to the eigenvalue problemdf + fdm = 0; f(0–) = 0. In this note we show that the set of functionsQ which are holomorphic in ⁺ and such that the kernel has negative squares of ⁺ and ImzQ(z) 0 ifz ⁺ is the principal Titchmarsh-Weyl coefficient of a generalized string, which is described by the eigenvalue problemdf +f dm + ² fdD = 0 on [0,L),f(0–) = 0. Here is the number of pointsx whereD increases or 0 >m(x + 0) –m(x – 0) –; outside of these pointsx the functionm is locally non-decreasing and the functionD is constant.To the memory of M.G. Krein with deep gratitude and affection.This author is supported by the Fonds zur Förderung der wissenschaftlichen Forschung of Austria, Project P 09832     

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.     

Milnor numbers and the topology of polynomial hypersurfaces

Summary LetF: n + 1 be a polynomial. The problem of determining the homology groupsH _q (F ^–1 (c)), c , in terms of the critical points ofF is considered. In the best case it is shown, for a certain generic class of polynomials (tame polynomials), that for allc,F ^–1 (c) has the homotopy type of a bouquet of - ^c n-spheres. Here is the sum of all the Milnor numbers ofF at critical points ofF and ^c is the corresponding sum for critical points lying onF ^–1 (c). A second best case is also discussed and the homology groupsH _q (F ^–1 (c)) are calculated for genericc. This case gives an example in which the critical points at infinity ofF must be considered in order to determine the homology groupsH _q (F ^–1 (c)).     

Twistor holomorphic Lagrangian surfaces in the complex projective and hyperbolic planes

We completely classify all the twistor holomorphic Lagrangian immersions in the complex projective plane ², i.e. those Lagrangian immersions such that their twistor lifts to the twistor space over ² are holomorphic. This classification provides a one-parameter family of examples of Lagrangian spheres in ².Research partially supported by a DGICYT grant No. PB91-0731.     

Differentiable structures with zero entropy on simply connected 4-manifolds

We show that a closed 4-dimensional simply connected topological manifoldM admits a differentiable structure with aC Riemannian metric whose geodesic flow has zero topological entropy if and only ifM is homeomorphic toS ⁴, ²,S ²×S ², or ²#².     

On (-<Emphasis Type="Italic">P</Emphasis> ⋅ <Emphasis Type="Italic">P</Emphasis>)-constant deformations of Gorenstein surface singularities

Let : X T be a small deformation of a normal Gorenstein surface singularity X ₀ = ^-1(0) over the complex number field . Suppose that T is a neighborhood of the origin of and that X ₀ is not log-canonical. We show that if a topological invariant -P _t P _t of X _t = ^-1(t) is constant, then, after a suitable finite base change, admits a simultaneous resolution f : M X which induces a locally trivial deformation of each maximal string of rational curves at an end of the exceptional set of M ₀ X ₀; in particular, if X ₀ has a star-shaped resolution graph, then admits a weak simultaneous resolution (in other words, is an equisingular deformation).