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.     

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     

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].     

On the Volume of a Domain Obtained by a Holomorphic Motion Along a Complex Curve

Let D be a domain obtained by a holomorphic motion of a domain D _p M _p ^n–1 along a complex curve P in a complex space form M ⁿ . We prove that, if n= 2, the volume of D depends only on the geometry of D _p and the intrinsic geometry of P, but not on the extrinsic geometry of P. When M is closed (compact without boundary), then the dependence on P is only through its topology. When n > 2, and for arbitrary domains D _p, a similar result holds only for Frenet motions, but when D _p has certain integral symmetries (and only in this case) this result is still true for any motion .     

Quasisimilarity of model contractions with unequal defects

Let T and T be C₁₀ contractions with characteristic functions H (ⁿⁿ⁺¹), H (^mm+1). The fundamental result is: T and T are quasisimilar if and only if The paper contains an analysis of this condition; examples are given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 149, pp. 24–37, 1986.     

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.     

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     

A note on the classification of holomorphic harmonic morphisms

In this note we give a complete classification of those holomorphic maps :U ⁿ defined on open and connected subsets of ^m which are harmonic morphisms.The first author was supported by the Icelandic Science Fund.     

Three algebraic structures of quantum projective [sl(2, C)-invariant] field theory

Systematic studies are made of three algebraic structures of quantum projective [sl(2, )-invariant] field theory: the operator algebra Vert(sl(2, )), the infinite-dimensionalR matrixR _proj(u), and the deformationT () of the algebraT() of weighted-shift operators, which is associated with expansion of the renormalized pointwise product of vertex operator fields.State Geological and Prospecting Academy, Moscow; Department of Mathematics in the Research Institute of System Investigations (Information Technologies), Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 97, No. 3, pp. 336–347, December, 1993.     

Embeddings of Danielewski surfaces

A Danielewski surface is defined by a polynomial of the form P=x ^nz –p(y). Define also the polynomial P =x ^nz –r(x)p(y) where r(x) is a non-constant polynomial of degree n–1 and r(0)=1. We show that, when n2 and deg p(y)2, the general fibers of P and P are not isomorphic as algebraic surfaces, but that the zero fibers are isomorphic. Consequently, for every non-special Danielewski surface S, there exist non-equivalent algebraic embeddings of S in ³. Using different methods, we also give non-equivalent embeddings of the surfaces xz=(y ^d _n >–1) for an infinite sequence of integers d _n . We then consider a certain algebraic action of the orthogonal group on ⁴ which was first considered by Schwarz and then studied by Masuda and Petrie, who proved that this action could not be linearized. This was done by comparing the strata of this action to those of the induced tangent space action. Inequivalent embeddings of a certain singular Danielewski surface S in ³ are found. We generalize their result and show how this leads to an example of two smooth algebraic hypersurfaces in ³ which are algebraically non-isomorphic but holomorphically isomorphic. Partially supported by NSF Grant DMS 0101836.     

Ordnungseigenschaften holomorpher Funktionen

A linear geometric order theory for holomorphic mappings F: ^nm is given if a family of linear subspaces L of ⁿ×^m is specified; the theory then is concerned with the order number of connected parts of the intersections L (F) of the L's with the graph (F) of F. For m=1 (i.e. case of one function) it is proved that the local linear order number of a holomorphic function is finite in every point. If the L's are real-linear subspaces then real differential geometric methods lead to the proof, if the L's are complex-linear then ideal-theoretical means do.     

Biquadratic Eisenstein Series

Biquadratic metaplectic Eisenstein series on the group SL ₃ are studied, and the sl(2)-triples technique for the computation of the Fourier coefficients of these series is used. The biquadratic theta function on the group SL ₃ is also constructed. Bibliography: 10 titles.     

Harmonic morphisms from complex projective spaces

In this paper we study harmonic morphisms :U P ^m N ² from open subsets of complex projective spaces to Riemann surfaces. We construct many new examples of such maps which are not holomorphic with respect to the standard Kähler structure on P ^m.The research leading to this paper was supported by the Icelandic Science Fund and the Danish National Science Fund.     

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.     

A perturbation result for the asymptotic behavior of matrix powers

We prove a perturbation result for the asymptotic behavior of the sequence (A ⁿ c) _nN , whereAG|(d), the space of invertibled×d matrices, andc ^d .     

Maximal arcs in the complex projective plane and completely 2-valent functions

In this note we construct maximal arcs in the projective plane over the complex numbers ₂() that are essentially sliced open 2-spheres. In doing so we rely heavily on results from the theory of multivalent analytic functions.     

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.     

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.