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     

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     

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.     

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.     

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.     

A geometrical interpretation of the shape invariant for geodesic triangles in complex projective spaces

We show that the shape invariant of a triangle in the complex projective space P ⁿ , see [B], can be obtained by integrating the Kählerian form of P ⁿ over a domain parametrized by geodesics and bounded by a geodesic loop formed with sides of the triangle.The second author was supported by a grant from INDAM-Rome.     

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.     

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].     

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.     

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.     

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.     

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)).     

Projections on Lp-spaces of polyanalytic functions

The fundamental result: for an arbitrary bounded, simply connected domain in , the subspace L_n,m ^p() of the space L^p(, ) ( is the plane Lebesgue measure, p 1), consisting of the (m, n)-analytic functions in , is complemented in LP(, ) (a function f is said to be (m, n)-analytic if (^m+n/¯Z^mZⁿ)f=0 in ). Consequently, by virtue of a theorem of J. Lindenstrauss and A. Pelczyski, the space L_n,m ^P() is linearly homeomorphic to lP. In particular, for m=n=1 we obtain that the space of all harmonic L^P-functions in is complemented in L^P(, ). This result has been known earlier only for smooth domains.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 190, pp. 15–33, 1991.     

New examples of Kobayashi hyperbolic surfaces in ℂℙ3

We show that a generic small deformation of the union of two generic cones in ³ of degree 4 is Kobayashi hyperbolic. Hence we obtain new examples of hyperbolic surfaces in ³ of any given degree d 8.__________Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 39, No. 1, pp. 90–94, 2005Original Russian Text Copyright © by B. Shiffman and M. ZaidenbergThe second authors research was supported in part by NSF grant No. DMS-0100474.Translated by B. Shiffman and M. Zaidenberg     

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.     

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     

Equilibrium triangulations of the complex projective plane

Starting with the well-known 7-vertex triangulation of the ordinary torus, we construct a 10-vertex triangulation of P² which fits the equilibrium decomposition of P² in the simplest possible way. By suitable positioning of the vertices, the full automorphism group of order 42 is realized by a discrete group of isometries in the Fubini-Study metric. A slight subdivision leads to an elementary proof of the theorem of Kuiper-Massey which says that P² modulo conjugation is PL homeomorphic to the standard 4-sphere. The branch locus of this identification is a 7-vertex triangulation P ₂ ⁷ of the real projective plane. We also determine all tight simplicial embeddings of P ₂ ¹⁰ and P ₂ ⁷ .     

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     

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.     

Multiplication operator on a matrix polynomial

It is proved that the study of a perturbed multiplication operator on a matrix polynomial in the space L₂(, ⁿ) may be reduced to the study of a perturbed multiplication operator with independent variable in the space L₂(, , ^N) with weight satisfying the Mackenhaupt condition.Translated from Ukrayins'kyy Matemarychnyy Zhurnal, Vol. 44, No. 9, pp. 1287–1289, September, 1992.