Algebraicity of Global Real Analytic Hypersurfaces

Let X be an algebraic manifold without compact component and let V be a compact coherent analytic hypersurface in X, with finite singular set. We prove that V is diffeotopic (in X) to an algebraic hypersurface in X if and only if the homology class represented by V is algebraic and singularities are locally analytically equivalent to Nash singularities. This allows us to construct algebraic hypersurfaces in X with prescribed Nash singularities.     

Approximation on Nash sets with monomial singularities

This paper is devoted to the approximation of differentiable semialgebraic functions by Nash functions. Approximation by Nash functions is known for semialgebraic functions defined on an affine Nash manifold M, and here we extend it to functions defined on Nash sets  X⊂M$X\subset M$whose singularities are monomial. To that end we discuss first finiteness and weak normality for such sets X. Namely, we prove that (i) X is the union of finitely many open subsets, each Nash diffeomorphic to a finite union of coordinate linear varieties of an affine space, and (ii) every function on X which is Nash on every irreducible component of X extends to a Nash function on M. Then we can obtain approximation for semialgebraic functions and even for certain semialgebraic maps on Nash sets with monomial singularities. As a nice consequence we show that m-dimensional affine Nash manifolds with divisorial corners which are class k   semialgebraically diffeomorphic, for k>m²$k>m^2$, are also Nash diffeomorphic.     

Semialgebraic sets and some versions of Tarski-Seidenberg-MacIntyre theorem

LetK be a local field of characteristic zero. We give a new definition of semialgebraic sets, which is then ettended to subsets of K-points of algebraicK-varieties. For algebraically closed fields, this notion is shown to be analogous to the concept of a constructible set. This follows from the various versions of the title theorem, according to which a projection .of any semiaigebraic set is again a semialgebraic set. One of such versions isTHEOREM 3. LetN be a quasiprojective algebraicK-variety andp: N → M a regularK-rational map ofN into the projectiveK-varietyM. Under this map~ then, the imageN(K) is a semialgebraic set inM(K). Using Hironaka's results on the resolution of singularities, we obtain a new proof of Theorem 3. Translated fromAlgebra i Logika, Vol. 34, No. 3, pp. 329-346, May-June, 1995. Supported by the Russian Foundation for Fundamental Research, grant No. 93-011-01520.     

Euler characteristics of algebraic varieties

This note studies the behavior of Euler characteristics and of intersection homology Euler characteristics under proper morphisms of algebraic (respectively, analytic) varieties. The methods also yield, for algebraic (respectively, analytic) varieties, formulae comparing these two kinds of Euler characteristics. The main results are direct consequences of the calculus of constructible functions and Grothendieck groups of constructible sheaves. Similar formulae for Hodge‐theoretic invariants of algebraic varieties under morphisms were announced by the first and third authors in [5, 14]. © 2007 Wiley Periodicals, Inc.     

Analytic cell decomposition and analytic motivic integration

The main results of this paper are a Cell Decomposition Theorem for Henselian valued fields with analytic structure in an analytic Denef-Pas language, and its application to analytic motivic integrals and analytic integrals over Fq((t)) of big enough characteristic. To accomplish this, we introduce a general framework for Henselian valued fields K with analytic structure, and we investigate the structure of analytic functions in one variable, defined on annuli over K. We also prove that, after parameterization, definable analytic functions are given by terms. The results in this paper pave the way for a theory of analytic motivic integration and analytic motivic constructible functions in the line of R. Cluckers and F. Loeser [Fonctions constructible et intégration motivique I, Comptes rendus de l'Académie des Sciences 339 (2004) 411-416].     

On the irreducible components of globally defined semianalytic sets

In this work we present the concept of amenable C-semianalytic subset of a real analytic manifold M and study the main properties of this type of sets. Amenable C-semianalytic sets can be understood as globally defined semianalytic sets with a neat behavior with respect to Zariski closure. This fact allows us to develop a natural definition of irreducibility and the corresponding theory of irreducible components for amenable C-semianalytic sets. These concepts generalize the parallel ones for: complex algebraic and analytic sets, C-analytic sets, Nash sets and semialgebraic sets.     

Nilpotent and soluble groups with respect to a homomorph

For an excellent ring Awhose real spectrum satisfies some connectedness condition, we give a sensible notion of real analytic component for a Zariski closed subset of Specr A(such a closed subset will also be called a locally Nash set).Indeed we show that the locally Nash sets are the closed subsets of a noetherian topology on an abstract new space G which we introduce.This generalizes the geometric notion of global real analytic component when Ais the ring of global Nash functions on an affine Nash manifold.     

A comparison theorem for l-adic cohomology

We show that, for certain types of rigid analytic varieties X and constructible l-adic sheaves (F_n)_n on, one has . As an application we obtain that, for an algebraic variety X and associated rigid analytic variety X^rig, the l-adic cohomology of X and X^rig agree.     

On normal surface singularities and a problem of enriques

We construct families of normal surface singularities with the following property: given any fiat projective connected family V →B of smooth, irreducible, minimal algebraic surfaces, the general singularity in one of our families cannot occur, analytically, on any algebraic surfaces which is Irrationally equivalent to a surface in V→B. In particular this holds for V→B consisting of a single rational surface, thus answering negatively to a long standing problem posed by F. Enriques. In order to prove the above mentioned results, wo develop a general, though elementary, method, based on the consideration of suitable correspondences, for comparing a given family of minimal surfaces with a family of surface singularities. Specifically the method in question gives us the possibility of comparing the parameters on which the two families depend, thus leading to the aforementioned results.     

Initial segments of the degrees of constructibility

We show that any constructible, constructibly countable, (dual) algebraic lattice is isomorphic to the degrees of constructibility of reals in some generic extension ofL. Research partially supported by NSF grant DMS-8601777. Research partially supported by NSF grant DMS-8601048 and grant 84-00067 from the U.S.-Israel Binational Science Foundation. Thanks are also due to Uri Abraham and Mati Rubin for helpful discussions about initial segments of the degrees of constructibility and to Bill Lampe for information on algebraic lattices.     

On topological Radon transformations

We construct a functor, which we call the topological Radon transform, from a category of complex algebraic varieties with morphisms given by divergent diagrams, to constructible functions. The topological Radon transform is thus the composition of a pull-back and a push-forward of constructible functions. We show that the Chern-Schwartz-MacPherson transformation makes the topological Radon transform of constructible functions compatible with a certain homological Verdier-Radon transform. We use this set-up to prove, given a projective variety X, a formula for the Chern-Mather class of the dual variety in terms of that of X.     

Algebraically constructible functions and signs of polynomials

LetW be a real algebraic set. We show that the following families of integer-valued functions onW coincide: (i) the functions of the formω →λ(X _ω ), where X _ω are the fibres of a regular morphismf :X →W of real algebraic sets, (ii) the functions of the formω →χ(X _ω ), where X _ω are the fibres of a proper regular morphismf :X →W of real algebraic sets, (iii) the finite sums of signs of polynomials onW. Such functions are called algebraically constructible onW. Using their characterization in terms of signs of polynomials we present new proofs of their basic functorial properties with respect to the link operator and specialization. Research partially supported by an Australian Research Council Small Grant. Second author also partially supported by KBN 610/P3/94.     

An algebraic characterization of holomorphic nondegeneracy for real algebraic hypersurfaces and its application to CR mappings

We give a new algebraic characterization of holomorphic nondegeneracy for embedded real algebraic hypersurfaces in , . We then use this criterion to prove the following result about real analyticity of smooth CR mappings: any smooth CR mapping H between a real analytic hypersurface and a rigid polynomial holomorphically nondegenerate hypersurface is real analytic, provided the map H is not totally degenerate in the sense of Baouendi and Rothschild. Received September 19, 1997     

Operators Commuting with the Volterra Operator are not Weakly Supercyclic

We prove that any bounded linear operator on L _p [0, 1] for 1 ≤ p < ∞, commuting with the Volterra operator V, is not weakly supercyclic, which answers affirmatively a question raised by Léon-Saavedra and Piqueras-Lerena. It is achieved by providing an algebraic type condition on an operator which prevents it from being weakly supercyclic and is satisfied for any operator commuting with V.     

Classification of real Nash fibre bundles

Let G be a compact subgroup of an orthogonal group and X an affine, real, semialgebraic Nash variety. A principal Nash G-bundle over X is said to be strongly Nash if it is induced, up to Nash equivalences, of some universal bundle under a Nash map. Not all Nash bundles are strongly Nash and we denote by S(X, G) the class of strongly Nash G-bundles over X. The principal aim of this paper is to prove the following classification theorem: two bundles of S(X, G) are Nash equivalent if and only if they are topologically equivalent; more,there exists a bijection between the family of the classes of Nash equivalent bundles of S(X, G) and , where is the sheaf of germs of the continous maps from X to G. This result leads to find the largest class of principal Nash G-bundles over X in which the topological equivalence always implies the Nash one. Well, we prove that this class is exactly S(X, G). Research partially supported by M.I.U.R.     

Characteristic exponents of semialgebraic singularities

Let X be a semialgebraic (or algebraic) set and let x₀ ∈ X be a singular point. There are some topological cycles of different dimensions contained in a small neighbourhood of x₀ in X. All these cycles vanish in x₀. The paper is devoted to “vanishing rates” of these cycles, which we call “characteristic exponents”. We prove that the characteristic exponents are invariant under bi‐Lipschitz transformations. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiplicity mod 2 as a Metric Invariant

We study the multiplicity modulo 2 of real analytic hypersurfaces. We prove that, under some assumptions on the singularity, the multiplicity modulo 2 is preserved by subanalytic bi-Lipschitz homeomorphisms of ℝ ⁿ . In the first part of the paper, we find a subset of the tangent cone which determines the multiplicity mod 2 and prove that this subset of S ⁿ is preserved by the antipodal map. The study of such subsets of S ⁿ enables us to deduce the subanalytic metric invariance of the multiplicity modulo 2 under some extra assumptions on the tangent cone. We also prove a real version of a theorem of Comte, and yield that the multiplicity modulo 2 is preserved by arc-analytic bi-Lipschitz homeomorphisms.     

Search for shortest path around semialgebraic obstacles in the plane

An algorithm is described which reduces in polynomial time the problem of constructing a shortest path (between two points) around semialgebraic obstacles in the plane to the problem of constructing a path of smallest weight in a graph the weights of whose vertices are integrals of positive algebraic functions (without singularities). As consequences of this construction one can get algorithms of polynomial complexity for constructing approximations to shortest paths. One such algorithm is given in the paper.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 192, pp. 163–173, 1991.     

The de-Rham theorem and Shapiro lemma for Schwartz functions on Nash manifolds

In this paper we continue our work on Schwartz functions and generalized Schwartz functions on Nash (i.e. smooth semi-algebraic) manifolds. Our first goal is to prove analogs of the de-Rham theorem for de-Rham complexes with coefficients in Schwartz functions and generalized Schwartz functions. Using that we compute the cohomologies of the Lie algebra g of an algebraic group G with coefficients in the space of generalized Schwartz sections of G-equivariant bundle over a G-transitive variety M. We do it under some assumptions on topological properties of G and M. This computation for the classical case is known as the Shapiro lemma.