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1.
We prove that a continuous map from a compact nonsingular real algebraic variety X into the unit 2-sphere can be approximated by regular maps if and only if it is homotopic to a continuous map which is regular in the complement of a Zariski closed subvariety A of X of codimension at least 3. The assumption on the codimension of A is essential. 相似文献
2.
Riccardo Ghiloni 《Mathematische Annalen》2006,335(4):751-766
Let R be a real closed field and let X be an affine algebraic variety over R. We say that X is universally map rigid (UMR for short) if, for each irreducible affine algebraic variety Z over R, the set of nonconstant rational maps from Z to X is finite. A bijective map from an affine algebraic variety over R to X is called weak change of the algebraic structure of X if it is regular and φ−1 is a Nash map, which preserves nonsingular points. We prove the following rigidity theorem: every affine algebraic variety
over R is UMR up to a weak change of its algebraic structure. Let us introduce another notion. Let Y be an affine algebraic variety over R. We say that X and Y are algebraically unfriendly if all the rational maps from X to Y and from Y to X are trivial, i.e., Zariski locally constant. From the preceding theorem, we infer that, if dim (X)≥1, then there exists a set of weak changes of the algebraic structure of X such that, for each t,s ∈ R with t≠s, and are algebraically unfriendly. This result implies the following expected fact: For each (nonsingular) affine algebraic variety
X over R of positive dimension, the natural Nash structure of X does not determine the algebraic structure of X. In fact, the moduli space of birationally nonisomorphic (nonsingular) affine algebraic varieties over R, which are Nash isomorphic to X, has the same cardinality of R. This result was already known under the special assumption that R is the field of real numbers and X is compact and nonsingular.
The author is a member of GNSAGA of CNR, partially supported by MURST and European Research Training Network RAAG 2002–2006
(HPRN–CT–00271). 相似文献
3.
W. Kucharz 《Archiv der Mathematik》2008,90(6):572-578
Every compact smooth manifold M is diffeomorphic to a nonsingular real algebraic set, called an algebraic model of M. We study modulo 2 homology classes represented by rational algebraic surfaces in X, as X runs through the class of all algebraic models of M.
Received: 16 June 2007 相似文献
4.
5.
Yildiray Ozan 《Journal of Pure and Applied Algebra》2002,170(2-3)
Assume that X is a compact connected orientable nonsingular real algebraic variety with an algebraic free S1-action so that the quotient Y=X/S1 is also a real algebraic variety. If π : X→Y is the quotient map then the induced map between reduced algebraic K-groups, tensored with
,
is onto, where
,
denoting the ring of entire rational (regular) functions on the real algebraic variety X, extending partially the Bochnak–Kucharz result that
for any real algebraic variety X. As an application we will show that for a compact connected Lie group G
. 相似文献
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6.
Wojciech Kucharz 《Selecta Mathematica, New Series》2011,17(4):855-878
According to the Nash–Tognoli theorem, each compact smooth manifold M is diffeomorphic to a nonsingular real algebraic set, called an algebraic model of M. It is interesting to investigate to what extent algebraic and differential topology of compact smooth manifolds can be transferred
into the algebraic-geometric setting. Many results, examples and counterexamples depend on the detailed study of the homology
classes represented by algebraic subsets of X, as X runs through the class of all algebraic models of M. The present paper contains several new results concerning such algebraic homology classes. In particular, a complete solution
in codimension 2 and strong results in codimensions 3 and 4. 相似文献
7.
Let M be a smooth submanifold of dimension m of a nonsingular real algebraic set X. If M can be approximated by nonsingular algebraic subsets of X, then the homology class in represented by M is algebraic. The converse, investigated in this paper, is true only in some exceptional cases. 相似文献
8.
We show that for all i?0 the i-th mod 2 Betti number of compact nonsingular real algebraic varieties has a unique extension to a virtual Betti numberβi defined for all real algebraic varieties, such that if Y is a closed subvariety of X then βi(X)=βi(X?Y)+βi(Y). We show by example that there is no natural weight filtration on the -cohomology of real algebraic varieties with compact supports such that the virtual Betti numbers are the weighted Euler characteristics. To cite this article: C. McCrory, A. Parusiński, C. R. Acad. Sci. Paris, Ser. I 336 (2003). 相似文献
9.
S.M. Natanzon 《Mathematische Zeitschrift》2003,243(2):391-407
Let Y be a complex algebraic curve and let be the set of all real algebraic curves with complexification , such that the real points divide . We find all such families [Y]. According to Harnak theorem a number of connected components of satisfies by the inequality , where g is the genus of Y. We prove that and these estimates are exact.
Received: 15 November 2001; in final form: 28 April 2002/Published online: 2 December 2002 相似文献
10.
Let X and Y be affine nonsingular real algebraic varieties.A general problem in real algebraic geometry is to try to decidewhen a continuous map f: X Y can be approximated by regularmaps in the space of c0 mappings from X to Y, equipped withthe c0 topology. This paper solves this problem when X is theconnected component containing the origin of the real part ofa complex Abelian variety and Y is the standard 2-dimensionalsphere. 相似文献
11.
A. Beridze 《Journal of Mathematical Sciences》2014,197(6):741-752
In this paper, we define coherent morphisms of chain maps and homology groups of morphisms of this type. We construct strong homology groups of continuous maps of compact metric spaces and prove that for each continuous map f : X?→?Y , there exists a long exact homological sequence. Moreover, we show that for each inclusion i : A?→?X of compact metric spaces, there exists an isomorphism $ {{\bar{H}}_n}(i)\approx {{\bar{H}}_n}\left( {X,A} \right) $ . 相似文献
12.
Wojciech Kucharz 《Archiv der Mathematik》2014,102(3):257-261
Investigated are continuous rational maps from a compact nonsingular real algebraic set into unit spheres. Special attention is devoted to such maps with values in the unit 2-sphere. 相似文献
13.
Lucia Beretta 《Annali dell'Universita di Ferrara》1998,44(1):9-25
LetX be a topological space,Y a closed subspace and π:x→T, ψ:Y→T be two continuous maps. We shall say that ψ can be extended by π if there exists a continuous man η=ν(π, ψ):X→T such that: η| x?y ?π, η| Y =ψ. Clearly a similar definition can be given in the category of real or complex algebraic varietes. In this paper we give some sufficient conditions to ensure that map ψ can be extended by π. In particular we study the topological and the real algebraic case. It seems that the last setting is the more interesting. 相似文献
14.
Juliana Coelho 《Journal of Pure and Applied Algebra》2010,214(8):1319-1333
Recently, the first Abel map for a stable curve of genus g≥2 has been constructed. Fix an integer d≥1 and let C be a stable curve of compact type of genus g≥2. We construct two d-th Abel maps for C, having different targets, and we compare the fibers of the two maps. As an application, we get a characterization of hyperelliptic stable curves of compact type with two components via the second Abel map. 相似文献
15.
Daniel Greb 《Advances in Mathematics》2010,224(2):401-431
Given an action of a complex reductive Lie group G on a normal variety X, we show that every analytically Zariski-open subset of X admitting an analytic Hilbert quotient with projective quotient space is given as the set of semistable points with respect to some G-linearised Weil divisor on X. Applying this result to Hamiltonian actions on algebraic varieties, we prove that semistability with respect to a momentum map is equivalent to GIT-semistability in the sense of Mumford and Hausen. It follows that the number of compact momentum map quotients of a given algebraic Hamiltonian G-variety is finite. As further corollary we derive a projectivity criterion for varieties with compact Kähler quotient. 相似文献
16.
Aleksander Czarnecki 《Journal of Pure and Applied Algebra》2019,223(3):1161-1166
We prove that every maximal ideal in the ring of k-regulous functions, , on a smooth real affine algebraic variety of dimension is not finitely generated. 相似文献
17.
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. 相似文献
18.
David Spring 《Topology and its Applications》1984,18(1):89-94
For reasonable spaces (including topological manifolds) X, Y, we characterize compact subsets of the space of continuous maps from X to Y, topologized with the fine (Whitney) C0-topology. In the case of smooth manifolds, we characterize also compact subsets of the space of Cr maps in the Whitney Cr topology. 相似文献
19.
Ivan PaninSerge Yagunov 《Journal of Pure and Applied Algebra》2002,172(1):49-77
In the following paper we introduce the notion of orientable functor (orientable cohomology theory) on the category of projective smooth schemes and define a family of transfer maps. Applying this technique, we prove that with finite coefficients orientable cohomology of a projective variety is invariant with respect to the base-change given by an extension of algebraically closed fields. This statement generalizes the classical result of Suslin, concerning algebraic K-theory of algebraically closed fields. Besides K-theory, we treat such examples of orientable functors as etale cohomology, motivic cohomology, algebraic cobordism. We also demonstrate a method to endow algebraic cobordism with multiplicative structure and Chern classes. 相似文献
20.
Consider a non-singular real algebraic varietyM together with a codimension 1 real algebraic setY M. SupposeY=–1(0) for a smooth function :M and denote by the signature induced by onM–Y. The following results are proved.For compactM, is induced by a regular functionf R(M) if and only if the setY
c, where changes sign, is the union of the (d–1)-dimensional parts of some irreducible components ofY if and only if can be approximated by regular functions with the same zero-set. For non-compactM this is true only ifR(M) is a factorial ring. Similar results are proved whenM andY are real analytic instead of algebraic.Dedicated to the memory of our friend Mario RaimondoThe authors are members of GNSAGA of CNR. This work is partially supported by MURST. 相似文献