Nonsingular Real Algebraic Variety Research Articles

Dominating real algebraic morphisms

Let X and Y be nonsingular real algebraic varieties, dim⁡X≥dim⁡Y. Assume that the variety Y is malleable, compact and connected. Our main result implies that each regular map from X to Y is homotopic to a surjective regular map. The class of malleable varieties includes all homogeneous spaces for linear real algebraic groups.

On approximation of maps into real algebraic homogeneous spaces

Let X be a real algebraic variety (resp. nonsingular real algebraic variety) and let Y be a homogeneous space for some linear real algebraic group. We prove that a continuous (resp. C∞) map f:X→Y can be approximated by regular maps in the C0 (resp. C∞) topology if and only if it is homotopic to a regular map. Taking Y=Sp, the unit p-dimensional sphere, we obtain solutions of several problems that have been open since the 1980's and which concern approximation of maps with values in the unit spheres. This has several consequences for approximation of maps between unit spheres. For example, we prove that for every positive integer n every C∞ map from Sn into Sn can be approximated by regular maps in the C∞ topology. Up to now such a result has only been known for five special values of n, namely, n=1,2,3,4 or 7.

Bohr Almost Periodic Sets of Toral Type

A locally finite multiset \((\Lambda ,c),\) \(\Lambda \subset {\mathbb {R}}^n, c : \Lambda \mapsto \{1,...,b\}\) defines a Radon measure \(\mu := \sum _{\lambda \in \Lambda } c(\lambda )\, \delta _\lambda \) that is Bohr almost periodic in the sense of Favorov if the convolution \(\mu *f\) is Bohr almost periodic for every \(f \in C_c({\mathbb {R}}^n).\) If it is of toral type: the Fourier transform \({\mathfrak {F}} \mu \) equals zero outside of a rank \(m < \infty \) subgroup, then there exists a compactification \(\psi : {\mathbb {R}}^n \mapsto {\mathbb {T}}^m\) of \(\mathbb R^n,\) a foliation of \({\mathbb {T}}^m,\) and a pair \((K,\kappa )\) where \(K := \overline{\psi (\Lambda )}\) and \(\kappa \) is a measure supported on K such that \({\mathfrak {F}} \kappa = ({\mathfrak {F}} \mu ) \circ \widehat{\psi }\) where \({\widehat{\psi }} : \widehat{{\mathbb {T}}^m} \mapsto \widehat{{\mathbb {R}}^n}\) is the Pontryagin dual of \(\psi .\) For \((\Lambda ,c)\) uniformly discrete, we prove that every connected component of K is homeomorphic to \({\mathbb {T}}^{m-n}\) embedded transverse to the foliation and the homotopy of its embedding is a rank \(m-n\) subgroup S of \({\mathbb {Z}}^m,\) and we compute its density as a function of S and \(\psi .\) For \(n = 1\) and K, a nonsingular real algebraic variety, this construction gives all Fourier quasicrystals recently characterized by Olevskii and Ulanovskii.

Some conjectures on continuous rational maps into spheres

Recently continuous rational maps between real algebraic varieties have attracted the attention of several researchers. In this paper we continue the investigation of approximation properties of continuous rational maps with values in spheres. We propose a conjecture concerning such maps and show that it follows from certain classical conjectures involving transformation of compact smooth submanifolds of nonsingular real algebraic varieties onto subvarieties. Furthermore, we prove our conjecture in a special case and obtain several related results.

Continuous rational maps into spheres

Let X be a compact nonsingular real algebraic variety. We prove that if a continuous map from X into the unit p-sphere is homotopic to a continuous rational map, then, under certain assumptions, it can be approximated in the compact-open topology by continuous rational maps. As a byproduct, we also obtain some results on approximation of smooth submanifolds by nonsingular subvarieties.

COMPLEX CYCLES AS OBSTRUCTIONS ON REAL ALGEBRAIC VARIETIES

AbstractLetYbe a compact nonsingular real algebraic variety of positive dimension. Then one can find a compact connected nonsingular real algebraic varietyX, which admits a continuous map intoYthat is not homotopic to any regular map. It is hard to determine the minimum dimension of such a varietyX. In this paper, new upper bounds for dimXare obtained. The main role in the constructions is played by complex algebraic cycles onY.

Approximation by continuous rational maps into spheres

Investigated are continuous rational maps of nonsingular real algebraic varieties into spheres. In some cases, necessary and sufficient conditions are given for a continuous map to be approximable by continuous rational maps. In particular, each continuous map between unit spheres can be approximated by continuous rational maps.

Regular versus continuous rational maps

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.

Homology of real algebraic varieties and morphisms to spheres

Let X and Y be affine nonsingular real algebraic varieties. One of the classical problems in real algebraic geometry is whether a given C ∞ mapping f: X → Y can be approximated by entire rational mappings in the space of C ∞ mappings. In this work, we obtain some sufficient conditions in the case when Y is the standard sphere S n .

On bordisms of real algebraic M-varieties

Any morphism of nonsingular complete real algebraic varieties F: Y → X determines a holomorphic mapping of the sets of complex points F ‒: Y (‒) → X(‒) as well as a differentiable mapping of the sets of real points F ℝ: Y(ℝ) → X(ℝ). These two mappings determine classes of nonoriented bordisms [F ‒] ∈ MO2m (X((‒)), [F ℝ] ∈ M Om (X(ℝ)), where m = dim Y. The paper describes relationship between these two classes of bordisms.

Real algebraic morphisms on 2-dimensional conic bundles

Abstract Given two nonsingular real algebraic varieties V and W, we consider the problem of deciding whether a smooth map f : V → W can be approximated by regular maps in the space of 𝒞∞ mappings from V to W in the 𝒞∞ topology. Our main result is a complete solution to this problem in case W is the usual 2-dimensional sphere and V is a real algebraic surface of negative Kodaira dimension.

On cohomology of invariant submanifolds of Hamiltonian actions

In [5] the author proved that if there is a free algebraic circle action on a nonsingular real algebraic variety X then the fundamental class is trivial in any nonsingular projective complexification i : X → XC. The Kahler forms on C and CP naturally induce symplectic structures on complex algebraic affine or projective varieties and in case they are defined over reals, their real parts, if not empty, are Lagrangian submanifolds. The following result can be considered as a symplectic equivalent of author’s above result on real algebraic varieties.

Relative topology of real algebraic varieties in their complexifications

We investigate, for a given smooth closed manifoldM, the existence of an algebraic model X forM (i.e., a nonsingular real algebraic variety di.eomorphic to M) such that some non-singular projective complexi.cation i : X ?? XC of X admits a retraction r : XC ?? X. If such an X exists, we show that M must be formal in the sense of Sullivan's minimal models, and that all rational Massey products on M are trivial. We also study the homomorphism on cohomology induced by I for algebraic models X of M. Using ?etale cohomology, we see that mod p Steenrod powers give an obstruction for the induced map on cohomology, i. : Hk(XC,Zp) ?? Hk(X, Zp), to be onto, if we require that X is de.ned over rational numbers.

On Homotopy Groups of Real Algebraic Varieties and their Complexifications

Let X0 be a topological component of a nonsingular real algebraic variety and i:X → XC is a nonsingular projective complexification of X. In this paper, we will study the homomorphism on homotopy groups induced by the inclusion map i:X0 → XC and obtain several results using rational homotopy theory and other standard tools of homotopy theory.

Real algebraic morphisms and Del Pezzo surfaces of degree $2$

Let X and Y be affine nonsingular real algebraic varieties. A general problem in Real Algebraic Geometry is to try to decide when a smooth map f : X -> Y can be approximated by regular maps in the space of smooth mappings from X to Y, equipped with the compact-open topology. In this paper we give a complete solution to this problem when the target space is the usual 2-dimensional sphere and the source space is a geometrically rational real algebraic surface. The approximation result for real algebraic surfaces rational over R is due to J. Bochnak and W. Kucharz. Here we give a detailed description of the more interesting case, namely a real Del Pezzo surfaces of degree 2.

Virtual Betti numbers of real algebraic varieties

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 number β i defined for all real algebraic varieties, such that if Y is a closed subvariety of X then β i ( X)= β i ( X⧹ Y)+ β i ( Y). We show by example that there is no natural weight filtration on the Z 2 -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).

REAL ABELIAN VARIETIES WITH MANY LINE BUNDLES

Let X and Y be affine nonsingular real algebraic varieties. A general problem in real algebraic geometry is to try to decide when a continuous map f: X → Y can be approximated by regular maps in the space of c0 mappings from X to Y, equipped with the c0 topology. This paper solves this problem when X is the connected component containing the origin of the real part of a complex Abelian variety and Y is the standard 2-dimensional sphere.

On algebraic K-theory of real algebraic varieties with circle action

Assume that X is a compact connected orientable nonsingular real algebraic variety with an algebraic free S 1-action so that the quotient Y= X/ S 1 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 Q , π ∗ : K ̃ 0( R(Y, C))⊗ Q→ K ̃ 0( R(X, C))⊗ Q is onto, where R(X, C)= R(X)⊗ C , R(X) denoting the ring of entire rational (regular) functions on the real algebraic variety X, extending partially the Bochnak–Kucharz result that K ̃ 0( R(X×S 1, C))= K ̃ 0( R(X, C)) for any real algebraic variety X. As an application we will show that for a compact connected Lie group G K ̃ 0( R(G, C))⊗ Q=0 .

On homology of real algebraic varieties

Let R be a commutative ring with unity and X an R-oriented compact nonsingular real algebraic variety of dimension n. If i: X -? X? is any nonsingular complexification of X, then the kernel, which we will denote by KHk (X, R), of the induced homomorphism i*: Hk (X, R) -Hk (X?, R) is independent of the complexification. In this work, we study KHk (X, R) and give some of its applications.

Sur les cycles algébriques d'une variété algébrique réelle obtenue à partir d'une modification

Let X be a real algebraic variety, we can construct another real algebraic variety by a modification of X, i.e., a sequence of blow-ups and blow-downs. In this paper, after recalling some results about the cohomology of blow-ups, we study algebraic cycles on a nonsingular real algebraic variety obtained by blow-down. Then we will consider the general case of modifications. This will allow us to prove, in particular, that the cohomology of real compact nonsingular toric varieties is algebraic.