Real Algebraic Morphisms 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.

Real algebraic morphisms represent few homotopy classes

We study the problem of representing homotopy classes of maps between real algebraic varieties of regular maps.

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.

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.

Smooth Maps and Real Algebraic Morphisms

AbstractLet X be a compact nonsingular real algebraic variety and let Y be either the blowup of along a linear subspace or a nonsingular hypersurface of of bidegree (1, 1). It is proved that a C1 map f : X → Y can be approximated by regular maps if and only if , where is the subgroup of H1(X, Z/2) generated by the cohomology classes of algebraic hypersurfaces in X. This follows from another result on maps into generalized flag varieties.

The Weierstrass approximation theorem and a characterization of the unit circle

We study real algebraic morphisms from nonsingular real algebraic varieties X X with dim ⁡ X ≥ 1 \dim X \geq 1 into nonsingular real algebraic curves C C . We show, among other things, that the set of real algebraic morphisms from X X into C C is never dense in the space of all C ∞ \mathcal C^\infty maps from X X into C C , unless C C is biregularly isomorphic to a Zariski open subset of the unit circle.

Approximation of Smooth Maps by Real Algebraic Morphisms

AbstractLet𝔾p,q(𝔽) be the Grassmann space of allq-dimensional𝔽-vector subspaces of𝔽p, where𝔽stands forℝ,ℂorℍ(the quaternions). Here𝔾p,q(𝔽) is regarded as a real algebraic variety. The paper investigates which C∞maps from a nonsingular real algebraic varietyXinto𝔾p,q(𝔽) can be approximated, in theC∞compact-open topology, by real algebraic morphisms.

The Homotopy Groups of Some Spaces of Real Algebraic Morphisms

The Homotopy Groups of Some Spaces of Real Algebraic Morphisms

Elliptic curves and real algebraic morphisms into the 2-sphere

Elliptic curves and real algebraic morphisms into the 2-sphere

Global almost analytic algebraicity of analytic sets

1. The main result. In this paper real algebraic varieties and morphisms between them are understood in the sense of Serre [15] (Serre considers algebraic varieties over an algebraically closed field but his basic definitions make sense over any field). The reader may consult a detailed exposition [7] for properties of real algebraic varieties, especially in connection with real algebraic blow-ups (cf. also [1]). All subvarieties will be assumed closed but not necessarily irreducible. Let Y be an aίfine real algebraic variety and let Z be a subvariety of Y. Then the algebraic blow-up π:B —• Y of Y along Z has the following properties: B is an affine real algebraic variety, π is a real algebraic morphism whose restriction to B\π~λ(Z) is an algebraic isomorphism onto Y\Z, and π is a proper map if B and Y are equipped with the Euclidean topology. Moreover, B is nonsingular and π is surjective if Y and Z are nonsingular varieties. Let X and X be affine nonsingular real algebraic varieties and let D be a subvariety of X. An algebraic morphism π: X —• X is called a k-fold algebraic multiblowup of X along D if π is the composition π = m o o nk , where v v π* Y %k~x π2 Y πi Y V

On Real Algebraic Morphisms Into Even-Dimensional Spheres

Given affine real algebraic varieties X and Y let us denote by R( X, Y) the set of regular mappings (real algebraic morphisms) from X into Y (for definitions and notions of real algebraic geometry see [2], where the theory of real algebraic varieties is treated systematically). Our aim in this paper is to study the set M(X,S') of regular mappings from affine nonsingular real algebraic varieties X into S ', the unit sphere EZn+?1x2 = 1. Earlier we obtained several results in this direction [3], [4]; cf. also [2], Chap. 13. In particular, it was shown in [4] that given a compact connected nonsingular orientable real algebraic subset X of RP, of odd dimension k, either each smooth mapping from X to Sk is homotopic to a regular mapping, or precisely those mappings of even topological degree have this property. Now we shall study the case of regular mappings into even-dimensional spheres. Strangely enough the situation then is radically different from that mentioned above for odd-dimensional spheres. For a very large class of compact smooth manifolds of even dimension 2k, most algebraic models X of these manifolds have the property that every regular mapping from X into S2k is null homotopic (the meaning of most will be made precise later; cf. Remark 1.6, Theorem 2.1, Example 2.3). This class of manifolds contains all compact Co hypersurfaces of R2k?1*