Morphisms, line bundles and moduli spaces in real algebraic geometry

Bochnak, Jacek; Kucharz, Wojciech; Silhol, Robert

doi:10.1007/bf02698900

Cited by 17 publications

(15 citation statements)

References 25 publications

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“…[2][3][4][5][6]11,12,14,21,22,25,27,32,42,[47][48][49]77,79,80]. Unless the variety V is quite exceptional, one can find a topological vector bundle on V that is not topologically isomorphic to any algebraic vector bundle.…”

Section: Piecewise-algebraic Vector Bundlesmentioning

confidence: 99%

“…Topological properties of regular maps and their applications to algebraization of topological vector bundles were investigated in numerous papers [2][3][4][5][6][7][8][9][10][11][12][13][14][16][17][18][19][20][21][22][23][24][25]27,[32][33][34][35]37,38,[40][41][42][43][47][48][49]51,53,56,64,65,68,71,72,[76][77][78][79][80]82,83]. In gene...…”

Section: Introductionmentioning

confidence: 99%

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Piecewise-regular maps

Kucharz

2017

Math. Ann.

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Let V , W be real algebraic varieties (that is, up to isomorphism, real algebraic sets), and X ⊆ V some subset. A map from X into W is said to be regular if it can be extended to a regular map defined on some Zariski locally closed subvariety of V that contains X . Furthermore, a continuous map f : X → W is said to be piecewise-regular if there exists a stratification S of V such that for every stratum S ∈ S the restriction of f to each connected component of X ∩ S is a regular map. By a stratification of V we mean a finite collection of pairwise disjoint Zariski locally closed subvarieties whose union is equal to V . Assuming that the subset X of V is compact, we prove that every continuous map from X into a Grassmann variety or a unit sphere can be approximated by piecewise-regular maps. As an application, we obtain a variant of the algebraization theorem for topological vector bundles. If the variety V is compact and nonsingular, we prove that each continuous map from V into a unit sphere is homotopic to a piecewise-regular map of class C k , where k is an arbitrary nonnegative integer. Mathematics Subject Classification 14P05 · 14P99 · 57R22Communicated by Ngaiming Mok.

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Section: Piecewise-algebraic Vector Bundlesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Piecewise-regular maps

Kucharz

2017

Math. Ann.

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“…Similarly, ξ is said to admit a stratified-algebraic structure if it is isomorphic to a stratified-algebraic F-vector bundle on X. These two types of F-vector bundles have been extensively investigated in [3,4,6,7,9,10,11,13] and [30,29], respectively. In general, their behaviors are quite different, cf.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Comparison of stratified-algebraic and topological K-theory

Kucharz¹,

Kurdyka²

2015

Preprint

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Stratified-algebraic vector bundles on real algebraic varieties have many desirable features of algebraic vector bundles but are more flexible. We give a characterization of the compact real algebraic varieties X having the following property: There exists a positive integer r such that for any topological vector bundle ξ on X, the direct sum of r copies of ξ is isomorphic to a stratifiedalgebraic vector bundle. In particular, each compact real algebraic variety of dimension at most 8 has this property. Our results are expressed in terms of K-theory.

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“…The question which continuous or C ∞ maps are homotopic to regular ones has been investigated in numerous works [3,5,[8][9][10]14,15,[25][26][27]37,42,44,[48][49][50]55,58]. By applying Theorem 1.1 or Corollary 1.5, some of these results can now be translated into the results on approximation by regular maps.…”

Section: Introductionmentioning

confidence: 99%

On approximation of maps into real algebraic homogeneous spaces

Bochnak¹,

Kucharz²

2020

Preprint

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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 C 0 (resp. C ∞ ) topology if and only if it is homotopic to a regular map. Taking Y = S p , 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 S n into S n 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.

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Morphisms, line bundles and moduli spaces in real algebraic geometry

Cited by 17 publications

References 25 publications

Piecewise-regular maps

Piecewise-regular maps

Comparison of stratified-algebraic and topological K-theory

On approximation of maps into real algebraic homogeneous spaces

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