Hostname: page-component-6587cd75c8-67gbf Total loading time: 0 Render date: 2025-04-23T14:46:40.664Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximation and homotopy in regulous geometry

Part of: Polynomials, rational functions Differential topology

Published online by Cambridge University Press:  09 November 2023

Wojciech Kucharz Open the ORCID record for Wojciech Kucharz [Opens in a new window]
Wojciech Kucharz*
Affiliation:
Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland Wojciech.Kucharz@im.uj.edu.pl
Get access Rights & Permissions [Opens in a new window]

Abstract

Let $X$, $Y$ be nonsingular real algebraic sets. A map $\varphi \colon X \to Y$ is said to be $k$-regulous, where $k$ is a nonnegative integer, if it is of class $\mathcal {C}^k$ and the restriction of $\varphi$ to some Zariski open dense subset of $X$ is a regular map. Assuming that $Y$ is uniformly rational, and $k \geq 1$, we prove that a $\mathcal {C}^{\infty }$ map $f \colon X \to Y$ can be approximated by $k$-regulous maps in the $\mathcal {C}^k$ topology if and only if $f$ is homotopic to a $k$-regulous map. The class of uniformly rational real algebraic varieties includes spheres, Grassmannians and rational nonsingular surfaces, and is stable under blowing up nonsingular centers. Furthermore, taking $Y=\mathbb {S}^p$ (the unit $p$-dimensional sphere), we obtain several new results on approximation of $\mathcal {C}^{\infty }$ maps from $X$ into $\mathbb {S}^p$ by $k$-regulous maps in the $\mathcal {C}^k$ topology, for $k \geq 0$.

Keywords

real algebraic varietyregular mapk-regulous mapapproximationhomotopymalleable varietyunit sphere

MSC classification

Primary: 14P05: Real algebraic sets 14P25: Topology of real algebraic varieties 26C15: Rational functions 57R99: None of the above, but in this section
Type
Research Article
Information
Compositio Mathematica , Volume 160 , Issue 1 , January 2024 , pp. 1 - 20
Check for updates
Copyright
© 2023 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Akbulut, S. and King, H., On approximating submanifolds by algebraic sets and a solution to the Nash conjecture, Invent. Math. 107 (1992), 87–97.CrossRefGoogle Scholar
Benoist, O., On the subvarieties with nonsingular real loci of a real algebraic variety, Geom. Topol., to appear. Preprint (2020), arXiv:2005.06424.Google Scholar
Benoist, O. and Wittenberg, O., The tight approximation property, J. Reine Angew. Math. 776 (2021), 151–200.CrossRefGoogle Scholar
Bilski, M., Kucharz, W., Valette, A. and Valette, G., Vector bundles and regulous maps, Math. Z. 275 (2013), 403–418.CrossRefGoogle Scholar
Bochnak, J., Coste, M. and Roy, M.-F., Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 36 (Springer, 1998).CrossRefGoogle Scholar
Bochnak, J. and Kucharz, W., Algebraic approximation of mappings into spheres, Michigan Math. J. 34 (1987), 119–125.CrossRefGoogle Scholar
Bochnak, J. and Kucharz, W., Realization of homotopy classes by algebraic mappings, J. Reine Angew. Math. 377 (1987), 159–169.Google Scholar
Bochnak, J. and Kucharz, W., On real algebraic morphisms into even-dimensional spheres, Ann. of Math. (2) 128 (1988), 415–433.CrossRefGoogle Scholar
Bochnak, J. and Kucharz, W., Algebraic models of smooth manifolds, Invent. Math. 97 (1989), 585–611.CrossRefGoogle Scholar
Bochnak, J. and Kucharz, W., Elliptic curves and real algebraic morphisms, J. Algebraic Geom. 2 (1993), 635–666.Google Scholar
Bochnak, J. and Kucharz, W., The Weierstrass approximation theorem for maps between real algebraic varieties, Math. Ann. 314 (1999), 601–612.CrossRefGoogle Scholar
Bochnak, J. and Kucharz, W., Algebraic approximation of smooth maps, Univ. Iagel. Acta Math. 48 (2010), 9–40.Google Scholar
Bochnak, J. and Kucharz, W., On approximation of maps into real homogeneous spaces, with Appendix by J. Kollár, J. Math. Pures Appl. 161 (2022), 111–134.CrossRefGoogle Scholar
Bochnak, J., Kucharz, W. and Silhol, R., Morphisms, line bundles and moduli spaces in real algebraic geometry, Publ. Math. Inst. Hautes Études Sci. 86 (1997), 5–65; Erratum in Publ. Math. Inst. Hautes Études Sci. 92 (2000), 195.10.1007/BF02698900CrossRefGoogle Scholar
Bogomolov, F. and Böhning, C., On uniformly rational varieties, in Topology, geometry, integrable systems, and mathematical physics, vol. 234 (American Mathematical Society, 2014), 33–48.Google Scholar
Borel, A. and Haefliger, A., La classe d'homologie fondamentale d'un espace analytique, Bull. Soc. Math. France 89 (1961), 461–513.CrossRefGoogle Scholar
Bröcker, T. and Jänich, K., Introduction to differential topology (Cambridge University Press, Cambridge, 1982).Google Scholar
Bredon, G. E., Topology and geometry (Springer, 1993).CrossRefGoogle Scholar
Comessatti, A., Sulla connessione delle superficie razionali reali, Ann. Math. Pura Appl. (4) 23 (1914), 215–283.CrossRefGoogle Scholar
Czarnecki, A., Maximal ideals of regulous functions are not finitely generated, J. Pure Appl. Algebra 223 (2019), 1161–1166.CrossRefGoogle Scholar
Fernando, J., Fichou, G., Quarez, R. and Ueno, C., On regulous and regular images of Euclidean spaces, Q. J. Math. 69 (2018), 1327–1351.Google Scholar
Fichou, G., Huisman, J., Mangolte, F. and Monnier, J.-P., Fonctions régulues, J. Reine Angew. Math. 718 (2016), 103–151.CrossRefGoogle Scholar
Fichou, G., Monnier, J.-P. and Quarez, R., Continuous functions on the plane regular after one blowing-up, Math. Z. 285 (2017), 287–323.CrossRefGoogle Scholar
Fichou, G., Monnier, J.-P. and Quarez, R., Substitution property for the ring of continuous rational functions, Singularities—Kagoshima 2017, in Proceedings of the 5th Franco-Japanese-Vietnamese Symposium on Singularities (World Scientific Publishing, 2020), 71–93.CrossRefGoogle Scholar
Fichou, G., Monnier, J.-P. and Quarez, R., Integral closures in real algebraic geometry, J. Algebraic Geom. 30 (2021), 253–285.CrossRefGoogle Scholar
Fichou, G., Monnier, J.-P. and Quarez, R., Weak normalization and seminormalization in real algebraic geometry, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 22 (2021), 1511–1558.Google Scholar
Forstnerič, F., Stein manifolds and holomorphic mappings: The homotopy principle in complex analysis, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 56, second edition (Springer, Cham, 2017).CrossRefGoogle Scholar
Ghiloni, R., On the space of morphisms into generic real algebraic varieties, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 5 (2006), 419–438.Google Scholar
Ghiloni, R., Rigidity and moduli space in real algebraic geometry, Math. Ann. 335 (2006), 751–766.CrossRefGoogle Scholar
Ghiloni, R., Second order homological obstructions on real algebraic manifolds, Topology Appl. 154 (2007), 3090–3094.CrossRefGoogle Scholar
Gromov, M., Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2 (1989), 851–897.Google Scholar
Hirsch, M. W., Differential topology, Graduate Texts in Mathematics, vol. 33 (Springer, New York, 1997).Google Scholar
Ivanov, N. V., Approximation of smooth manifolds by real algebraic sets, Russian Math. Surveys 37 (1982), 1–59.CrossRefGoogle Scholar
Joglar-Prieto, N., Rational surfaces and regular maps into the 2-dimensional sphere, Math. Z. 234 (2000), 399–405.CrossRefGoogle Scholar
Joglar-Prieto, N. and Mangolte, F., Real algebraic morphisms and del Pezzo surfaces of degree $2$, J. Algebraic Geom. 13 (2004), 269–285.CrossRefGoogle Scholar
Kollár, J., The topology of real algebraic varieties, Current Developments in Mathematics, 2000 (International Press, Somerville, MA, 2001), 197–231.Google Scholar
Kollár, J., Kucharz, W. and Kurdyka, K., Curve-rational functions, Math. Ann. 370 (2018), 39–69.CrossRefGoogle Scholar
Kollár, J. and Nowak, K., Continuous rational functions on real and $p$-adic varieties, Math. Z. 279 (2015), 85–97.CrossRefGoogle Scholar
Kucharz, W. and Kurdyka, K., Some conjectures on continuous rational maps into spheres, Topology Appl. 208 (2016), 17–29.CrossRefGoogle Scholar
Kucharz, W. and Kurdyka, K., Linear equations on real algebraic surfaces, Manuscripta Math. 154 (2017), 285–295.CrossRefGoogle Scholar
Kucharz, W. and Kurdyka, K., Stratified-algebraic vector bundles, J. Reine Angew. Math. 745 (2018), 105–154.CrossRefGoogle Scholar
Kucharz, W. and Kurdyka, K., From continuous rational to regulous functions, in Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018, vol. II, Invited lectures (World Scientific Publishing, Hackensack, NJ, 2018), 719–747.CrossRefGoogle Scholar
Kucharz, W., Algebraic morphisms into rational real algebraic surfaces, J. Algebraic Geom. 8 (1999), 369–379.Google Scholar
Kucharz, W., Rational maps in real algebraic geometry, Adv. Geom. 9 (2009), 517–539.CrossRefGoogle Scholar
Kucharz, W., Complex cycles on algebraic models of smooth manifolds, Math. Ann. 346 (2010), 829–856.CrossRefGoogle Scholar
Kucharz, W., Regular versus continuous rational maps, Topology Appl. 160 (2013), 1375–1378.CrossRefGoogle Scholar
Kucharz, W., Approximation by continuous rational maps into spheres, J. Eur. Math. Soc. (JEMS) 16 (2014), 1555–1569.CrossRefGoogle Scholar
Kucharz, W., Continuous rational maps into the unit $2$-sphere, Arch. Math. (Basel) 102 (2014), 257–261.CrossRefGoogle Scholar
Kucharz, W., Some conjectures on stratified-algebraic vector bundles, J. Singul. 12 (2015), 92–104.Google Scholar
Kucharz, W., Continuous rational maps into spheres, Math. Z. 283 (2016), 1201–1215.CrossRefGoogle Scholar
Kucharz, W., Stratified-algebraic vector bundles of small rank, Arch. Math. (Basel) 107 (2016), 239–249.CrossRefGoogle Scholar
Kucharz, W., Nash regulous functions, Ann. Polon. Math. 119 (2017), 275–289.CrossRefGoogle Scholar
Kucharz, W., On continuous rational functions, Singularities—Kagoshima 2017, in Proceedings of the 5th Franco-Japanese-Vietnamese Symposium on Singularities (World Scientific Publishing, 2020), 41–68.CrossRefGoogle Scholar
Kucharz, W. and Zieliński, M., Regulous vector bundles, Math. Nachr. 291 (2018), 2252–2271.CrossRefGoogle Scholar
Lee, J. M., Introduction to smooth manifolds (Springer, 2003).CrossRefGoogle Scholar
Mangolte, F., Real algebraic morphisms on 2-dimensional conic bundles, Adv. Geom. 6 (2006), 199–213.10.1515/ADVGEOM.2006.011CrossRefGoogle Scholar
Mangolte, F., Real rational surfaces, in Real Algebraic Geometry, Panoramas et Synthèses, vol. 51 (Société Mathématique de France, 2017), 1–26.Google Scholar
Mangolte, F., Real algebraic varieties, Springer Monographs in Mathematics (Springer, 2020).CrossRefGoogle Scholar
Monnier, J.-P., Semi-algebraic geometry with rational continuous functions, Math. Ann. 372 (2018), 1041–1080.CrossRefGoogle Scholar
Ozan, Y., On entire rational maps in real algebraic geometry, Michigan Math. J. 42 (1995), 141–145.CrossRefGoogle Scholar
Ozan, Y., On algebraic K-theory of real algebraic varieties with circle action, J. Pure Appl. Algebra 170 (2002), 287–293.10.1016/S0022-4049(01)00129-3CrossRefGoogle Scholar
Silhol, R., Real algebraic surfaces, Lecture Notes in Mathematics, vol. 1392 (Springer, Berlin, 1989).CrossRefGoogle Scholar
Wall, C. T. C., Differential topology (Cambridge University Press, Cambridge, 2016).CrossRefGoogle Scholar
Zieliński, M., Homotopy properties of some real algebraic maps, Homology Homotopy Appl. 18 (2016), 287–294.10.4310/HHA.2016.v18.n1.a15CrossRefGoogle Scholar
Zieliński, M., Approximation of maps into spheres by regulous maps, Arch. Math. (Basel) 110 (2018), 29–34.CrossRefGoogle Scholar