An addendum on algebraic models of smooth manifolds
type: Item
instance of: scholarly article
title: An addendum on algebraic models of smooth manifolds
zbMATH Open document ID: 0726.58006
DOI: 10.1007/BF00181196
author: E. Ballico
published in: Geometriae Dedicata
publication date: 1991-01-01T00:00:00Z
review text: \textit{J. Bochnak} and \textit{W. Kucharz} [Invent. Math. 97, 585-611 (1989; Zbl 0687.14023)] proved that for every \(C^{\infty}\) m-dimensional compact manifold M there exist countably many algebraic manifolds \(X_ i\) diffeomorphic to M, where \(X_ i\) are irreducible \({\mathbb{R}}\)-algebraic subsets of \({\mathbb{R}}^{2m+1}\) and for \(i\neq j\), \(X_ i\) is not birationally equivalent to \(X_ j\). They conjectured that in fact there exist uncountably many such birationally inequivalent algebraic models of M. The author of the present paper proves this conjecture roughly in the following way. For a compact m-dimensional \(C^{\infty}\) submanifold M of \({\mathbb{R}}^{2m+1}\) he constructs a holomorphic map between complex spaces p: \(X\to T\) defined over \({\mathbb{R}}\), where locally for \(U\subset T\), \(p^{-1}(U)\hookrightarrow U\times {\mathbb{P}}^ N({\mathbb{C}})\) for some N (and p commutes with the Cartesian projection) such that for \(t\in T\), the real part of \(X_ t=p^{-1}(t)\) is diffeomorphic to M, \(X_ t\) is not birationally equivalent to \(X_ u\) if \(t\neq u\), moreover there exists a family of \(C^{\infty}\) embeddings \(g_ t: M\hookrightarrow {\mathbb{R}}^{2m+1}\) (arbitrarily close to the identity) such that for every \(t\in T\), \(X_ t\) is a complexification of \(g_ t(M)\). In the proof the author uses the proofs of the paper by J. Bochnak and W. Kucharz, and standard methods of complex algebraic geometry, e.g. the theory of deformation and moduli. The discussed conjecture has been independently proved by J. Bochnak in another way.
Mathematics Subject Classification ID: 58A07
Mathematics Subject Classification ID: 14P15
Mathematics Subject Classification ID: 14C05
Mathematics Subject Classification ID: 14J15
Mathematics Subject Classification ID: 14E30
Mathematics Subject Classification ID: 32G05
zbMATH DE Number: 4199066
zbMATH Keywords: algebraic model of a \(C^{\infty }\) manifold
zbMATH Keywords: real algebraic
zbMATH Keywords: coarse moduli scheme
zbMATH Keywords: Kuranishi family
zbMATH Keywords: Hilbert scheme
zbMATH Keywords: polarized manifold
zbMATH Keywords: algebraic space
zbMATH Keywords: Hilbert polynomial
reviewed by: Krzysztof Dabrowski
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00181196
OpenAlex ID: W1997119286
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description: scientific article; zbMATH DE number 4199066