An addendum on algebraic models of smooth manifolds
\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.
algebraic model of a \(C^{\infty }\) manifold
scientific article; zbMATH DE number 4199066