Nash's 1952 questions on real algebraic approximation of smooth manifolds: the surviving codimension-one case of the biregular strengthening of the optimal ambient dimension problem

In his 1952 paper "Real Algebraic Manifolds", John Nash posed a cluster of questions on how far smooth manifolds can be described by real algebraic equations: whether every closed smooth manifold is the entire real point set of a compact nonsingular real algebraic variety (a "pure representation"); the structure of the "sheets" (maximal analytically connected pieces) of real algebraic varieties; whether every closed manifold admits an algebraic model with a rational parametrization; and whether his construction of algebraic representations, which placed the model variety in an ambient space of dimension $2r+1$ for an $r$ -dimensional manifold, could be pushed down to the original ambient dimension. Later literature settles most of this: Tognoli (1973) achieved whole-variety algebraic models; Whitney and Wallace (1957-1960) developed the sheet theory; Comessatti's 1914 surface classification and Kollár's 1998-2000 work on threefolds disproved the rational parametrization conjecture; and Banecki (2026) proved that every smooth compact submanifold of $\mathbb{R}^n$ can be approximated, up to small isotopy, by the real locus of a nonsingular complex algebraic subset of $\mathbb{C}^n$ defined over $\mathbb{R}$ , removing the ambient-dimension restriction. The surviving open core, explicitly conjectured by Banecki and out of reach of Akbulut and King's 1992 techniques, is the codimension-one case of the stronger biregular statement: whether a smooth embedding of a nonsingular compact real algebraic set $W$ into $\mathbb{R}^{\dim W + 1}$ can be approximated in the $C^\infty$ topology by regular maps that are biregular isomorphisms onto nonsingular algebraic sets with nonsingular complexifications.

Background

A real algebraic set is the common zero locus in $\mathbb{R}^n$ of a finite family of polynomial equations with real coefficients. Its complexification is the smallest complex algebraic set containing it, equivalently its Zariski closure in $\mathbb{C}^n$ (the closure in the topology whose closed sets are zero loci of polynomial families). Following the conventions of the standard monograph of Bochnak, Coste, and Roy, and of Banecki's paper, a real algebraic subset of $\mathbb{R}^n$ is called nonsingular when its Zariski closure in $\mathbb{C}^n$ is smooth at every point of the real locus; the real locus is then a smooth manifold. The nonsingular locus of an algebraic set is the set of points near which the set is itself a smooth manifold. A regular map between real algebraic sets is a map whose coordinate functions are quotients of polynomials with no poles on the domain, and a biregular isomorphism is a bijection that is regular in both directions, the algebraic counterpart of a diffeomorphism. Approximation throughout this literature is up to small isotopy: the approximating set must lie in an arbitrarily small neighbourhood of the original one and be related to it by a continuous deformation through embeddings of the ambient space.

Seifert (1936) proved that smooth compact hypersurfaces can be approximated by nonsingular real algebraic hypersurfaces. Nash (1952) then proved that every smooth compact connected $M \subset \mathbb{R}^n$ with $n > 2\dim M$ can be approximated by a nonsingular connected component of a real algebraic set, and constructed algebraic models for arbitrary compact smooth $r$ -dimensional manifolds in ambient dimension $2r+1$ ; as recorded by Banecki, Nash conjectured that both the extra connected components and the dimension restriction could be removed. Nash also asked two structural questions. First, on sheets: a sheet of a real algebraic (or real analytic) variety is a maximal analytically connected subset, that is, a subset in which every two points can be joined by a real analytic arc lying in the subset, maximal with this property; Nash asked whether varieties decompose into finitely many such sheets, whether every point lies on one, and whether they are closed. Second, on rational models: a rational parametrization of a real algebraic set $X \subset \mathbb{R}^N$ is a rational map $\Phi: \mathbb{R}^n \dashrightarrow X$ , given by quotients of polynomials on a dense open set, with dense image, in Nash's version injective on a dense open subset; Nash asked whether every closed manifold is diffeomorphic to the real point set of a smooth real algebraic set admitting such a parametrization.

The subsequent development is central to real algebraic geometry. Tognoli (1973) removed the extra components, proving that every compact smooth manifold is diffeomorphic to the entire real point set of a nonsingular real algebraic set in $\mathbb{R}^n$ for $n > 2\dim M$ (the Nash-Tognoli theorem). Ivanov (1984) improved the ambient bound to $(2n-1)/3 > \dim M$ . Akbulut and King (1992) approximated any smooth compact submanifold $M \subset \mathbb{R}^n$ by the nonsingular locus of a possibly singular real algebraic subset of the same $\mathbb{R}^n$ , and by a nonsingular real algebraic subset of $\mathbb{R}^{n+1}$ , but could not remove the extra dimension; their 1993 construction of transcendental submanifolds shows that the analogous statement with nonsingular Zariski closure in complex projective space $\mathbb{CP}^n$ is false. For hypersurfaces, Bochnak, Buchner, and Kucharz (1989, Theorem 7.1) had already achieved approximation by algebraic hypersurfaces with nonsingular projective closure. On the rational side, Comessatti's 1914 classification, as presented by Kollár, shows that a compact surface admits a rational algebraic model only if it is a sphere, a torus, or nonorientable, which disproves Nash's rational conjecture in dimension 2; Kollár (1998, 2000) disproved it in dimension 3 using the minimal model program (the birational classification machinery for complex algebraic varieties of dimension at least three), which forces the real locus of a rationally parametrized threefold to be a connected sum of copies of $\mathbb{RP}^3$ and $S^1 \times S^2$ with a standard sphere; and a symplectic obstruction, stated by Kollár as an unpublished theorem of Viterbo and Eliashberg (the real locus is a Lagrangian submanifold of the complex points, a submanifold on which the natural symplectic form vanishes, which constrains its curvature), excludes hyperbolic 3-manifolds. The sheet questions were resolved by Whitney (1957) and Wallace (1958, 1960): every point of a real analytic variety lies on a proper sheet, each regular point lies on exactly one sheet, and sheets of maximal dimension are closed. Most recently, Banecki (2026) removed the ambient-dimension restriction: every smooth compact submanifold of $\mathbb{R}^n$ is approximable up to small isotopy by the real locus of a nonsingular complex algebraic subset of $\mathbb{C}^n$ defined over $\mathbb{R}$ , and, when the codimension is at least two, by algebraic sets biregularly isomorphic to a prescribed nonsingular algebraic model (his Theorem 1.4, proved for ambient dimension $n \ge \dim W + 2$ ). What his method cannot reach is exactly codimension one with the biregular conclusion, which he leaves as an explicit conjecture; this is the unresolved target.

Problem Statement

Settle the surviving codimension-one case of the biregular strengthening of the optimal ambient dimension problem for algebraic approximation of smooth manifolds, the case that Akbulut and King (1992) could not reach and that Banecki (2026) states as an explicit conjecture (his Theorem 1.4 with $n = \dim W + 1$ ). Let $W \subset \mathbb{R}^m$ be a nonsingular compact real algebraic set, let $n = \dim W + 1$ , and let $i: W \to \mathbb{R}^n$ be a smooth embedding. Can $i$ be approximated arbitrarily well in the $C^\infty$ topology by regular maps $j: W \to \mathbb{R}^n$ such that: (1) $j(W)$ is a nonsingular real algebraic subset of $\mathbb{R}^n$ and $j: W \to j(W)$ is a biregular isomorphism; and (2) the Zariski closure of $j(W)$ in $\mathbb{C}^n$ is a nonsingular complex algebraic set? Since approximation in the affine category is already known even in codimension one, the open requirement here is exactly the biregular isomorphism (1) together with the nonsingular complexification (2), and not approximation by a nonsingular real algebraic hypersurface alone. This is the last gap in the program Nash began in 1952, whether the ambient-dimension restriction on representing smooth manifolds by real algebraic sets is necessary, which is otherwise settled affirmatively: whole-variety models exist in ambient dimension $n > 2\dim M$ (Tognoli 1973), and same-ambient-dimension approximation holds in the affine category for all codimensions (Banecki 2026, building on Bochnak-Buchner-Kucharz 1989 for hypersurfaces).

References

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