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Article Dans Une Revue Geometria. Dedicata 119 Année : 2006

Algebraicity of global real analytic hypersurfaces

Résumé

The aim of this work is to prove the following result in real algebraic geometry: Let $X$ be a non-singular affine real algebraic variety of dimension $n$ with no compact connected component and let $V$ be a compact coherent analytic hypersurface of $X$ having only isolated singular points. Then the following conditions are equivalent: \roster \item"(i)" There exists an analytic diffeomorphism $\varphi:X\rightarrow X$, diffeotopic to the identity, such that $\varphi(V)$ is an algebraic hypersurface of $X$. \item"(ii)" The fundamental class $[V]_X\in H_{n-1}(X,\Bbb{Z}/2)$ of $V$ is represented by the fundamental class of an algebraic hypersurface (i.e., $[V]_X$ is contained in the subgroup $H^{\rm alg}_{n-1}(X,\Bbb{Z}/2)$ and, for each singular point $x$ of $V$, the germ $V_x$ of $V$ at $x$ is analytically equivalent to the germ of a Nash set in some neighbourhood of $x$ in $X$. \endroster The fundamental class $[Y]_X\in H_\ast(X,\Bbb{Z}/2)$ of a compact analytic subset $Y$ of $X$ was defined in [A. Borel and A. Haefliger, Bull. Soc. Math. France 89 (1961), 461--513; MR0149503 (26 #6990)], the subgroup $H^{\rm alg}_\ast(X,\Bbb{Z}/2)\subset H_\ast(X,\Bbb{Z}/2)$ being the subgroup consisting of the fundamental classes of compact algebraic subsets of $X$. On the other hand, a Nash set in an open semi-algebraic set $U\subset X$ is a semi-algebraic subset $Z\subset U$ defined by the zero locus of finitely many semi-algebraic functions of class $\scr{C}^\infty$ in $U$. As a corollary, under certain conditions, a compact algebraic hypersurface of $X$ can be constructed with a finite number of prescribed singularities. The implication (i)$\Rightarrow$(ii) is clear. The two conditions in (ii) are necessary. The proof of (ii)$\Rightarrow$(i) is very readable, and the authors give detailed references for the results which they use.
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Dates et versions

hal-00368879 , version 1 (17-03-2009)

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  • HAL Id : hal-00368879 , version 1

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Krzysztof Kurdyka, Wojciech Kucharz. Algebraicity of global real analytic hypersurfaces. Geometria. Dedicata 119, 2006, 119, pp.141-149. ⟨hal-00368879⟩
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