Minimal generation of basic open semianalytic sets. (English) Zbl 0655.32011

The question of how many analytic functions are needed to describe basic open semianalytic sets i.e. sets of the form \[ Z=\{z\in {\mathbb{R}}^ n;\quad f_ 1(z)>0,...,f_ r(z)>0\} \] has been open for a long time. So far all efforts to obtain any result concerning this problem by the means of elementary calculus haveasure on the product space \({\mathcal M}et\times {\mathcal E}\), where \({\mathcal M}et\) is the space of metrics on M and \({\mathcal E}\) is the space of maps from M into \({\mathbb{R}}^ D\). S(g,x) is invariant under the group of motions of \({\mathbb{R}}^ D\), the group of diffeomorphisms of M and the conformal group of M. The integral over x actually reduces to a Gaußian integral because the action is quadratic in x, and so one is lead to construct a natural measure on \({\mathcal M}et\), conformally invariant and invariant under the group of diffeomorphisms, in other words a natural measure on \({\mathcal M}_ p\), the space of moduli of Riemann surfaces of genus p.
As the integral above doesn’t make sense, one considers first a finite dimensional model, giving a formula for a measure on a finite dimensional \({\mathcal C}^{\infty}\)-manifold, satisfying corresponding invariance requirements. Then, transplanting formally this formula to the Fréchet manifold \({\mathcal M}et\) one finds a measure on \({\mathcal M}et\) which descends to the Teichmüller space \({\mathcal T}_ p\) (and actually \({\mathcal M}_ p)\) if and only if a certain expression involving infinite dimensional determinants of the Laplacian \(\Delta_ g\) (or equivalently \({\bar \partial}^*{\bar \partial})\) of M and the positive differential operator \({\bar \partial}^*_ T{\bar \partial}_ T\) on the holomorphic tangent bundle T on M, depends only on the image of g in \({\mathcal T}_ p\). To give a rigorous meaning to these determinants one uses so-called \(\zeta\)-function regularization. The measure is conformally invariant if and only if \(D=26\) (formulas of conformal anomalies).
The formulas of conformal anomalies can be derived from a local Riemann- Roch-Grothendieck (RRG) theorem for a holomorphic family of compact Riemann surfaces \(\pi\) : \(X\to S\) and tensor powers of the relative tangent bundle \(T=T_{X/S}\) equipped with a \({\mathcal C}^{\infty}\) hermitian metric. To have a globally defined \({\mathcal C}^{\infty}\)-metric one uses the Quillen metric \(\| \|_ Q\) on the determinant bundle DET \({\bar \partial}_ T\otimes n\). The RRG theorem then gives an expression on the level of differential forms for the first Chern class \(c_ 1(DET {\bar \partial}_ T\otimes n\), \(\| \|_ Q)\). The critical dimension \(D=26\) appears as an avatar of the coefficient 1/12 of the Todd genus occurring in the expression for \(c_ 1.\)
A basic result for the rest of the underlying article is a theorem of Mumford which says that for a family of connected Riemann surfaces as above, one has an isomorphism: \(M_{\pi}: DET {\bar \partial}_{T_{X/S}}\cong (DET {\bar \partial}_{{\mathcal O}})^{\otimes 13}\), compatible with base change and such that if \(\pi\) is an algebraic morphism of quasi-projective complex varieties, then \(M_{\pi}\) is algebraic. A result of Beilinson and Drinfeld, using Mumford’s theorem, implies that the Polyakov measure on \({\mathcal T}_ p\) coincides (up to a multiplicative constant) with the one defined by the metric on the canonical line bundle on \({\mathcal T}_ p\), thus establishing the “algebraic” nature of Polyakov’s measure.
Mumford’s theorem can also be applied to obtain a more or less explicit description of the Polyakov measure in the vecinity of the divisor at infinity \(\Delta =\bar {\mathcal M}_ p-{\mathcal M}_ p\), where \(\bar {\mathcal M}_ p\) is the compactification of \({\mathcal M}_ p\) by stable curves.
Explicit formulas for the Polyakov measure on \({\mathcal T}_ p\) for \(p=1,2,3\) are derived. On \({\mathcal T}_ 1\) Dedekind’s \(\eta\)-function is involved, whereas on \({\mathcal T}_ 2\) and \({\mathcal T}_ 3\) \(\vartheta\)- functions appear.
Reviewer: W.W.J.Hulsbergen


32B20 Semi-analytic sets, subanalytic sets, and generalizations
16W99 Associative rings and algebras with additional structure
14Pxx Real algebraic and real-analytic geometry
Full Text: DOI EuDML


[1] [Be] Becker, E.: Hereditarily pythagorean fields and orderings of higher level. IMPA Lecture Notes29, 1978
[2] [Be-A] Becker, E., Andradas, C.: On the real stability index of rings and its application to semialgebraic geometry. Sem. Logique et Algèbre Réelle Paris VII, 1984/85
[3] [Be-Brö] Becker, E., Bröcker, L.: On the description of the reduced With ring. J. Algebra52, 328-346 (1978) · Zbl 0396.10012
[4] [Be-Kö] Becker, E., Köpping, E.: Reduzierte quadratische Formen und Semiordnungen reeller Körper. Abh. Math. Semin. Univ. Hamb.46, 143-177 (1977) · Zbl 0365.12011
[5] [B-C-R] Bochnak, J., Coste, M., Roy, M.F.: Géométrie Algébrique Réelle. Ergeb. Math., Berlin Heidelberg New York: Springer 1987
[6] [Brö1] Bröcker, L.: Zur Theorie der quadratischen Formen über formal reellen Körpern, Math. Ann.210, 233-256 (1974) · Zbl 0284.13020
[7] [Brö2] Bröcker, L.: Über die Anzahl der Anordnungen eines kommutativen Körpers. Arch. Math.29, 458-464 (1977) · Zbl 0368.12014
[8] [Brö3] Bröcker, L.: Minimale Erzeugung von Positivbereichen. Geom. Dedicata16, 335-350 (1984) · Zbl 0546.14016
[9] [Brö4] Bröcker, L.: Spaces of orderings and semialgebraic sets. Can. Math. Soc., Conf. Proc.4, 231-248 (1984) · Zbl 0547.14015
[10] [C] Cartan, H.: Variétés analytiques réelles et variétés analytiques complexes. Bull. Soc. Math. Fr.85, 77-99 (1957) · Zbl 0083.30502
[11] [F] Frisch, J.: Points de platitude d’un morphisme d’espaces analytiqués complexes. Invent. Math.4, 118-138 (1967) · Zbl 0167.06803
[12] [G] Gamboa, J.M.: Constructibles du spectre réel a adhérance non constructible (Preprint 1985)
[13] [Kn] Knebusch, M.: On the local theory of signatures and reduced quadratic forms. Abh. Math. Semin. Univ. Hamb.51, 149-195 (1981) · Zbl 0469.10008
[14] [Ma] Mahé, L.: Théorème de Pfister pour les variétés et anneaux de Witt réduits. Invent. Math.85, 53-72 (1986) · Zbl 0601.14019
[15] [Mr] Marshall, M.: Spaces of orderings, IV. Can. J. Math.32, 603-627 (1980) · Zbl 0433.10009
[16] [M] Matsumura, H.: Commutative Algebra. Second edition, Amsterdam: W.A. Benjamin Co., 1980 · Zbl 0441.13001
[17] [Rb] Robson, R.: Local semianalytic geometry (Preprint 1986)
[18] [Rt] Rotthaus, Ch.: On the approximation property of excellent rings: Invent. Math.88, 39-63 (1987) · Zbl 0614.13014
[19] [Rz1] Ruiz, J.M.: On Hilbert’s 17th problem and real nullstellensatz for global analytic functions. Math. Z.190, 447-454 (1985) · Zbl 0579.14021
[20] [Rz2] Ruiz, J.M.: Cônes locaux et complétions. CR. Acad. Sci. Paris Série I,302, 67-69 (1986)
[21] [Sch] Schwartz, N.: Der Raum der Zusammenhangskomponenten einer reellen Varietät. Geom. Dedicata13, 361-396 (1983) · Zbl 0508.14015
[22] [Sh] Shiota, M.: Equivalence of differentiable mappings and analytic mappings. IHES54, 37-122 (1981) · Zbl 0516.58012
[23] [T] Tougeron, J.C.: Idéaux de fonctions différentiables. Ergeb. Math.71, Berlin Heidelberg New York: Springer 1972
[24] [W-B] Whitney, H., Bruhat, F.: Quelques proporiétés fondamentales des ensembles analytiquesréels. Comment. Math. Helv.33, 132-160 (1959) · Zbl 0100.08101
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