## 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

### MSC:

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

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