Monday, May 11, 2015

Model theory and algebraic geometry, 3 — Real closed fields and o-minimality

In this third post devoted to some interactions between model theory and algebraic geometry, we describe the concept of o-minimality and the o-minimal complex analysis of Peterzil and Starchenko.

1. Real closed fields and the theorem of Tarski-Seidenberg

To begin with, we work in the language LorL_{\mathrm{or}} of ordered rings which is the language of rings Lr={+,,,0,1}L_{\mathrm r}=\{+,-,\cdot,0,1\} enlarged with an order relation \leq.

Let us recall the definition of a real closed field: this is an field KK endowed with an ordering which is compatible with the field laws (the sum of positive elements is positive and the product of positive elements is positive) which satisfies the intermediate value theorem for polynomials: for every polynomial PK[T]P\in K[T], any pair (a,b)(a,b) of elements of KK such that a<ba<b, P(a)<0P(a)<0 and P(b)>0P(b)>0, there exists cKc\in K such that P(c)=0P(c)=0 and a<c<ba<c<b. Observe that this property can be expressed by a sequence of first-order formulas, one for each degree.

The field R\mathbf R of real numbers is real closed, but there are many other. For example, the field of formal Puiseux series with real coefficients is also real closed.

A theorem of Artin-Schreier asserts that a field KK is real closed if and only if 1∉K\sqrt{-1}\not\in K and K(1)K(\sqrt{-1}) is an algebraic closure of KK. This is also equialent to the fact that “the” algebraic closure of KK is a finite non-trivial extension of KK. While the algebraic notion adapted to the language of rings is that of an algebraically closed field, the notion of a real closed field is the one which is adapted to the language of ordered rings. In model theoretic terms, the theory of real closed fields is the model companion of the theory of ordered fields.

The analogue of the theorem of Chevalley is the classical theorem of Tarski-Seidenberg:

Theorem (Tarski-Seidenberg). — The theory of real closed fields eliminates quantifiers in the language of ordered rings.

There is a very classical example of this theorem, namely, the resolution of polynomial equation of degree 2. Indeed, in a real closed field, every positive element has a square root (if a>0a>0, then the polynomial T2aT^2-a is negative at 00 and positive at max(a,1)\max(a,1), so that it admits a positive root). The usual algebraic computation thus shows that the formula x,x2+ax+b=0\exists x, x^2+ax+b=0 is equivalent to the formula a24b0a^2-4b\geq 0.

Corollary 1. — If MM is a real closed field and AA is a subset of AA, then Def(Mn,A)\mathop{\rm Def}(M^n,A) is the set of all semi-algebraic subsets of MM defined by polynomials with coefficients in AA.

Corollary 2. — If MM is a real closed field, the definable subsets of MM are the finite unions of intervals (open, closed or half-open, ]a;b[\mathopen]a;b\mathclose[, ]a;b]\mathopen]a;b], [a;b[[\mathopen a;b\mathclose[, [a;b][a;b], possibly unbounded, possibly reduced to singletons).

2. O-minimality

The seemingly innocuous property stated in corollary 2 leads to a definition which is surprisingly important and powerful.

Definition. — Let TT be the theory of a real closed field MM in an expansion LL of the language of ordered rings. One says that TT is o-minimal if the definable subsets of MM are the finite unions of intervals.

It is a non-trivial result that the o-minimality is indeed a property of the theory TT, and not a property of the model MM: if it holds, then for every elementary extension NN of MM, the definable subsets of NN still are finite unions of intervals.

By the theorem of Tarski-Seidenberg, the theory of real closed fields is o-minimal. The discovery of more complicated o-minimal theories is a remarkable fact from the 80s.

Example. — Let Lan,expL_{\mathrm{an},\mathrm{exp}} be the language obtained by adjoining to the language LorL_{\mathrm{or}} of ordered rings symbols of functions exp\exp and ff, for every real analytic function f ⁣:[0;1]nRf\colon [0;1]^n\to\mathbf R. The field of real numbers is viewed as a structure for this language by interpreting exp\exp as the exponential function from R\mathbf R to R\mathbf R, and every function symbol ff as the function from Rn\mathbf R^n to R\mathbf R that maps xx to f(x)f(x) if x[0;1]nx\in [0;1]^n, and to 00 otherwise. The theory (denoted Ran,exp)\mathbf R_{\mathrm{an},\mathrm{exp}})) of R\mathbf R in this language is o-minimal.

This is a thorem of van den Dries and Miller; the case of LanL_{\mathrm{an}} (without the exponential function) had been established Denef and van den Dries, while the case of LexpL_{\mathrm{exp}} is due to Wilkie.

To give a non-example, let us consider the language obtained by adjoining a symbol sin\sin and view R\mathbf R as a structure for this language, the symbol sin\sin being interpreted as the sine function from R\mathbf R to R\mathbf R. Then the theory of R\mathbf R in this language is not o-minimal. Indeed, the set 2πZ2\pi\mathbf Z is definable by the formula sin(x)=0\sin(x)=0, but 2πZ2\pi\mathbf Z has infinitely many connected components, so is not a finite union of intervals.

One motivation for o-minimality is that it realizes (part of) Grothendieck quest towards tame topology as described in his Esquisse d'un programme. Indeed, sets which are definable in an o-minimal structure have many tameness properties:
  • The interior, the closure, the boundary of a definable set is definable.
  • Every definable set is homeomorphic to (the topological realization) of a simplicial complex
  • Every definable set has a celllular decomposition. Precisely, let us call a cell of Rn+1\mathbf R^{n+1} any subset CC of the following form: one is given a definable subset AA of Rn\mathbf R^n and definable functions f,g ⁣:ARf,g\colon A\to\mathbf R such that f(x)<g(x)f(x)<g(x) for every xAx\in A, and the set CC is defined by the condition xAx\in A, and by one of the conditions t<f(x)t<f(x), or t=f(x)t=f(x), or f(x)<t<g(x)f(x)<t<g(x), or t>f(x)t>f(x).  Then for every finite family (Bi)(B_i) of definable subsets of Rn+1\mathbf R^{n+1}, there is a finite partition of Rn+1\mathbf R^{n+1} into cells such that every BiB_i is a union of cells.
  • Every definable function is piecewise smooth.
  • Definable continuous functions are definably piecewise trivial (theorem of Hardt): for every function f ⁣:XYf\colon X\to Y between definable sets which is definable and continuous, there is a finite partition (Yi)(Y_i) of YY into definable subsets such that the map fi ⁣:f1(Yi)Yif_i\colon f^{-1}(Y_i)\to Y_i deduced from ff by restriction is isomorphic to a projection Yi×SiYiY_i\times S_i\to Y_i.

Recently, o-minimality has had spectacular and fantastic applications via the approach of Pila-Zannier to the conjecture of Pink, leading to new proofs of the Manin-Mumford conjecture (Pila-Zannier), and to proofs of the André-Oort conjecture (Pila, Pila-Tsimerman, Klingler-Ullmo-Yafaev), and, more recently, to partial results towards the conjecture of Pink (Gao, Habegger-Pila,...). However, this is not the goal of that post, so let me refer the interested reader to Tom Scanlon's Bourbaki talk on that topic.

3. O-minimal complex analysis

The standard identification of the field C\mathbf C of complex numbers with R2\mathbf R^2 (associating with a complex number its real and imaginary parts) allows to talk of complex valued functions (on a subset of Cn\mathbf C^n) which are definable in a given language. In a remarkable series of papers, Peterzil and Starchenko have shown that holomorphic functions which are definable in an o-minimal structure possess very rigid properties. Let us quote some of their theorems.

So we fix an expansion of the language LorL_{\mathrm{or}} of which the field R\mathbf R is a structure whose theory is o-minimal. By “definable”, we mean definable in that language. The typical language considered in the applications here is the language Lan,expL_{\mathrm{an},\mathrm{exp}}.

Theorem. — Let AA be a finite subset of C\mathbf C and let f ⁣:CACf\colon \mathbf C\setminus A\to \mathbf C be a holomorphic function. If ff is definable, then it is a rational function.

Theorem. — Let VCnV\subset\mathbf C^n be a closed analytic subset. If VV is definable, then VV is algebraic.

Corollary (Theorem of Chow). — Let VPn(C)V\subset\mathbf P^n(\mathbf C) be a closed analytic subset. Then VV is algebraic.

Indeed, working on the standard charts of Pn(C)\mathbf P^n(\mathbf C), we see that VV is locally definable by analytic functions. By compactness of Pn(C)\mathbf P^n(\mathbf C), it is thus definable in the language LanL_{\mathrm{an}}. Since the theory of R\mathbf R in this language is o-minimal, the corollary is a consequence of the previous theorem.

Let us finally give an important example. Let XX be an bounded symmetric domain. This means that XX is a bounded open subset of Cn\mathbf C^n such that for every point pXp\in X, there exists a biholomorphic involution f ⁣:XXf\colon X\to X such that pp is an isolated fixed point of ff. This implies that XX is a homogeneous space G/KG/K under a semisimple Lie group GG which acts by holomorphisms, and KK is a maximal compact subgroup of GG. Moreover, XX has a canonical Kähler metric which is invariant under GG.

The most classical example is given by the Poincaré upper half-plane on which PGL(2,R)\mathrm{PGL}(2,\mathbf R) acts by homographies; of course, the upper half-plane is not bounded, but is biholomorphic to the open unit disk.

A more sophisticated example is given by the Siegel upper half-plane or, rather, its bounded version. That is, XX is the set of n×nn\times n symmetric complex matrices ZZ such that InZZ\mathrm I_n-Z^* Z is positive definite. It is a homogeneous space for the symplectic group Sp(2n,R)\mathrm{Sp}(2n,\mathbf R); the fixator of Z=0Z=0 is the unitary group U(n)U(n).

Let now Γ\Gamma be an arithmetic subgroup of Sp(2n,R)\mathrm{Sp}(2n,\mathbf R); for example, let us take Γ\Gamma be a subgroup of finite index of Sp(2n,Z)\mathrm{Sp}(2n,\mathbf Z). Then the quotient S=X/ΓS=X/\Gamma admits a structure of an analytic set and the projection p ⁣:XSp\colon X\to S is an analytic map. If Γ\Gamma is “small enough” (torsion free, say), then SS is even complex manifold manifold, and pp is a covering. An important and difficult theorem of Baily-Borel asserts that SS is an algebraic variety.

In fact, it is classical in this context that there exist Siegel sets, which are explicit subsets FF of XX such that ΓF=X\Gamma\cdot F=X and such that the set of γΓ\gamma\in\Gamma such that γFF\gamma\cdot F\cap F\neq\emptyset is finite. So Siegel sets are almost fundamental domains. An important remark is that they are semi-algebraic, that is, definable in the language of ordered rings. For example in the upper half-plane, one may take FF to be the set of all zCz\in\mathbf C such that 12(z)12-\frac12\leq \Re(z)\leq \frac12 and (z)3/2\Im(z)\geq \sqrt 3/2. One may even take “fundamental sets” (which are fundamental domains up to something of empty interior) such as the one defined by the inequalities 12(z)12-\frac12\leq \Re(z)\leq\frac12 and z1\lvert z\rvert \geq1.

Peterzil and Starchenko have proved that there restriction to FF of the projection pp is definable in the language Lan,expL_{\mathrm{an},\mathrm{exp}}. An immediate consequence is that SS is definable in this language, hence is algebraic.

These results have been generalized by Klinger, Ullmo and Yafaev to any bounded symmetric domain. This is an important technical part of their proof of the hyperbolic Ax-Lindemann conjecture.

Link to Part 4 — Elimination of imaginaries

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