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

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

Let us recall the definition of a real closed field: this is an field $K$ 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 $P\in K[T]$, any pair $(a,b)$ of elements of $K$ such that $a<b$, $P(a)<0$ and $P(b)>0$, there exists $c\in K$ such that $P(c)=0$ and $a<c<b$. Observe that this property can be expressed by a sequence of first-order formulas, one for each degree.

The field $\mathbf R$ 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 $K$ is real closed if and only if $\sqrt{-1}\not\in K$ and $K(\sqrt{-1})$ is an algebraic closure of $K$. This is also equialent to the fact that “the” algebraic closure of $K$ is a finite non-trivial extension of $K$. 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 $a>0$, then the polynomial $T^2-a$ is negative at $0$ and positive at $\max(a,1)$, so that it admits a positive root). The usual algebraic computation thus shows that the formula $\exists x, x^2+ax+b=0$ is equivalent to the formula $a^2-4b\geq 0$.

**Corollary 1. —**

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

**Corollary 2. —**

*If $M$ is a real closed field, the definable subsets of $M$ are the finite unions of intervals (open, closed or half-open, $\mathopen]a;b\mathclose[$, $\mathopen]a;b]$, $[\mathopen a;b\mathclose[$, $[a;b]$, 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 $T$ be the theory of a real closed field $M$ in an expansion $L$ of the language of ordered rings. One says that $T$ is*

**o-minimal**if the definable subsets of $M$ are the finite unions of intervals.It is a non-trivial result that the o-minimality is indeed a property of the theory $T$, and not a property of the model $M$: if it holds, then for every elementary extension $N$ of $M$, the definable subsets of $N$ 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 $L_{\mathrm{an},\mathrm{exp}}$ be the language obtained by adjoining to the language $L_{\mathrm{or}}$ of ordered rings symbols of functions $\exp$ and $f$, for every real analytic function $f\colon [0;1]^n\to\mathbf R$. The field of real numbers is viewed as a structure for this language by interpreting $\exp$ as the exponential function from $\mathbf R$ to $\mathbf R$, and every function symbol $f$ as the function from $\mathbf R^n$ to $\mathbf R$ that maps $x$ to $f(x)$ if $x\in [0;1]^n$, and to $0$ otherwise.

*The theory*

*(denoted $\mathbf R_{\mathrm{an},\mathrm{exp}})$)*of $\mathbf R$ in this language is o-minimal.This is a thorem of van den Dries and Miller; the case of $L_{\mathrm{an}}$ (without the exponential function) had been established Denef and van den Dries, while the case of $L_{\mathrm{exp}}$ is due to Wilkie.

To give a non-example, let us consider the language obtained by adjoining a symbol $\sin$ and view $\mathbf R$ as a structure for this language, the symbol $\sin$ being interpreted as the sine function from $\mathbf R$ to $\mathbf R$. Then the theory of $\mathbf R$ in this language is not o-minimal. Indeed, the set $2\pi\mathbf Z$ is definable by the formula $\sin(x)=0$, but $2\pi\mathbf Z$ 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 $\mathbf R^{n+1}$ any subset $C$ of the following form: one is given a definable subset $A$ of $\mathbf R^n$ and definable functions $f,g\colon A\to\mathbf R$ such that $f(x)<g(x)$ for every $x\in A$, and the set $C$ is defined by the condition $x\in A$, and by one of the conditions $t<f(x)$, or $t=f(x)$, or $f(x)<t<g(x)$, or $t>f(x)$. Then for every finite family $(B_i)$ of definable subsets of $\mathbf R^{n+1}$, there is a finite partition of $\mathbf R^{n+1}$ into cells such that every $B_i$ is a union of cells.
- Every definable function is piecewise smooth.
- Definable continuous functions are definably piecewise trivial (theorem of Hardt): for every function $f\colon X\to Y$ between definable sets which is definable and continuous, there is a finite partition $(Y_i)$ of $Y$ into definable subsets such that the map $f_i\colon f^{-1}(Y_i)\to Y_i$ deduced from $f$ by restriction is isomorphic to a projection $Y_i\times S_i\to Y_i$.

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 $\mathbf C$ of complex numbers with $\mathbf R^2$ (associating with a complex number its real and imaginary parts) allows to talk of complex valued functions (on a subset of $\mathbf C^n$) 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 $L_{\mathrm{or}}$ of which the field $\mathbf R$ 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 $L_{\mathrm{an},\mathrm{exp}}$.

**Theorem. —**

*Let $A$ be a finite subset of $\mathbf C$ and let $f\colon \mathbf C\setminus A\to \mathbf C$ be a holomorphic function. If $f$ is definable, then it is a rational function.*

**Theorem. —**

*Let $V\subset\mathbf C^n$ be a closed analytic subset. If $V$ is definable, then $V$ is algebraic.*

**Corollary**(Theorem of Chow). —

*Let $V\subset\mathbf P^n(\mathbf C)$ be a closed analytic subset. Then $V$ is algebraic.*

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

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

The most classical example is given by the Poincaré upper half-plane on which $\mathrm{PGL}(2,\mathbf R)$ 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, $X$ is the set of $n\times n$ symmetric complex matrices $Z$ such that $\mathrm I_n-Z^* Z$ is positive definite. It is a homogeneous space for the symplectic group $\mathrm{Sp}(2n,\mathbf R)$; the fixator of $Z=0$ is the unitary group $U(n)$.

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

In fact, it is classical in this context that there exist Siegel sets, which are explicit subsets $F$ of $X$ such that $\Gamma\cdot F=X$ and such that the set of $\gamma\in\Gamma$ such that $\gamma\cdot F\cap F\neq\emptyset$ 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 $F$ to be the set of all $z\in\mathbf C$ such that $-\frac12\leq \Re(z)\leq \frac12$ and $\Im(z)\geq \sqrt 3/2$. One may even take “fundamental sets” (which are fundamental domains up to something of empty interior) such as the one defined by the inequalities $-\frac12\leq \Re(z)\leq\frac12$ and $\lvert z\rvert \geq1$.

Peterzil and Starchenko have proved that there restriction to $F$ of the projection $p$ is definable in the language $L_{\mathrm{an},\mathrm{exp}}$. An immediate consequence is that $S$ 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.