Dimensions and homogeneity in

mathematical structures B. Zilber University of Oxford To assess the expressive power of a formal language study the structures which can be described in full (up to isomorphism) by means of the language.

This property of the structure is called (absolute) categoricity.

We restrict the language by choosing a set of names Σ for relations and operations which have meaning in a structure

M = (M, Σ),

where M is the universum of the structure. Th(M), the theory of M, is the set of all sentences formulated in the language Σ which are true in M. Any structure satisfying the theory Th(M) is called a model of the theory.

Theorem (Lowenheim, Skolem) The first order description Th(M) of M is absolutely categorical iff M is finite.

Uninteresting once the description is of the same size as the structure it describes.

Definition A structure M is said to be categorical in cardinality λ if there is exactly one, up to isomorphism, structure M′ of cardinality λ satisfying Th(M).

Of special interest is the case of uncountable cardinality λ.

Theorem (M.Morley 1964). If a first-order countable theory is categorical in one uncountable λ, it is categorical in all uncountable cardinalities.

We have then a large structure which has a concise (countable) categorical description.

This could be viewed as a formalization of what we would like to see as a canonical structure.

Definition A subset S ⊆ M^{n} in structure M is said to be a definable (using parameters from X ⊆ M) subset if there is a first order property (formula) φ(x_{1}, …, x_{n}) in language Σ with x_{1}, …, x_{n} free variables such that

S = {⟨a_{1}, …, a_{n}⟩ ∈ M^{n} :

φ(a_{1}, …, a_{n}) holds in M}.

If S ⊆ M^{n} is a definable subset and E an equivalence relation on S which is definable too, then the quotient-set S/E is said to be a definable set in M.

Suppose sets S and relations R_{1} ⊆ S^{m_{1}}, ...R_{k} ⊆ S^{m_{k}} all are definable in M. Then the structure S on the set S given by the relations R_{1}, …, R_{k} is said to be definable in M.

Basic examples of uncountably categorical structures in countable languages:

(1) Trivial structures (Σ contains only the equality); Graph of a countable group, uncountably many copies.

(2) Abelian divisible torsion-free groups; Abelian groups of prime exponent; Vector spaces over a countable division ring.

(3) Algebraically closed fields.

Given these structures one can construct in a definable way more complicated uncountably categorical structures.

Let K be an algebraically closed field and G(K) an algebraic group over K, (the Kpoints of the group). G(K) is uncountably categorical in a language witnessing the way G(K) is definable in K.

Theorem Assume G is a simple algebraic group. Then G(K), in the pure language of groups, is uncountably categorical.

No structural theory of algebraic groups is needed!

For G simple algebraic, the field K is definable in G(K).

Need to know that maximal solvable connected (Borel) subgroups are non-abelian.

Corollary(Special case of Borel-Tits Thm) Any (abstract) automorphism of G(K) is a composition of an automorphism induced by a field automorphism K → K and a biregular automorphism G → G.

Conjecture(G.Cherlin 1979, B.Zilber 1977) Any simple uncountably categorical group is isomorphic to an algebraic group G(K), with K algebraically closed.

Status: open.

Main logical problem

What properties of M make it categorical?

Why do certain structures allow an uncountably categorical description?

The answer (M.Morley, S.Shelah and others):

The key factor is the measurability by a dimension in the presence of homogeneity. The dimension notion satisfies certain geometric properties.

This is given by Geometric Stability Theory. When applied to fields, Stability Theory approach in many respects is very close to Algebraic Geometry.

Dimension notions for finite X ⊂ M :

(1) Trivial structures: size of X

(2) Abelian divisible torsion-free groups; Abelian groups of prime exponent; Vector spaces over a division ring:

linear dimension of spanX

(3) Algebraically closed fields:

transcendense degree tr.d.(X).

Dually, the dimension of an algebraic variety V ⊆ K^{n} :

dim V = max{tr.d.(x̄) | x̄ ∈ V }.

Atlas of Stability

13

Geometries

A closure operator acl(X) on M :

y ∈ acl(X) iff there is a finite subset Y ⊆ M which is definable using parameters in X such that y ∈ Y.

For any M the following property of the algebraic closure hold for any finite X ⊆ M, y, z ∈ M :

(i) X ⊆ acl(X); (ii) acl(acl(X)) = acl(X) ; A structure M is said to be a combinatorial pregeometry if the exchange principle holds: (iii) z ∈ acl(X, y) and z ∉ acl(X) ⇒ y ∈ acl(X, z).

M is said to be a combinatorial geometry if acl({x}) = {x} for any singleton.

M is said to be a homogeneous pregeometry if for any finite X ⊆ M and y, y′ ∈ M \ acl(X) there is an automorphism fixing acl(X) and sending y → y′.

acl(X) = the subspace generated by X.

In combinatorial pregeometries the notions of independence, basis, dimension are well defined.

Theorem (Baldwin-Lachlan, Zilber, Harrington-Cherlin-Lachlan and others) An uncountably categorical structure is coordinatizable by a unique homogeneous combinatorial geometry. Any ω-stable structure is coordinatizable by a collection of combinatorial geometries.

Weak Trichotomy Theorem (Zilber 1980) Given an uncountably categorical M, one and only one of the following holds:

(i) the geometry of M is trivial;

(ii) the geometry of M is isomorphic to the geometry of a projective or affine space over a division ring;

(iii) there is a pseudoplane definable in M.

Definition (Lachlan) A pseudoplane is a two-sorted structure (P, L), P is said to be the set of points and L the set of lines, with a binary relation I ⊆ P × L (I(x, y) is read as “point x belongs to line y”) and satisfying the following: any line contains infinitely many points; through any point pass infinitely many lines; any two distinct points belong to at most finitely many lines; any two distinct lines intersect in at most finitely many points.

Theorem If M is locally finite (acl(X) is finite for finite X) then there is no pseudoplane in M.

Proof 1 (Zilber 84) Imitate algebraic geometry on the ’surface’ P and develop intersection theory. ’Bezout Theorem’ leads to a contradiction.□

Proof 2 (Cherlin, Mills, P.Newmann 84) Use the classification of finite doubly transitive permutation groups, based on the classification of finite simple groups. □

Corollaries

(1) An infinite locally finite homogeneous combinatorial geometry is either trivial or an affine or projective geometry over a finite field.

(2) Ultrahomogeneous graphs are coordinatizable by trivial geometries.

Both can be finitized.

In (1) dim ≥ 7 suffices.

(2) leads to a full classification of families of ultrahomogeneous finite structures (Lachlan, Cherlin, Shelah)

(1)&(2) is at the background of the classification of Finite Strictures with few Types (oncoming book by G.Cherlin and E.Hrushovski) http://www.math.rutgers.edu/cherlin/ Trichotomy Conjecture Given an uncountably categorical structure M one and only one of the following holds:

(i) the combinatorial geometry of M is trivial;

(ii) the combinatorial geometry of M is isomorphic to an affine or projective geometry over a countable division ring;

(iii) M is a structure of algebraic geometry over an algebraically closed field K. (M is definably equivalent to the field K.)

The Trichotomy conjecture was refuted in general (E.Hrushovski, 1989) proved under extra Zariski assumptions (E.Hrushovski and B.Zilber, 1994) Also, proved in the o-minimality – real algebraic geometry context (Y.Peterzil, S.Starchenko, 1996).

II. The fundamental trichotomy of

Geometric Stability Theory, Zariski structures and Diophantine

geometry

B.Zilber

University of Oxford The definition of Zariski structure M assumes:

- Weak topology on a structure M, with all basic relations assumed closed and satisfying the descending chain condition for closed subsets of M^{n}.

- The projection pr : M^{n+1} → M^{n} maps irreducible closed sets to closed-minus-closed sets.

- Dimension, a number is assigned to any closed set, with natural geometric properties, e.g. for any closed irreducible S ⊆ M^{n+1}

dim pr(S) = dim S − dim S_{a},

S_{a} a fiber of minimal dimension.

- Pre-smoothness. Given closed irreducible S_{1}, S_{2} ⊆ M^{n} and a point a ∈ S_{1} ∩ S_{2}, the dimension of an irreducible component containing a satisfies

dim_{a} S_{1} ∩ S_{2} ≥ dim S_{1} + dim S_{2} − dim M^{n}.

Theorem 2.1 Zariski structures are of finite Morley rank. More precisely Mdim A ≤ dim A for any definable A.

Examples.

1. Algebraic structure on a smooth algebraic variety M over an algebraically closed field is Zariski

n-ary basic relations P = algebraic subvarieties P ⊆ M^{n}.

2. Compact complex manifods are Zariski

n-ary basic relations P = analytic subsets P ⊆ M^{n}.

3. Proper analytic varieties in rigid analytic geometry are Zariski.

4. Finite Morley rank substructures of differentially closed fields F = (F, +, ·, D) of characteristic zero.

(Hrushovski, Sokolovich) A definable substructure A of F of Morley rank one in an induced language is up to finite number of points a Zariski structure.

Examples

4.1 C = {x ∈ F : Dx = 0} the field of constants of F.

4.2 (Yu.Manin) A = {(x, y) ∈ F^{2} satisfying:

y^{2} = x(x − 1)(x − t); Dt = 1;
y^{3} + 2(2t − 1)(x − t^{2})y · Dx −

−2t(t − 1)(yD^{2}x − 2Dx · D^{2}y) = 0}, a group containing C(t)-rational points on the elliptic curve y^{2} = x(x − 1)(x − t).

Theorem (Hrushovski, Zilber 1994) Zariski structures satisfy the Trichotomy conjecture.

Proof Let M be a Zariski structure,

dim M = 1.

Imitate algebraic geometry in M.

1. Develope an intersection theory (multiplicity of intersection in a point, tangency e.c.). Prove Bezout Theorem.

2. Assuming M × M is a pseudoplane, use tangency to define jets of curves through (0, 0). This forms a group under the composition operation. This group gives rise to a multiplicative and then to the additive group of a field K. K coordinatizes M.

3. The proof that the coordinatization is given by algebraic functions amounts to proving the Chow Theorem: any closed subset of ℙ^{n}(K) is algebraic.

4. M is an algebraic curve C(K) with relations on it given by polynomial equations over K. □

Diophantine Geometry context

Hrushovski, Manin: The solutions of Manin’s equations form a subgroup A ⊆ 𝔸(F) of the abelian variety, and

(1) A ⊇ Γ, the group of rational points over a function field,

(2) Mdim A is finite, A satisfies the Trichotomy, thus either

(i) A does not have non-linear relations induced from the whole F, or

(ii) A = G(C′), the C′-points of an algebraic group over a field C′ ⊆ F^{n}, Mdim C′ = 1. In fact C = C′. Equivalently, 𝔸 is isogeneous over the function field to an abelian variety over C.

Corollary (Holds in general once the linear case (i) has been identified). If V ⊆ 𝔸 is an algebraic subvariety then

Γ ∩ V = Γ ∩ 𝔹_{1} ∪ … ∪ Γ ∩ 𝔹_{k}

for some abelian subvarieties 𝔹_{i} ⊆ 𝔸.

This is the statement of Mordell-Lang conjecture for function fields of characteristic 0 for 𝔸 not isogeneous to an abelian variety over the field of constants.

Similar scheme works for the function field case in positive charactersitics. The model theory of differentially closed fields is replaced by the model theory of separably closed fields of positive characteristic.

Similar scheme works for solving the Manin-Mumford conjecture (Γ the torsion subgroup of a semiabelian variety) the general case. The model theory of difference fields is used by Hrushovski.

Similar scheme works to prove a strong uniform version of Manin-Mumford conjecture. T.Scanlon. Based on the structure of Witt vectors with an automorphism (A.Macintyre, L.Belair, T.Scanlon).

1.Model Theory puts the diophantine problems into a new geometric context:

Geometric Stability Theory - Zariski Geometries - the Trichotomy.

2. Model Theory works here by synthesizing new structures with prescribed properties to solve (or give a geometrical meaning to ) mathematical problems: Tame structures in which the problem is definable and analyzable.

More on compact complex manifolds(CCM). A.Pillay, R.Moosa, T.Scanlon and others: Model theory of CCM ↭

Classification theory of CCM. (K.Ueno, A.Fujiki, F.Kampana...)

Zariski Geometries links

Compact  ?  Finite-Morley-rank
complex  ↭  substructures of a

manifolds diff. closed field Useful informal analogies (A.Pillay) Problem Find formal links.

Schanuel conjecure in differential fields

Using the equation

(∗) Dy = yDx

in differential fields J.Ax (1972) proved a function field case of Schanuel Conjecture:

If x_{1}, y_{1}, … x_{n}, y_{n} satisfy the equation (*) and Dx_{1}, …, Dx_{n} are ℚ-lineraly independent, then

tr.d._{C}(x_{1}, y_{1}, … x_{n}, y_{n}) > n.

W.D.Brownawell and K.K.Kubota (1977) extended this result to Weierstrass functions and Abelian integrals.

Problem Find a general model theoretic principle behind these results.

One more example

A powered algebraically closed field (with operations raising to non-standard powers)

(F, +, ·, x^{η})_{η∈ℕ̂} ℕ̂ ≻ ℕ.

Theorem Powered a.c.fields of characteristic 0 have superstable theories if the following is true: Conjecture Let W ⊆ ℂ^{n} be an irreducible algebraic variety over ℚ. There is finite family τ(W) of proper tori in ℂ^{n} such that for any torus T ⊆ ℂ^{n}, given an irreducible component S ⊆ W ∩ T with dim S > dim W + dim T − n, there is A ∈ τ(W) such that S ⊆ A.

This can be generalized to semiabelian varieties. In this general form The conjecture implies the Mordell-Lang conjecture over number and over function fields of characteristic 0.

III. Pseudo-analytic structures and Transcendental Number Theory

B. Zilber University of Oxford Hrushovski construction of new structures

Given a class of structures M with a dimension notion ∂ we want to consider a new function f on M. Then on (M, f) we can calculate a predimension

δ(X) = ∂(X ∪ f(X)) − size(X).

Consider now subclass of structures (M, f) which satisfy (Hrushovski inequality):

δ(X) ≥ 0 for any finite X ⊂ M.

Amalgamate all such structures to get a universal and homogeneous structure in the class. The resulting structure (M, f) will have a good dimension notion Δ(X) and a nice geometry.

Are Hrushovski structures mathematical pathologies?

Observation: If M is a field and we want f = ex to be a group homomorphism

ex(x_{1} + x_{2}) = ex(x_{1}) · ex(x_{2})

then the corresponding predimension must be

δ(X) = tr.d.(X ∪ ex(X)) − l.d.(X) ≥ 0.

The Hrushovski inequality, under assumption that X is ℚ-linearly independent, is equivalent to

tr.d.(X ∪ ex(X)) ≥ size(X),

which in case M = (ℂ, +, ·) and ex = exp is known as the Schanuel conjecture.

for linearly independent x_{1}, …, x_{n} ∈ ℂ

tr.d.(x_{1}, …, x_{n}, e^{x_{1}}, …, e^{x_{n}}) ≥ n.

Pseudo-exponentiation

Consider class F_{ex} of fields of characteristic 0 with a function ex K_{ex} = (K, +, ·, ex) satisfying EXP1: ex(x_{1} + x_{2}) = ex(x_{1}) · ex(x_{2}); EXP2: ker ex = πℤ;

and the Hrushovski inequality

SCH: for any finite X ⊂ F

δ(X) = tr.d.(X ∪ ex(X)) − l.d.(X) ≥ 0 Applying Fraisse amalgamation to F_{ex} one gets algebraically closed fields with pseudo-exponentiation K_{ex}.

K_{ex} satisfy algebraic-exponential closedness assumption.

EC: For any free and non-overdetermined irreducible system of polynomial equations

P (x_{1}, …, x_{n}, y_{1}, …, y_{n}) = 0

there exists a generic solution satisfying

y_{i} = ex(x_{i}) i = 1, …, n.

And Countable closure property CC: ’Analytic’ subsets of K^{n} of dimension 0 are countable.

Theorem Given an uncountable cardinal λ, there is a unique model of axioms ACF_{0} + EXP + SCH + EC + CC of cardinality λ.

This is a consequence of Theorems A and B:

Theorem A The L_{ω_{1},ω}(Q)-sentence ACF_{0} + EXP + SCH + EC + CC is axiomatizing a quasi-minimal excellent class.

Theorem B (S.Shelah 1983) A quasi-minimal excellent class has models and is categorical in any uncountable cardinality.

The proof of Theorem A is essentially the proof of the following fact of field arithmetic (plus Galois (Kummer) theory) Theorem C Given algebraically closed subfields L_{1}, …, L_{n} ⊆ ℂ and elements a_{1}, …, a_{m} ∈ ℂ^{∗} the multiplicative group of the field

k = ℚ(L_{1}, …, L_{n}, a_{1}, …, a_{m})

can be represented as

A · T · L_{1}^{∗} · . . . · L_{n}^{∗},

where A is free abelian, T the torsion subgroup (when n = 0) and L_{i}^{∗} the multiplicative groups of L_{i}.

In case of Abelian variety 𝔸(ℂ) in place of ℂ^{∗}, this corresponds to the similar properties of the group of k-rational points 𝔸(k), the impossibility of an infinite descent in the group.

The proof of Theorem C uses

1. Theory of fractional ideals of number fields.

2. Theory of Weil divisors and the normalization theorem.

Theorem B2 (S.Shelah) The assumptions of excellency are necessary for an L_{ω_{1},ω}- axiomatizable class to be categorical in all uncountable cardinalities.

Corollary The statement of Theorem C is essentially a form of model theoretic properties of categorical classes.

Conjecture. The field of complex numbers with the genuine exponentiation ℂ_{exp} is isomorphic to the canonical field with exponentiation K_{ex} of cardinality 2^{ℵ_{0}}.

Two crucial properties of K_{ex}.

SCH: The Schanuel condition

EC: The exponential-algebraic closedness: Any non-obviously-contradictory system of equations in +, ·, ex has a solution in K_{ex}.

A test for the main conjecture

Consider the class of fields with a function f on them satisfying ’the absolute Schanuel condition’

δ(X) = tr.d.(X ∪ f(X)) − size(X) ≥ 0.

Construct, using Hrushovski-Fraisse method, the canonical structure K_{f} = (K, +, ·, f) of cardinality 2^{ℵ_{0}}.

Problem Is there an analytic function f on ℂ such that ℂ_{f} is isomorphic to K_{f}?

It follows from the general theory that the first-order theory of K_{f} is ω-stable and is determined by two sets of axioms: (SCH) stating the absolute Schanuel condition (EC) stating the condition of f-algebraic closedness.

Any such ℂ_{f} would also satisfy CC and thus will be quasi-minimal excellent.

Theorem (A.Wilkie + P.Koiran, 2002) There is an entyre function f on ℂ satisfying SCH + EC