Analytic Zariski structures

B. Zilber

Logic and Mathematics: connections

and interactions Urbana-Champaign, 24 May 2003 1.1. We want to understand analytic (dimension) theory as an extension of stability theory.

1.2 What is the real analytic theory?

1.3. What is the non-Archimedean analytic theory?

1.4. Can we understand non-commutative algebraic geometry model-theoretically?

2. Want to extend Zariski geometries context to a non-compact, local context.

3. We want to understand Hrushovski’s examples of new stable structures.

Hrushovski’s counter-examples.

Given a class of structures M with a dimension notions d_{1} and d_{2} we want to consider a new function f on M. On (M, f) we can calculate a predimension

δ(X) = d_{1}(X ∪ f(X)) − d_{2}(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 D(X) and nice geometry.

Pseudo-exponentiation Apply Hrushovski’s construction to the class 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.deg(X∪ex(X))−lin.dim_{ℚ}(X) ≥ 0 Hrushovski’s inequality

SCH : tr.deg(X∪ex(X)) ≥ lin.dim_{ℚ}(X) for ex = exp, is equivalent to the Schanuel conjecture:

if x_{1}, …, x_{n} ∈ ℂ are ℚ-linearly independent, then tr.deg(x_{1}, …, x_{n}, e^{x_{1}}, …, e^{x_{n}}) ≥ n.

A resultung structure K_{ex} satisfies existential closedness condition

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

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

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

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

Pseudo-exponentiation structure K_{ex} is an example of a pseudo-analytic structure.

Typical

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

Other pseudo-analytic structures

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

(K, +, ·, x^{η})_{η∈ℕ̂} ℕ̂ ≻ ℕ.

A powered algebraically closed field with operations raising to irrational powers

(K, +, ·, x^{r})_{r∈ℚ(a_{1},...,a_{n})}

Theorem Both types of powered algebraicall closed fields ( char= 0) have superstable theories if the ’Schanuel’ conjecture above is true.

Analytic examples.

A Liouville function f : ℂ → ℂ is given as

f(x) = ∑_{i≥0} c_{i}x_{i}

c_{i}^{−1} = a_{i} are integers satisfying: for every l ≥ 1, for all sufficiently large i, |a_{i+1}| > |a_{i}|^{i^{l}}.

Theorem (A.Wilkie, P.Koiran, B.Z.) The first-order theory T_{f} of (ℂ, +, ·, f) is ω-stable, and

T_{f}+ CC

is categorical in all uncountable cardinals.

Real analytic examples.

Let ϵ = 1 + \sqrt{−1}

P = {f(ϵt + q) : t ∈ ℝ, t ≠ 0, q ∈ ℚ}

G = {exp(ϵt + q) : t ∈ ℝ, q ∈ ℚ}.

Theorem The first-order theories of (ℂ, +, ·, P) and (ℂ, +, ·, G) are ω-stable;

rkℂ = ω × 2, rkP = rkG = ω;

(ℂ, +, ·, P) is the Poizat’s bad field with black points and (ℂ, +, ·, G) is the bad field with green points (Schanuel conjecture)

The ’connected component’ of G

G^{0} = {e^{ϵt} = e^{t} cos t+ı·e^{t} sin t : t ∈ ℝ}

is the Miller-Speissegger spiral.

Theorem (Miller-Speissigger) The theory of

(ℝ, +, ·, {(e^{t} cos t, e^{t} sin t) : t ∈ ℝ}

allows QE to the level of ∃-formulas and is d-minimal. Any definable subset is of a low complexity in Borel hierarchy.

Let U(x, y) be the binary predicate on ℝ saying that x + ıy is a root of 1 of some power.

Theorem The theory of

(ℝ, +, ·, U)

allows QE to the level of nice ∃-formulas. Any definable subset is of a low complexity in Borel hierarchy.

(ℂ, +, ·, P) and (ℂ, +, ·, G)

rkℂ = ω × 2, rkP = rkG = ω.

Problem How can we get

rkℂ = ω × 3, rkP = ω?

Is this the theory of Hausdorff dimension?

Is this the bridge between stability and o-minimality?

Between the complex and real model theory?

Are (ℂ, +, ·, P), (ℂ, +, ·, G) and (ℝ, +, ·, U) objects of non-commutative geometry?

Analytic-Zariski structures with Nick Peatfield AZ-assumptions (sketchy):

M^{n} can be definably extended to P^{n}(M), all n s.t.

A collection of definable subsets of P^{n}(M)’s form a basis of a compact (non-Noetherian) topology.

Given an open U ⊆ P^{n}(M), some subsets closed in U are called analytic and have the properties:

- an analytic set has a finite dimension

- an analytic set has countably many irreducible analytic components;

- the addition and other dimension formulas for analytic sets; Pre-smoothness condition for analytic sets;

- ....

3. Near model completeness. Any definable set is a Boolean combination of projections of analytic subsets of open sets AZ-examples and semi-examples:

1. (M, R^{3}) Hrushovski’s ab initio triple relation rkM = ω

2. (K, +, ·, R^{3}) free triple relation on a.c.field K

3. (ℂ, +, ·, f) f Liouville (with nonstandard topology)

4. (ℂ, +, ·, f^{(n)})_{n∈ℤ} f^{(n)} Liouville and its derivatives, standard topology 5. (ℂ, +, ·, P) Poizat’s black points 6. (ℂ, +, ·, G) Poizat’s green points 7. (K, +, ·, x^{a}) powered a.c.field char= 0, a ∈ acl(ℚ)

8. (K, +, ·, x^{r}) powered a.c.field char= 0, any r

9. (K, +, ·, ex) pseudo-exponentiation 10. (ℂ, +, ·, exp) exponentiation.

AZ-assumptions (sketchy):

M^{n} can be definably extended to P^{n}(M), all n s.t.

A collection of definable subsets of P^{n}(M)’s form a basis of a compact (non-Noetherian) topology.

Given an open U ⊆ P^{n}(M), some subsets closed in U are called analytic and have the properties:

- an analytic set has a finite dimension

- an analytic set has countably many irreducible analytic components;

- the addition and other dimension formulas for analytic sets; AZ-assumptions (extended):

M^{n} can be definably extended to P^{n}(M), all n s.t.

A collection of definable subsets of P^{n}(M)’s form a basis of a compact (non-Noetherian) topology.

Given an open U ⊆ P^{n}(M), some subsets closed in U are called analytic and have the properties:

- an analytic set has a finite dimension

- an analytic set has boundedly many irreducible analytic components;

- the addition and other dimension formulas for analytic sets;