Analytic Zariski structures, predimensions and non-elementary stability

Boris Zilber University of Oxford

25 August 2008

The notion of an analytic Zariski structure was introduced in [1] by the author and N.Peatfield in a form slightly different from the one presented here. Analytic Zariski generalises the previously known notion of a Zariski structure (see [2] for one-dimensional case and [3], [4] for the general definition) mainly by dropping the requirement of Noetherianity and weakening the assumptions on the projections of closed sets. This leads to interesting new fenomena, in particular, the family of closed-in-open subsets forms a hierarchy which starts with analytic sets and continues by induction to more complex ones, called in [5] generalised analytic sets (defined classically on the complex numbers and in the context of rigid analytic geometry).

In [1] we assumed that the Zariski structure is compact (or compactifiable), here we drop this assumption (some interesting structures are not compactifiable in the strict sense).

The class of analytic Zariski structures is much broader and geometrically more rich than the class of Noetherian Zariski structures. The main examples come from two sources:

(i) structures which are constructed in terms of complex analytic functions and relations;

(ii) “new stable structures” introduced by Hrushovski’s construction; in many cases these objects exhibit properties similar to those of class (i).

Although there are concrete examples for both (i) and (ii), in many cases we can only conjecturally identify a particular structure as an analytic Zariski one. In particular despite some attempts the conjecture that ℂ_{exp} is analytic Zariski is still open (even assuming this is the same as pseudoexponentiation).

Note that the results of [1] are valid in the present context, in particular, if M is compact and C^{0} is the subfamily of closed sets S ⊆ M^{n}, all n, which are analytic in M^{n}, then (M, C^{0}) is a Noetherian Zariski structure. Hence, by [3], (M, C^{0}) has elimination of quantifiers and is of finite Morley rank.

The aim of this paper is to carry out a model-theoretic analysis of the whole M = (M, C). We do it in the spirit of the theory of abstarct elementary classes. We start by introducing a suitable countable fragment of the family of basic Zariski relations and a correspondent substructure of constants over which all the further analysis is carried out. Then we proceed to the analysis of the notion of dimension, already present in Zariski setting. We define a more delicate notions of the predimension and dimension of a tuple in M. In fact by doing this we reinterprete dimensions which are present in every analytic structure in terms familiar to many from Hrushovski’s construction, thus establishing once again conceptual links between classes (i) and (ii).

Our main results are proved under assumption that M is one-dimensional (as an analytic Zariski structure) and irreducible. No assumption on presmoothness is needed. We prove for such an M, in the terminology closely related to [6] and [7]:

1. M is quasiminimal with regards to a closure operator cl associated with the predimension;

2. M is homogeneous over countable submodels (see [6]) in the ∃-definable expansion of the language;

3. there are at most countably many L_{∞,ω}-types over any countable submodel realised in M (ω-stability).

It is tempting to conjecture that submodels of M form an excellent quasiminimal class. The condition for this is the amalgamation over independent submodels. This property is very difficult to prove in concrete cases (needs a lot of concrete algebra) but on the other hand we do not know counterexamples to this conjecture even in the present very general setting. We prove that if the class is indeed excellent then we can uniquely lift M to any uncountable cardinality.

I would like to remark that M.Gavrilovich in his DPhil thesis [8] proved 2 and 3 above for a very interesting class of actual complex analytic structures which includes universal covers of Abelian varieties. His method also produces a natural countable language for each case and exhibits deep relations of the model-theoretic conditions assumed below with some facts and conjectures of complex geometry.

Finally I want to mention that some natural questions in this context are widely open. In particular, we have no classification for presmooth analytic Zariski groups (with the graph of multiplication analytic).

I want to express my thanks to Assaf Hasson who saw a very early version of this work and made many useful comments.

1 Analytic Zariski structures

Let M = (M, C) a structure with primitives C. We assume that M^{n}, for all n ≥ 1, is endowed with a topology and relations of C define the closed subsets of the M^{n}’s. More precisely we assume for the primitives of the language:

(L)

1. arbitrary intersections of closed sets are closed;

2. finite unions of closed sets are closed;

3. the domain of the structure is closed;

4. the graph of equality is closed;

5. any singleton of the domain is closed;

6. Cartesian products of closed sets are closed;

7. the image of a closed S ⊆ M^{n} under a permutation of coordinates is closed;

8. for a ∈ M^{k} and S a closed subset of M^{k+ℓ} defined by a predicate S(x^{⌢}y), x ∈ M^{k}, y ∈ M^{ℓ}, the fibre over a, S(a, M) := {y ∈ M^{ℓ} : a^{⌢}y ∈ S}, is closed.

L6 needs some clarification. If S_{1} ⊆ M^{n} and S_{2} ⊆ M^{m} are closed the assumption states that S_{1} × S_{2} canonically identified with a subset of M^{n+m} is closed in the latter.

Remark 1.1 Note that it follows that projections pr_{i_{1},...,i_{m}} : M^{n} → M^{m} are continuous in the sense that the inverse image of a closed set under a projection is closed. Indeed, pr^{−1}S = S × M^{n−m}.

We write X ⊆_{op} V to say that X is open in V and X ⊆_{cl} V to say it is closed.

We say that X ⊆ M^{n} is C-constructible (usually omitting C) if X is a finite union of some sets S, such that S ⊆_{cl} U ⊆_{op} M^{n}.

A subset P ⊆ M^{n} will be called projective if P is a union of finitely many sets of the form pr S, for some S ⊆_{cl} U ⊆_{op} M^{n+k}, pr : M^{n+k} → M^{n}.

Note that any set S such that S ⊆_{cl} U ⊆_{op} M^{n+k}, is constructible, a projection of a constructible set is projective and that any constructible set is projective.

Dimension. To any nonempty projective S a non-negative integer called the dimension of S, dim S, is attached.

We assume:

(SI) (strong irreducibility) for an irreducible set S ⊆_{cl} U ⊆_{op} M^{n} (that is S is not a union of two proper closed subsets) and its closed subset S′ ⊆_{cl} S,

dim S′ = dim S ⇒ S′ = S;

(DP) (dimension of points) for a nonempty projective S, dim S = 0 if and only if S is at most countable.

(CU) (countable unions) If S = ⋃_{i∈ℕ} S_{i}, all projective, then dim S = max_{i∈ℕ} dim S_{i};

(WP) (weak properness) given irreducible S ⊆_{cl} U ⊆_{op} M^{n} and F ⊆_{cl} V ⊆_{op} M^{n+k} with the projection pr : M^{n+k} → M^{n} such that pr F ⊆ S and dim pr F = dim S, there exists D ⊆_{op} S such that D ⊆pr F.

Remark 1.2 (CU) in the presence of (DCC) implies the essential uncountability property (EU) usually assumed for Noetherian Zariski structures.

We postulate further, for S constructible irreducible:

(AF) dim pr S = dim S − min_{u∈pr S} dim(pr^{−1}(u) ∩ S); (FC) The set {a ∈ pr S : dim(pr^{−1}(a) ∩ S) ≥ k} is of the form T ∩ pr S

for some constructible T, and there exists an open set V such that V ∩ pr S ≠ ∅ and

min_{a∈pr S} dim(pr^{−1}(a) ∩ S) = dim(pr^{−1}(v) ∩ S), for any v ∈ V ∩ pr (S).

The following helps to understand the dimension of projective sets.

Lemma 1.3 Let P = pr S ⊆M^{n}, for S irreducible constructible, and U ⊆_{op} M^{n} with P ∩ U ≠ ∅. Then

dim P ∩ U = dim P.

Proof. We can write P ∩U = pr S′ = P′, where S′ = S∩pr^{−1}U constructible irreducible, dim S′ = dim S by (SI). By (FC), there is V ⊆_{op} M^{n} such that for all c ∈ V ∩ P,

dim pr^{−1}(c) ∩ S = min_{a∈P} dim pr^{−1}(a) ∩ S = dim S − dim P.

Note that pr^{−1}U ∩ pr^{−1}V ∩ S ≠ ∅, since S ir irreducible. Taking s ∈ pr^{−1}U ∩ pr^{−1}V ∩ S and c = pr s we get, using (AF) for S′,

dim pr^{−1}(c) ∩ S′ = dim pr^{−1}(c) ∩ S = min_{a∈P′} dim pr^{−1}(a) ∩ S = dim S − dim P′.

So, dim P′ = dim P. □

Analytic subsets.

Definition 1.4 A subset S, S ⊆_{cl} U ⊆_{op} M^{n}, is called analytic in U if for every a ∈ S there is an open V_{a} ⊆_{op} U such that S ∩ V_{a} is the union of finitely many closed in V_{a} irreducible subsets.

We postulate the following properties

(INT) (Intersections) If S_{1}, S_{2} ⊆_{an} U are irreducible then S_{1} ∩ S_{2} is analytic in U; (CMP) (Components) If S ⊆_{an}U and a ∈ S then there is S_{a} ⊆_{an}U, a finite

union of irreducible analytic subsets of U, and some S′_{a} ⊆_{an}U such that a ∈ S_{a} \ S′_{a} and S = S_{a} ∪ S′_{a};

Each of the irreducible subsets of S_{a} above is called an irreducible component of S containing a.

(CC) (Countability of the number of components) Any S ⊆_{an} U is a

union of at most countably many irreducible components. Remark 1.5 For S analytic and a ∈ pr S, the fibre S(a, M) is analytic. Lemma 1.6 If S ⊆_{an} U is irreducible, V open, then S ∩ V is an irreducible analytic subset of V and, if non-empty, dim S ∩ V = dim S. Proof. Immediate.□

Lemma 1.7 (i) ∅, any singleton and U are analytic in U; (ii) If S_{1}, S_{2} ⊆_{an} U then S_{1} ∪ S_{2} is analytic in U; (iii) If S_{1} ⊆_{an} U_{1} and S_{2} ⊆_{an} U_{2}, then S_{1} × S_{2} is analytic in U_{1} × U_{2}; (iv) If S ⊆_{an} U and V ⊆U is open then S ∩ V ⊆_{an} V;

(v) If S_{1}, S_{2} ⊆_{an} U then S_{1} ∩ S_{2} is analytic in U. Proof. Immediate. □

Definition 1.8 Given a subset S ⊆_{cl} U ⊆_{op} M^{n} we define the notion of the analytic rank of S in U, ark_{U}(S), which is a natural number satisfying

1. ark_{U}(S) = 0 iff S = ∅;

2. ark_{U}(S) ≤ k + 1 iff there is a set S′ ⊆_{cl} S such that ark_{U}(S′) ≤ k and with the set S^{0} = S \ S′ being analytic in U \ S′.

Obviously, any nonempty analytic subset of U has analytic rank 1.

Example 1.9 In [5] we have discussed the following notion of generalised analytic subsets of [P^{1}(ℂ)]^{n} and, more generally, of [P^{1}(K)]^{n} for K algebraically closed complete valued field.

Let F ⊆ℂ^{2} be a graph of an entire analytic function and F̄ its closure in [P^{1}(ℂ)]^{2}. It follows from Picar’s Theorem that F̄ = F ∪ {∞} × P^{1}(ℂ), in particular F̄ has analytic rank 2.

Generalised analytic sets are defined as the subsets of [P^{1}(ℂ)]^{n} for all n, obtained from classical (algebraic) Zariski closed subsets of [P^{1}(ℂ)]^{n} and F̄ by applying the positive operations: Cartesian products, finite intersections, unions and projections.

It has been proven in [5] (by a simple induction on the number of operation) that any generalised analytic set is of finite analytic rank.

The next assumptions guarantees that the class of analytic subsets explicitly determines the class of closed subsets in M.

(AS) [Analytic stratification] For any S ⊆_{cl} U ⊆_{op} M^{n}, ark_{U}S is defined and finite.

We also are going to consider the property

(PS) [Presmoothness] If S_{1}, S_{2} ⊆_{an} U ⊆_{op} M^{n} both S_{1}, S_{2} irreducible, then for any irreducible component S_{0} of S_{1} ∩ S_{2}

dim S_{0} ≥ dim S_{1} + dim S_{2} − dim U.

Definition 1.10 A topological structure satisfying axioms (L)-(AS) above will be called an analytic Zariski structure. An analytic Zariski structure will be called presmooth if it has the presmoothness property (PS).

2 Model theory of analytic Zariski structures

Definition 2.1 Let M_{0} be a nonempty subset of M and C_{0} a subfamily of C. We will say that (M_{0}, C_{0}) is a core substructure if

1. if {⟨x_{1}, …, x_{n}⟩} ∈ C_{0} (a singleton) then x_{1}, …, x_{n} ∈ M_{0}; 2. finite intersections of C_{0}-closed sets are in C_{0};

3. C_{0} satisfies (L1)-(L7), and (L8) with a ∈ M_{0}^{k};

4. C_{0} satisfies (WP), (AF), (FC) and (AS);

5. for any C_{0}-constructible S ⊆_{an} U ⊆_{op} M^{n}, every irreducible component S_{i} of S is C_{0}-constructible;

6. for any nonempty C_{0}-constructible U ⊆M, U ∩ M_{0} ≠ ∅.

Lemma 2.2 Given any countable N ⊆ M and countable C ⊆ C there exist countable M_{0} ⊇N and C_{0} ⊇C such that (M_{0}, C_{0}) is a core substructure.

Proof. Standard. □

We fix below a core substructure (M_{0}, C_{0}) with M_{0} and C_{0} countable.

Definition 2.3 For finite X ⊆M we define the C_{0}-predimension

δ(X) = min{dim S : X⃗ ∈ S, S ⊆_{an} U ⊆_{op} M^{n}, S is C_{0}-constructible}

and dimension

∂(X) = min{δ(XY) : finite Y ⊂ M}.

For X ⊆ M finite, we say that X is self-sufficient and write X ≤ M, if ∂(X) = δ(X).

For infinite A⊆M we say A ≤ M if for any finite X ⊆A there is a finite X ⊆X′ ⊆A such that X′ ≤ M.

We work now under Assumption dim M = 1 and M is irreducible.

Note that we then have

0 ≤ δ(Xy) ≤ δ(X) + 1, for any y ∈ M,

since X⃗y ∈ S × M.

Lemma 2.4 Given F ⊆_{an} U ⊆_{op} M^{k}, dim F > 0, there is i ≤ k such that for pr_{i} : (x_{1}, …, x_{k}) ↦ x_{i},

dim pr_{i}F > 0.

Proof. Use (AF) and induction on k.□

Proposition 2.5 Let P = pr S, for some C_{0}-constructible S ⊆_{an} U ⊆_{op} M^{n+k}, pr : M^{n+k} → M^{n}. Then

dim P = max{∂(x) : x ∈ P (M)}.  (1)

Moreover, this formula is true when S ⊆_{cl} U ⊆_{op} M^{n+k}.

Proof. We use induction on dim S.

We first note that by induction on ark_{U}S, if (1) holds for all analytic S of dimension less or equal to k then it holds for all closed S of dimension less or equal to k.

The statement is obvious for dim S = 0 and so we assume that dim S > 0 and for all analytic S′ of lower dimension the statement is true.

By (CU) and (CMP) we may assume that S is irreducible. Then by (AF)

dim P = dim S − dim S(c, M)  (2)

for any c ∈ P (M) ∩ V (M) (such that S(c, M) is of minimal dimension) for some open C_{0}-constructible V.

Claim 1. It is enough to prove the statement of the proposition for the projective set P ∩ V′, for some C_{0}-open V′ ⊆_{op} M^{n}.

Indeed,

P ∩ V′ = pr(S ∩ pr^{−1}V′), S ∩ pr^{−1}V′ ⊆_{cl} pr^{−1}V′ ∩ U ⊆_{op} M^{n+k}.

And P \V′ = pr(S ∩T), T = pr^{−1}(M^{n} \V′) ∈ C_{0}. So, P \V′ is the projection of a proper analytic subset, of lower dimension. By induction, for x ∈ P \V′, ∂(x) ≤ dim P \ V′ ≤ dim P and hence, using 1.3,

dim P ∩ V′ = max{∂(x) : x ∈ P ∩ V′} ⇒ dim P = max{∂(x) : x ∈ P }.

Claim 2. The statement of the proposition holds if dim S(c, M) = 0 in (2).

Proof. Given x ∈ P choose a tuple y ∈ M^{k} such that S(x^{⌢}y) holds. Then δ(x^{⌢}y) ≤ dim S. So we have ∂(x) ≤ δ(x^{⌢}y) ≤ dim S = dim P.

It remains to notice that there exists x ∈ P such that ∂(x) ≥ dim P. Consider the C_{0}-type

x ∈ P &{x ∉ R : dim R ∩ P < dim P and R is projective}.

This is realised in M, since otherwise P = ⋃_{R}(P ∩R) which would contradict (CU) because (M_{0}, C_{0}) is countable.

For such an x let y be a tuple in M such that δ(x^{⌢}y) = ∂(x). By definition there exist S′ ⊆_{an} U′ ⊆_{op} M^{m} such that dim S′ = δ(x^{⌢}y). Let P′ = pr S′, the projection into M^{n}. By our choice of x, dim P′ ≥ dim P. But dim S′ ≥ dim P′. Hence, ∂(x) ≥ dim P. Claim proved.

Claim 3. There is a C_{0}-constructible R ⊆_{an} S such that all the fibers R(c, M) of the projection map R → pr R are 0-dimensional and dim pr R = dim P.

Proof. We have by construction S(c, M)⊆M^{k}. Assuming dim S(c, M) > 0 on every open subset we show that there is a b ∈ M_{0} such that (up to the order of coordinates) dim S(c, M) ∩ {b} × M^{k−1} < dim S(c, M), for all c ∈ P ∩ V′ ≠ ∅, for some open V′ ⊆V and dim pr S(c, M) ∩ {b} × M^{k−1} = dim P. By induction on dim S this will prove the claim.

To find such a b choose a ∈ P ∩ V and note that by 2.4, up to the order of coordinates, dim pr_{1}S(a, M) > 0, where pr_{1} : M^{k} → M is the projection on the first coordinate.

Consider the projection pr_{M^{n},1} : M^{n+k} → M^{n+1} and the set pr_{M^{n},1}S. By (AF) we have

dim pr_{M^{n},1}S = dim P + dim pr_{1}S(a, M) = dim P + 1.

Using (AF) again for the projection pr^{1} : M^{n+1} → M with the fibers M^{n} × {b}, we get, for all b in some open subset of M,

1 ≥ dim pr^{1}pr_{M^{n},1}S = dim pr_{M^{n},1}S − dim[pr_{M^{n},1}S] ∩ [M^{n} × {b}] =
= dim P + 1 − dim[pr_{M^{n},1}S] ∩ [M^{n} × {b}].

Hence dim[pr_{M^{n},1}S] ∩ [M^{n} × {b}] ≥ dim P, for all such b, which means that the projection of the set S_{b} = S ∩ (M^{n} × {b} × M^{k−1}) on M^{n} is of dimension dim P, which finishes the proof if b ∈ M_{0}. But dim S_{b} = dim S − 1 for all b ∈ M ∩V′, some C_{0}-open V′, so for any b ∈ M_{0} ∩V′. The latter is not empty since (M_{0}, C_{0}) is a core substructure. This proves the claim.

Claim 4. Given R satisfying Claim 3,

P \ pr R⊆pr S′, for some S′ ⊆_{cl} S, dim S′ < dim S.

Proof. Consider the cartesian power

M^{n+2k} = {x^{⌢}y^{⌢}z : x ∈ M^{n}, y ∈ M^{k}, z ∈ M^{k}}

and its C_{0}-constructible subset

R&S := {x^{⌢}y^{⌢}z : x^{⌢}z ∈ R & x^{⌢}y ∈ S}.

Clearly R&S ⊆_{an} W ⊆_{op} M^{n+2k}, for an appropriate C_{0}-constructible W.

Now notice that the fibers of the projection pr_{xy} : x^{⌢}y^{⌢}z ↦ x^{⌢}y over pr_{xy}R&S are 0-dimensional and so, for some irreducible component (R&S)^{0} of the analytic set R&S, dim pr_{xy}(R&S)^{0} = dim S. Since pr_{xy}R&S ⊆ S and S irreducible, we get by (WP) D ⊆pr_{xy}R&S for some D ⊆_{op} S. Clearly

pr R = pr pr_{xy}R&S ⊇pr D

and S′ = S \ D satisfies the requirement of the claim.

Now we complete the proof of the proposition: By Claims 2 and 3

dim P = max_{x∈pr R} ∂(x).

By induction on dim S, using Claim 4, for all x ∈ P \ pr R,

∂(x) ≤ dim pr S′ ≤ dim P.

The statement of the proposition follows. □

In what follows a C_{0}-substructure of M is a C_{0}-structure on a subset N ⊇ M_{0}. Recall that C_{0} is purely relational.

Recall the following well-known Fact Given a, a′ ∈ M^{n} the L_{∞,ω}(C_{0})-types of the two n-tuples in M are equal if and only if they are back and forth equivalent that is there is a nonempty set I of isomorphisms of C_{0}-substructures of M such that a ∈ Dom f_{0} and a′ ∈ Range f_{0}, for some f_{0} ∈ I, and

(forth) for every f ∈ I and b ∈ M there is a g ∈ I such that f ⊆ g and b ∈ Dom g;

(back) For every f ∈ I and b′ ∈ M there is a g ∈ I such that f ⊆ g and b′ ∈ Range g.

Definition 2.6 For a ∈ M^{n}, the projective type of a over M is

{P (x) : a ∈ P, P is a projective set over C_{0}}∪
∪{¬P (x) : a ∉ P, P is a projective set over C_{0}}.

Lemma 2.7 Suppose X ≤ M, X′ ≤ M and the (first-order) quantifier-free C_{0}-type of X is equal to that of X′. Then the L_{∞,ω}(C_{0})-types of X and X′ are equal.

Proof. We are going to construct a back-and-forth system for X and X′.

Let S_{X} ⊆_{an} V ⊆_{op} M^{n}, S_{X} irreducible, all C_{0}-constructible, and such that X ∈ S_{X}(M) and dim S_{X} = δ(X).

Claim 1. The quantifier-free C_{0}-type of X (and X′) is determined by formulas equivalent to S_{X} ∩ V′, for V′ open such that X ∈ V′(M).

Proof. Use the stratification of closed sets (AS) to choose C_{0}-constructible S ⊆_{cl} U ⊆_{op} M^{n} such that X ∈ S and ark_{U}S is minimal. Obviously then ark_{U}S = 0, that is S ⊆_{an} U ⊆_{op} M^{n}. Now S can be decomposed into irreducible components, so we may choose S to be irreducible. Among all such S choose one which is of minimal possible dimension. Obviously dim S = dim S_{X}, that is we may assume that S = S_{X}. Now clearly any constructible set S′ ⊆_{cl} U′ ⊆_{op} M^{n} containing X must satisfy dim S′ ∩ S_{X} ≥ dim S_{X}, and this condition is also sufficient for X ∈ S′.

Let y be an element of M. We want to find a finite Y containing y and an Y′ such that the quantifier-free type of XY is equal to that of X′Y′ and both are self-sufficient in M. This, of course, extends the partial isomorphism X → X′ to XY → X′Y′ and will prove the lemma.

We choose Y to be a minimal set containing y and such that δ(XY) is also minimal, that is

1 + δ(X) ≥ δ(Xy) ≥ δ(XY) = ∂(XY)

and XY ≤ M.

We have two cases: δ(XY) = ∂(X) + 1 and δ(XY) = ∂(X). In the first case Y = {y}. By Claim 1 the quantifier-free C_{0}-type r_{Xy} of Xy is determined by the formulas of the form (S_{X} × M) \ T, T ⊆_{cl}M^{n+1}, T ∈ C_{0}, dim T < dim(S_{X} × M).

Consider

r_{Xy}(X′, M) = {z ∈ M : X′z ∈ (S_{X} × M) \ T, dim T < dim S_{X}, all T }.

We claim that r_{Xy}(X′, M) ≠ ∅. Indeed, otherwise M is the union of countably many sets of the form T (X′, M). But the fibers T (X′, M) of T are of dimension 0 (since otherwise dim T = dim S_{X} + 1, contradicting the definition of the T). This is impossible, by (CU).

Now we choose y′ ∈ r_{Xy}(X′, M) and this is as required.

In the second case, by definition, there is an irreducible R ⊆_{an} U ⊆_{op} M^{n+k}, n = |X|, k = |Y |, such that XY ∈ R(M) and dim R = δ(XY) = ∂(X). We may assume U ⊆V × M^{k}.

Let P = pr R, the projection into M^{n}. Then dim P ≤ dim R. But also dim P ≥ ∂(X), by 2.5. Hence, dim R = dim P. On the other hand, P ⊆ S_{X} and dim S_{X} = δ(X) = dim P. By axiom (WP) we have S_{X} ∩ V′ ⊆P for some C_{0}-constructible open V′.

Hence X′ ∈ S_{X} ∩ V′ ⊆ P (M), for P the projection of an irreducible analytic set R in the C_{0}-type of XY. By Claim 1 the quantifier-free C_{0}-type of XY is of the form

r_{XY} = {R \ T : T ⊆_{cl} R, dim T < dim R}.

Consider

r_{XY} (X′, M) = {Z ∈ M^{k} : X′Z ∈ R \ T, T ⊆_{cl} R, dim T < dim R}.

We claim again that r_{XY} (X′, M) ≠ ∅. Otherwise the set R(X′, M) = {X′Z : R(X′Z)} is the union of countably many subsets of the form T (X′, M). But dim T (X′, M) < dim R(X′, M) as above, by (AF).

Again, an Y′ ∈ r_{XY} (X′, M) is as required.□

Corollary 2.8 There is at most countably many L_{∞,ω}(C_{0})-types of tuples X ≤ M.

Indeed, any such type is determined uniquely by the choice of a C_{0}-constructible S_{X} ⊆_{an} U ⊆_{op} M^{n} such that dim S_{X} = ∂(X).

Lemma 2.9 Suppose, for finite X, X′ ⊆M, the projective C_{0}-types of X and X′ coincide. Then the L_{∞,ω}(C_{0})-types of the tuples are equal.

Proof. Choose finite Y such that ∂(X) = δ(XY). Then XY ≤ M. Let XY ∈ S ⊆_{an} U ⊆_{op} M^{n} be C_{0}-constructible and such that dim S is minimal possible, that is dim S = δ(XY). We may assume that S is irreducible. Notice that for every proper closed C_{0}-constructible T ⊆_{cl} U, XY ∉ T by dimension considerations.

By assumptions of the lemma X′Y′ ∈ S, for some Y′ in M. We also have X′Y′ ∉ T, for any T as above, since otherwise a projective formula would imply that XY′′ ∈ T for some Y′′, contradicting that ∂(X) > dim T.

We also have δ(X′Y′) = dim S. But for no finite Z′ it is possible that δ(X′Z′) < dim S, for then again a projective formula will imply that δ(XZ) < dim S, for some Z.

It follows that X′Y′ ≤ M and the quantifier-free types of XY and X′Y′ coincide, hence the L_{∞,ω}(C_{0})-types are equal, by 2.7.□

Definition 2.10 Set, for finite X ⊆M,

cl_{C_{0}}(X) = {y ∈ M : ∂(Xy) = ∂(X)}.

We fix C_{0} and omit the subscript below.

Lemma 2.11 b ∈ cl(A), for A⃗ ∈ M^{n}, if and only if b ∈ P (A⃗, M) for some projective P ⊆ M^{n+1} such that P (A⃗, M) is at most countable. In particular, cl(A) is countable for any finite A.

Proof. Let d = ∂(A) = δ(AV), and δ(AV) is minimal for all possible finite V ⊆ M. So by definition d = dim S_{0}, some analytic irreducible S_{0} such that A⃗V ∈ S_{0} and S_{0} of minimal dimension. This corresponds to a C_{0}-definable relation S_{0}(x, v), where x, v strings of variables of length n, m

First assume that b belongs to a countable P (A⃗, M). By definition

P (x, y) ≡ ∃w S(x, y, w),

for some analytic S ⊆ M^{n+1+k}, x, y, w strings of variables of length n, 1 and k and the fiber S(A⃗, b, M^{k}) is nonempty. We also assume that P and S are of minimal dimension, answering this description. By (FC), (AS) and minimality we may choose S so that dim S(A⃗, b, M^{k}) is minimal among all the fibers S(A⃗′, b′, M^{k}).

Consider the analytic set S^{♯} ⊆ M^{n+m+1+k} given by S_{0}(x, v) & S(x, y, w). By (AF), considering the projection of the set on (x, v)-coordinates,

dim S^{♯} ≤ dim S_{0} + dim S(A⃗, M, M^{k}),

since S(A⃗, M, M^{k}) is a fiber of the projection. Now we note that by countability dim S(A⃗, M, M^{k}) = dim S(A⃗, b, M^{k}), so

dim S^{♯} ≤ dim S_{0} + dim S(A⃗, b, M^{k}).

Now the projection pr_{w}S^{♯} along w (corresponding to ∃w S^{♯}) has fibers of the form S(X⃗, y, M^{k}), so by (AF)

dim pr_{w}S^{♯} ≤ dim S_{0} = d.

Projecting further along v we get dim pr_{v}pr_{w}S^{♯} ≤ d, but A⃗b ∈ pr_{v}pr_{w}S^{♯} so by Proposition 2.5 ∂(A⃗b) ≤ d. The inverse inequality holds by definition, so the equality holds. This proves that b ∈ cl(A).

Now, for the converse, we assume that b ∈ cl(A). So, ∂(A⃗b) = ∂(A⃗) = d. By definition there is a projective set P containing A⃗b, defined by the formula ∃w S(x, y, w) for some analytic S, dim S = d. Now A⃗ belongs to the projective set pr_{y}P (defined by the formula ∃y∃w S(x, y, w)) so by Proposition 2.5 d ≤ dim pr_{y}P, but dim pr_{y}P ≤ dim P ≤ dim S = d. Hence all the dimensions are equal and so, the dimension of the generic fiber is 0, but as above We may assume without loss of generality that all fibers are of minimal dimension, so

dim S(A⃗, M, M^{k}) = 0.

Hence, b belongs to a 0-dimensional set ∃w S(A⃗, y, w), which is projective and countable. □

Lemma 2.12 (i)

cl(∅) = cl(M_{0}) = M_{0}.

(ii) Given finite X ⊆M, y, z ∈ M,

z ∈ cl(X, y) \ cl(X) ⇒ y ∈ cl(X, z).

(iii)

cl(cl(X)) = cl(X).

Proof. (i) Clearly M_{0} ⊆cl(∅), by definition.

We need to show the converse, that is if ∂(y) = 0, for y ∈ M, then y ∈ M_{0}. By definition ∂(y) = ∂(∅) = min{δ(Y) : y ∈ Y ⊂ M} = 0. So, y ∈ Y, Y⃗ ∈ S ⊆_{an} U ⊆_{op} M^{n}, dim S = 0. The irreducible components of S are points (singletons), so {Y⃗ } is one and must be in C_{0}, since (M_{0}, C_{0}) is a core substructure. By 2.1.1, y ∈ M_{0}.

(ii) Assuming the left-hand side ∂(Xyz) = ∂(Xy) > ∂(X), ∂(Xz) > ∂(X). By the definition of ∂ then,

∂(Xy) = ∂(X) + 1 = ∂(Xz),

so ∂(Xzy) = ∂(Xz), y ∈ cl(Xz).

(iii) Immediate by 2.11. □

Below, if not stated otherwise, we use the language C_{0}^{∃} the primitives of which correspond to relations ∃-definable in M. Also, we call a submodel of M any C_{0}^{∃}-substructure closed under cl.

Theorem 2.13 (i) Every L_{∞,ω}(C_{0})-type realised in M is equivalent to a projective type, that is a type consisting of existential (first-order) formulas and the negations of existential formulas.

(ii) There are only countably many L_{∞,ω}(C_{0})-types realised in M.

(iii) (M, C_{0}^{∃}) is quasiminimal ω-homogeneous over countable submodels.

Proof. (i) Immediate from 2.9.

(ii) By 2.8 there are only countably many types of finite tuples Z ≤ M. Let N ⊆ M_{0} be a countable subset of M such that any finite Z ≤ M is L_{∞,ω}(C_{0})-equivalent to some tuple in N. Every finite tuple X ⊂ M can be extended to XY ≤ M, so there is a L_{∞,ω}(C_{0})-monomorphism XY → N. This monomorphism identifies the L_{∞,ω}(C_{0})-type of X with one of a tuple in N, hence there are no more than countably many such types.

(iii) Lemma 2.12 proves that cl defines a pregeometry on M. Letting G := cl_{C_{0}}(A) we we need to check, according to [6], that (a)for any X and X′, both cl-independent over G, of the same size n, a bijection ϕ : X → X′ is a G-monomorphism, and (b) given any G-monomorphism ϕ : X → X′ and y ∈ M we can extend ϕ so that y ∈ Dom ϕ.

Consider first (a). Note that GX ≤ M and GX′ ≤ M and so the types of X and X′ over G are C_{0}-quantifier-free. But there is no proper C_{0}-closed subset S ⊆_{cl} M^{n} such that X⃗ ∈ S or X⃗′ ∈ S. Hence the types are equal.

For (b) just use the fact that the G-monomorphism by our definition preserves ∃-formulas, so by 2.9 complete L_{∞,ω}(C_{0})-types of X and X′ coincide, so ϕ can be extended. □

Now we want to define an abstract elementary class A associated with M. Set

A_{0}(M) = { countable C_{0}^{∃}-structures N : N ≅ N′ ⊆M, cl(N′) = N′}

and define embedding N_{1} ≼ N_{2} in the class as an C_{0}-embedding f : N_{1} → N_{2} such that there are isomorphisms g_{i} : N_{i} → N′_{i}, N′_{1} ⊆N′_{2} ⊆M, all embeddings commuting and cl(N′_{i}) = N′_{i}.

Recall the construction from [9] (of a special set and quasi-minimal excellence) corresponding to Shelah’s notion of an independent system and excellence.

Let B ⊂ M be a countable cl-independent set and B_{1}, …, B_{k} ⊂ B, ⋃ B_{i} = B. Consider the structures N_{i} = cl(B_{i}), these obviously belong to A_{0}(M). A system of structures N_{i} of the form cl(B_{i}) as above will be called an independent system in A_{0}(M).

We say that A_{0}(M) is excellent if for any independent system {N_{i} : i ≤ k} and a finite X ⊂ C = cl(⋃_{i} N_{i}) the L_{∞,ω}(C_{0})-type of X over C is equivalent to a subtype over a finite C′ ⊂ C.

Now define A(M) to be the class of C_{0}-structures H with cl_{C_{0}} defined with respect to H and satisfying:

(i) A_{0}(H)⊆A_{0}(M) as classes with embeddings

and

(ii) for every finite X ⊆H there is N ∈ A_{0}(H), such that X ⊆N.

Note that in any H ∈ A(M), given a finite X ⊆ H, the set cl(X) is countable. That is A(M) satisfies the countable closure property. Indeed, by definition there exists a countable N ∈ A_{0}(H) with X ⊆N. Since cl_{H}(N) = N, also cl_{H}(X)⊆N.

Given H_{1} ⊆ H_{2}, H_{1}, H_{2} ∈ A(M), we define H_{1} ≼ H_{2} to hold in the class, if for every finite X ⊆ H_{1}, cl(X) is the same in H_{1} and H_{2}. More generallly, for H_{1}, H_{2} ∈ A(M) we define H_{1} ≼_{f} H_{2} to be an embedding f such that there are isomorphisms H_{1} ≅ H′_{1}, H_{2} ≅ H′_{2} such that H′_{1} ⊆H′_{2}, all embeddings commute, and H_{1} ≼ H_{2}.

Lemma 2.14 A(M) is closed under the unions of ascending ≼-chains.

Proof. Immediate from the fact that for infinite Y ⊆M,

cl(Y) = ⋃{cl(X) : X ⊆_{finite}Y }.

□

Theorem 2.15 Given an analytic Zariski structure M and a countable core substructure (M_{0}, C_{0}), assume that A_{0}(M) is excellent. Then the class A(M) contains structures of any infinite cardinality and is categorical in uncountable cardinals.

Proof. This is essentially the same proof that of Theorem 2 of [9], which is based on Theorem 1 of the same paper.

The starting point of the proof in [9] establishes the existence in the class of an H_{0} of the form H_{0} = cl(A_{0}), for A_{0} an infinite countable cl-independent set. Under the present assumptions this is immediate since such H_{0} exists already in A_{0}(M).

The rest of the proof just repeats that of Theorem 2 of [9]. One constructs an ascending chain of structures H_{α} ∈ A(M) of the form cl(A_{α}), for some cl-independent sets A_{α}, up to a given cardinal κ, so that A_{α+1} \ A_{α} = {a_{α}}. H_{α+1} is defined uniquely up to isomorphism by a bijection ψ_{α} : A_{α} → A_{α+1} and the fact proved in Theorem 1 of [9]: under assumptions of quasi-minimal excellence ψ_{α} can be extended to an isomorphism of cl(A_{α}) → cl(A_{α+1}). □

Problem Given an analytic Zariski structure M is there always a big enough countable core substructure (M_{0}, C_{0}) such that the class A_{0}(M) is excellent?

The example in the next section, of the cover of the algebraic torus K^{×}, for K an algebraically closed field, is excellent for a natural choice of C_{0} and some M_{0}. The proof even in this basic example is difficult. In other, more general cases, the answer is not known or known partially and is closely linked to difficult problems in arithmetic algebraic geometry, see [11].

We introduce one more condition which holds for many known analytic Zariski structures. We say that the language of M is essentially countable, (CL) for short, if there is a countable C_{base} ⊂ C such that every S ∈ C is of the form S = P (a, M), for some P ∈ C_{base} and a ∈ M^{k}.

Proposition 2.16 Under assumptions of Theorem 2.15, if also the language of M is essentially countable and assuming without loss of generality C_{base} ⊆C_{0}, any uncountable H ∈ A(M) is an analytic Zariski structure in the language C_{0} with parameters in H. Also H is presmooth if M is.

Proof. We define C(H) to consist of the subsets of H^{n} of the form P (a, H) for P ∈ C_{0} of arity k + n, a ∈ H^{k}. The assumption (L) is obviously satisfied.

Now note that the constructible and projective sets in C(H) are also of the form P (a, H) for some C_{0}-constructible or C_{0}-projective P.

Define dim P (a, H) = d if dim P (b, M) = d for some b ∈ M^{k} such that The C_{0}^{∃}-quantifier-free types of a and b are equal. This is well-defined by (FC) and the fact that the any C_{0}^{∃}-quantifier-free type realised in H is also realised in M. Moreover, we have the following. Claim. The set of C_{0}^{∃}-quantifier-free types realised in H is equal to that realised in M.

Indeed, this is immediate from the definition of the class A(M), stability of A(M) and the fact that the class is categorical in uncountable cardinalities.

The definition of dimension immediately implies (DP), (CU),(AF) and (FC) for H.

(SI): if P_{1}(a_{1}, H) ⊆_{cl} P_{0}(a_{0}, H), dim P_{1}(a_{1}, H) = dim P_{0}(a_{0}, H) and the two sets are not equal, then the same holds for P_{1}(b_{1}, M) and P_{0}(b_{0}, M) for equivalent b_{0}, b_{1} in M. Then, P_{0}(b_{0}, M) is reducible, tha is for some proper P_{2}(b_{2}, M) ⊆_{cl} P_{0}(b_{0}, M), P_{0}(b_{0}, M) = P_{1}(b_{1}, M) ∪ P_{2}(b_{2}, M). Now, by homogeneity we can choose a_{2} in H such that P_{0}(a_{0}, H) = P_{1}(a_{1}, H) ∪ P_{2}(a_{2}, H), a reducible representation.

This also shows that the notion of irreducibility is preserved by equivalent substituition of parameters. Then the same is true for the notion of analytic subset. Hence (INT), (CMP),(CC) and (PS) follow. For the same reason (AS) holds. Next we notice that the axioms (WP) follows by the homogeneity argument. □

3 Some examples

We consider the universal cover of ℂ^{×} as a topological structure and show that this is analytic Zariski.

This is a structure with the universe V identified with the set of complex numbers ℂ and we are going to use the additive structure on it. We also consider the usual exponentiation map

exp : V → ℂ^{×}

and want to take into our language and topology the usual Zariski topology (of algebraic varieties) on (ℂ^{×})^{n} as well as exp as a continuous map.

A model-theoretic analysis of this structure was carried out in [9], [10],

[11] and in the DPhil thesis [12] of Lucy Smith. The latter work, based on [9], provides the description of the topology C on V and proves that (V, C) is analytic Zariski. (It then addresses the issue of possible compacifications of the structure).

Note that the whole analysis below up to Corollary 3.9 uses only the first order theory of the structure (V, C), so one can wonder what changes if we replace ℂ and exp with its abstract analogues. The answer to this question is known in the form of the categoricity theorem proved in [9] and [6] (see some corrections in [10]): if ex : U → K^{×} is a group homomorphism, U a divisible torsion-free group, K an algebraically closed field of characteristic 0 and cardinality continuum and ker(ex) is cyclic, then the structure is isomorphic to the structure (V, C) on the complex numbers. More generally, any two covers of 1-tori over algebraically closed fields of characteristic 0 of the same uncountable cardinality and with cyclic kernels are isomorphic.

We follow [12] pp.17-25 with modifications and omission of some technical details.

The base of the PQF-topology (positive quantifier-free) on V and its cartesian powers V^{n} is, in short, the family of subsets of V^{n} defined by PQFformulae.

Definition 3.1 A PQF-closed set is defined as a finite union of sets of the form

L ∩ m · ln W  (3)

where W ⊆ (ℂ^{×})^{n} an algebraic subvariety and L is a ℚ-linear subspace of V^{n}, that is defined by equations of the form m_{1}x_{1} + … + m_{n}x_{n} = a, m_{i} ∈ ℤ, a ∈ V.

Slightly rephrasing the quantifier-elimination statement proved in [11] Corollary 2 of section 3, we have

Lemma 3.2 (i) Projection of a PQF-closed set is PQF-constructible, that is a boolean combination of PQF-closed sets.

(ii) The image of a constructible set under exponentiation is a Zariskiconstructible (algebraic) subset of (ℂ^{×})^{n}. The image of the set of the form (3) is Zariski closed.

The PQF_{ω}-topology is given by closed basic sets of the form

∪_{a∈I}(S + a)

where S is of the form (3) and I a subset of (ker exp)^{n}.

We define C to be the family of all PQF_{ω}-closed sets.

Corollary 3.3 C satisfies (L).

We assign dimension to a closed set of the form (3)

dim L ∩ m · ln W := dim exp (L ∩ m · ln W) .

using the fact that the object on the right hand side is an algebraic variety. We extend this to an arbitrary closed set assuming (CU), that is that the dimension of a countable union is the maximum dimension of its members.This immediately gives (DP). Using 3.2 we also get (WP).

For a variety W ⊆(ℂ^{×})^{n} consider the system of its roots

W^{\frac{1}{m}} = \{⟨x_{1}, …, x_{n}⟩ ∈ (ℂ^{×})^{n} : ⟨x_{1}^{m}, …, x_{n}^{m}⟩ ∈ W \}.

Let d_{W} (m) be the number of irreducible components of W^{\frac{1}{m}} . We say that the sequence W^{\frac{1}{m}} , m ∈ ℕ, stops branching if the sequence d_{W} (m) is eventually constant.

Obviously, in case W is a singleton, W = {⟨w_{1}, …, w_{n}⟩} ⊆ (ℂ^{×})^{n}, the sequence W^{\frac{1}{m}} does not stop branching as d_{W} (m) = m^{n}. This is the simplest case when W is contained in a coset of a torus, namely given by the equations ⋀_{i} x_{i} = 1. Similarly, if W is contained in a coset of an irreducible torus given by k independent equations of the form

x_{1}^{ℓ_{i1}} · . . . · x_{n}^{ℓ_{in}} = 1

then d_{W} (m) = m^{k} so does not stop branching.

Fact ([9], Theorem 2, case n = 1 and its Corollary) The sequence W^{\frac{1}{m}} stops branching if and only if W is not contained in a coset of a proper subtorus of (ℂ^{×})^{k}.

Lemma 3.4 Any irreducible closed subset of V^{n} is of the form (3), for W not contained in a coset of a proper torus, m ∈ ℤ.

In case W is contained in a coset of a proper torus T, note that T = exp L, for some L a ℚ-linear subspace of V^{n}. Also there is an obvious ℚ-linear isomorphism σ_{L} : L → V^{k}, k = dim L, which induces a biregular isomorphism σ_{T} : T → (ℂ^{×})^{k}. Now σ_{T} (W)⊆(ℂ^{×})^{k} is not contained in a coset of a proper torus and so σ_{T} (W)^{\frac{1}{m}} stops branching.

Note that L is defined up to the shift by a ∈ (ker exp)^{n}.

Proposition 3.5 Let W ⊆ (ℂ^{×})^{n} be an irreducible subvariety, T = exp L the minimal coset of a torus containing W and m is the level where σ_{T} (W^{\frac{1}{m}}) stops branching. Let σ_{T} (Wi^{\frac{1}{m}}) be an irreducible component of σ_{T} (W^{\frac{1}{m}}). Then

L ∩ mσ_{T}^{−1}σ_{T} (Wi^{\frac{1}{m}})  (4)

is an irreducible component of ln W. Moreover, any irreducible component of ln W has this form for some choice of L, exp L = T.

Remark 3.6 (i) The irreducible components of the form (4) for distinct choices of L do not intersect.

(ii) There are finitely many irreducible components of the form (4) for a fixed L and W.

Remark 3.7 Proposition 3.5 eventually provides a description of the irreducible decomposition for any set of the form (3), so for any closed set. Indeed, the irreducible components of the set L ∩ m · ln W are among irreducible components of ln X, for the algebraic variety X = exp(L ∩ m · ln W).

Corollary 3.8 Any closed subset of V^{n} is analytic in V^{n}.

It is easy now to check that (SI), (INT), (CMP), (CC), (AS) and (PS) are satisfied.

Corollary 3.9 The structure (V, C) is analytic Zariski one-dimensional and presmooth.

An inquisitive reader will notice that the analysis above treats only formal notion of analyticity on the cover ℂ of ℂ^{×} but does not address the classical one. In particular, is the formal analytic decomposition as described by 3.5 the same as the actual complex analytic one? In a private communication F.Campana answered this question in positive, using a cohomological argument. M.Gavrilovich proved this and much more general statement in his thesis (see [8], III.1.2) by a similar argument.

Now we look into yet another version of a cover structure which is proven to be analytic Zariski, a cover of the one-dimensional algebraic torus over an algebraically closed field of a positive characteristic.

Let (V, +) be a divisible torsion free abelian group and K an algebraically closed field of a positive characteristic p. We assume that V and K are both of the same uncountable cardinality. Under these assumptions it is easy to construct a surjective homomorphism

ex : V → K^{×}.

The kernel of such a homomorphism must be a subgroup which is p-divisible but not q-divisible for each q coprime with p. One can easily construct ex so that

ker ex ≅ ℤ[\frac{1}{p}],

the additive group (which is also a ring) of rationals of the form \frac{m}{p^{n}} , m, n ∈ ℤ, n ≥ 0. In fact in this case it is convenient to view V and ker ex as ℤ[\frac{1}{p}]- modules.

In this new situation Lemma 3.2 is still true, with obvious alterations, and we can use the definition 3.1 to introduce a topology and the family C as above. The above Fact (right before 3.1) for K^{×} is proved in [10]. Hence the corresponding versions of 3.5 - 3.9 follow.

Finally we mention a wide class of structures on ℂ considered by M.Gavrilovich in [8]. These are universal covers U of smooth complex algebraic varieties 𝔸 = 𝔸(ℂ), not necessarily of dimension one. Gavrilovich defines, quite cleverly, a family C of closed (analytic) subsets on V^{n} closely related to etale coverings of 𝔸. In case 𝔸 is ℂ^{×}, his C coincides with the one discussed above, but already for elliptic curves it can be richer. Gavrilovich then proves that this uncountable family can be defined in terms of a countable subfamily C_{0}.

He calls a complex variety 𝔸 Shafarevich, if V is holomorphically convex and the fundamental groups of 𝔸^{n} are subgroup separable (properties studied elsewhere). It is known that Abelian varieties satisfy these properties. Summarising the results of chapter III of [8] we get the following.

Theorem 3.10 (M.Gavrilovich) Assume 𝔸 is Shafarevich. Then, for its universal cover V, the structure (V, C) is analytic Zariski.

References

[1] N.Peatfield and B.Zilber, Analytic Zariski structures and the Hrushovski construction, Annals of Pure and Applied Logic, Vol 132/2-3 (2004) pp 127-180

[2] E.Hrushovski and B.Zilber, Zariski Geometries, Journal of AMS, 9(1996), 1-56

[3] B.Zilber, Model Theory and Algebraic Geometry. In Seminarberichte Humboldt Universitat zu Berlin, Nr 93-1, Berlin 1993, 202-222

[4] B.Zilber,Zariski Geometries forthcoming book, CUP

[5] B.Zilber, Generalized Analytic Sets Algebra i Logika, Novosibirsk, v.36, N 4 (1997), 361 - 380 (translation on the author’s web-page and published by Kluwer as ’Algebra and Logic’)

[6] B.Zilber, A categoricity theorem for quasi-minimal excellent classes. In: Logic and its Applications eds. Blass and Zhang, Cont.Maths, v.380, 2005, pp.297-306

[7] J.Baldwin, Categoricity authors web-page

[8] M.Gavrilovich, Model Theory of the Universal Covering Spaces of Complex Algebraic Varieties, DPhil Thesis, Oxford 2006, http://misha.uploads.net.ru/misha-thesis.pdf

[9] B.Zilber, Covers of the multiplicative group of an algebraically closed field of characteristic zero J. London Math. Soc. (2), 74(1):41–58, 2006

[10] M.Bays and B.Zilber, Covers of Multiplicative Groups of Algebraically Closed Fields of Arbitrary Characteristic April 2007, arXive math.AC/0401301

[11] B.Zilber, Model theory, geometry and arithmetic of the universal cover of a semi-abelian variety In: Model Theory and Applications, ed. L.Belair and el. ( Proc. of Conf. in Ravello 2000) Quaderni di Matematica, v.11, Series edited by Seconda Univli, Caserta, 2002

[12] L.Smith, Toric Varieties as Analytic Zariski Structures, DPhil Thesis, Oxford 2008