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APPEARED IN BULLETIN OF THE

arXiv:math/9304212v1 [math.AG] 1 Apr 1993APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 28, Number 2, April 1993, Pages 315-323ZARISKI GEOMETRIESEhud Hrushovski and Boris ZilberAbstract. We characterize the Zariski topologies over an algebraically closed fieldin terms of general dimension-theoretic properties.

Some applications are given tocomplex manifold and to strongly minimal sets.1. IntroductionThere is a class of theorems that characterize certain structures by their basictopological properties.

For instance, the only locally compact connected fields areR and C.These theorems refer to the classical topology on these fields.Thepurpose of this paper is to describe a similar result phrased in terms of the Zariskitopology.The results we offer differs from the one considered above in that we do notassume in advance that our structure is a field or that it carries an algebraic struc-ture of any kind. The identification of the field structure is rather a part of theconclusion.Because the Zariski topologies on two varieties do not determine the Zariskitopology on their product (and indeed the topology on a one-dimensional varietycarries no information whatsoever), the data we require consists not only of atopology on a set X, but also of a collection of compatible topologies on Xn foreach n. Such an object will be referred to here as a geometry.

It will be calleda Zariski geometry if a dimension can be assigned to the closed sets, satisfyingcertain conditions described below. Any smooth algebraic variety is then a Zariskigeometry, as is any compact complex manifold if the closed subsets of Xn are takento be the closed holomorphic subvarieties.If X arises from an algebraic curve, there always exist large families of closedsubsets of X2; specifically, there exists a family of curves on X2 such that throughany two points there is a curve in the family passing through both and anotherseparating the two.

An abstract Zariski geometry with this property is called veryample. By contrast, there are examples of analytic manifolds X such that Xn hasvery few closed analytic submanifolds and is not ample.

Precise definitions will begiven in the next section; the complex analytic case is discussed in §4. Our mainresult is1991 Mathematics Subject Classification.

Primary 03G30; Secondary 14A99.Received by the editors February 12, 1992 and, in revised form, August 24, 1992The first author was supported by NSF grants DMS 9106711 and DMS 8958511c⃝1993 American Mathematical Society0273-0979/93 $1.00 + $.25 per page1

2EHUD HRUSHOVSKI AND BORIS ZILBERTheorem 1. Let X be a very ample Zariski geometry.

Then there exists a smoothcurve C over an algebraically closed field F, such that X, C are isomorphic asZariski geometries. F and C are unique, up to a field isomorphism and an isomor-phism of curves over F.A weaker version is available in higher dimensions (and actually follows easilyfrom Theorem 1).

If X is a higher-dimensional Zariski geometry and there existsa family of curves on X passing through any two points and separating any two,then there exists a dense open subset of X isomorphic to a variety. Globally, Xprobably arises from an algebraic space in the sense of [A], but we do not provethis.

In this survey we will concentrate on the one-dimensional case (though, ofcourse, we must consider the geometry of arbitrary powers).2. Zariski GeometriesZariski geometries will be defined as follows.

Recall that a topological space isNoetherian if it has the descending chain condition on closed subsets. (See [Ha].

)A closed set is irreducible if it is not the union of two proper closed subsets. If Xis Noetherian, then every closed set can be written as a finite union of irreducibleclosed sets.

These are uniquely determined (provided that no one is a subset ofthe other) and are called the irreducible components of the given set. We say thatX has dimension n if n is the maximal length of a chain of closed irreducible setsCn ⊃Cn−1 ⊃· · · ⊃C0.We will use the following notation: if C ⊆D1 × D2, a ∈D1, we let C(a) = {b ∈D2 : (a, b) ∈C}.Definition 1.

A Zariski geometry on a set X is a topology on Xn for each n thatsatisfies the following:(Z0) Let fi be a constant map (fi(x1, . .

. , xn) = c) or a projection (fi(x1, .

. .

, xn) =xj(i)). Let f(x) = (f1(x), .

. .

, fm(x)). Then f : Xn →Xm is continuous.The diagonals xi = xj of Xn are closed.

(Z1) Let C be a closed subset of Xn, and let π be the projection to Xk. Thenthere exists a proper closed subset F of cl(πC) such that πC ⊇cl(πC) −F.

(Z2) X is irreducible and uniformly 1-dimensional: if C ⊇Xn × X is closed,then for some m, for all a ∈Xn, C(a) = X or |C(a)| ≤m. (Z3) (Dimension Theorem) Let U be a closed irreducible subset of Xn, andlet ∆ij be the diagonal xi = xj.

Then every component of U ∩∆ij hasdimension ≥dim(U) −1.Axiom (Z0) includes the obvious compatibility requirements on the topologies.X is called complete (or proper) if all projection maps are closed. Note that (Z1)is a weakening of completeness.

We prefer not to assume completeness axiomaticallybecause we do not wish to exclude affine models. (Z2) is the assumption of one-dimensionality that we make in this paper.

Theresults still have consequences in higher dimensions, as will be seen in §4.Axiom (Z3) is valid in smooth algebraic (or analytic) varieties. One still obtainsinformation on other varieties by removing the singular locus, applying the theorem,and going back.

It is partly for this reason that it is important not to assumecompleteness in (Z1).We now turn to the “ampleness” conditions. By a plane curve over X we meanan irreducible one-dimensional subset of X2.

A family of plane curves consists of a

ZARISKI GEOMETRIES3closed irreducible set E ⊆Xn (parametrizing the family) and a closed irreducibleC ⊆E × X2, such that C(e) is a plane curve for generic e ∈E.Definition 2. A Zariski geometry X is very ample if there exists a family C ⊆E × X2 of plane curves such that:(i) For generic a, b ∈X2 there exists a curve C(e) passing through a, b.

(ii) For any a, b ∈X2 there exists e ∈E such that C(e) passes through just oneof a, b.If only (i) holds, then X is called ample.Axioms (Z0)–(Z3) alone allow a degenerate geometry, in which the closed irre-ducible subsets of Xn are just those defined by equations of the form xi = ai. Moreinteresting are the linear geometries, where X is an arbitrary field (or division ring)and the closed subsets of Xn are given by linear equations.

These are nonampleZariski geometries. A thorough analysis of nonample Zariski geometries can becarried out; they are degenerate or else closely related to the above linear example.We will not describe this analysis here; see [HL].Theorem 1 describes the very ample Zariski geometries, while the nonampleZariski geometries were previously well understood.

This leaves a gap—the amplebut not very ample Zariski geometries. We can show:Theorem 2.

Let D be an ample Zariski geometry.Then there exists an alge-braically closed field K and a surjective map f : D →P1(K). f maps constructiblesets to (algebraically) constructible sets ; in fact offa certain finite set, f induces aclosed, continuous map on each Cartesian power.This represents D as a certain branched cover of P1.

Among complex analyticmanifolds, all such covers are algebraic curves.This is not true in the Zariskicontext; there are indeed ample, not very ample Zariski geometries, which do notarise from algebraic curves. We construct these as formal covers of P1, startingfrom any nonsplit finite extension G of a subgroup of Aut(P1) that cannot berealized as the automorphism group of any algebraic curve.

We obtain a finitecover of P1, with an action of G on it, and define the closed sets so as to includethe pullbacks of the Zariski closed subsets of P1 and the graphs of the G-operators.The automorphism group of the cover is then too large to arise from an algebraiccurve.The following is implicit in Theorem 1, but we wish to isolate it:Theorem 3. Let K be an algebraically closed field, and let X be a Zariski geometryon P1(K), refining the usual Zariski geometry.

Then the two geometries coincide.These results sprang from two sources, which we proceed to describe.3. Strongly minimal setsThe original goal was to characterize an algebraically closed field F in termsof the collection of constructible subsets of F n, rather than in terms of the closedsubsets.

The motivation was the importance in model theory of certain structures,called strongly minimal sets. (In particular they form the backbone of any structurecategorical in an uncountable power (see [BL]).)

4EHUD HRUSHOVSKI AND BORIS ZILBERDefinition 3. A structure is an infinite set D together with a collection of subsetsof Dn (n = 1, 2, .

. . ) closed under intersections, complements, projections and theirinverses, and containing the diagonals.

These are called the 0-definable sets. D isstrongly minimal if it satisfies:(SM) For every 0-definable C ⊆Dm+1 there exists an integer n such that forall a ∈Dn, letting C(a) = {b ∈D: (a, b) ∈C}, either |C(a)| ≤n or|D −C(a)| ≤n.These axioms can be viewed as analogs of (Z1), (Z2) for the class of constructiblesets.

Under these assumptions one can prove the existence of a well-behaved di-mension theory.In particular, one can state an axiom analogous to ampleness(non-local-modularity; strongly minimal sets not satisfying it are well understood).See [Z, HL]. However, it is not clear how to state (Z3) in terms of constructible setsalone.

In the absence of such an axiom, it was shown in [Hr1] that the analog ofTheorem 2 is false and in [Hr2] that the analog of Theorem 3 also fails.We note that Macintyre (see [Mac]) characterized the strongly minimal fields asthe algebraically closed fields. (These are strongly minimal sets with a definablefield structure, i.e., the graphs of addition and multiplication are 0-definable subsetsof D3.) A conjecture by Cherlin and the second author that simple groups definableover strongly minimal sets are algebraic groups over an algebraically closed fieldremains open.4.

Complex manifoldsLet X be a (reduced, Hausdorff) compact complex analytic space, and considerthe topology An on Xn whose closed sets are the closed analytic subvarieties of Xn.Remmert’s theorem then implies that the projection of a closed set is closed. Itcan be shown further that if U is a locally closed subset of Xn+1, i.e., the differenceof two An-closed sets, then the projection of U to Xn is itself a finite union oflocally closed sets, or an An-constructible set.

Thus the An-constructible sets forma structure in the sense of §3. If V is a minimal analytic subvariety of X, then Vwith this structure is strongly minimal; further, if one removes the singular locusof V , one obtains a Zariski geometry in the sense of §1.

In this section we discusssome consequences of this observation.In the analytic context, Theorem 2 resembles Riemann’s existence theorem (thepart stating that a compact complex manifold of dimension 1 is a finite cover of theprojective line). Indeed Riemann’s existence theorem would follow from Theorem2, but the hypothesis of Theorem 2 includes ampleness, which we do not know howto prove directly.

Instead we offer a variation in dimensions ≥2. Note that theassumption on M is true of a generic complex torus of dimension ≥2 (as will alsofollow from Proposition 3).Proposition 1.

Suppose M is a compact K¨ahler manifold of complex dimension≥2, with no proper infinite analytic subvarieties. If H1(M) ̸= 0, then M is acomplex torus.This follows from Theorem 2, as follows.

If M were ample, then it would be afinite branched cover of P1(K) for some field K, which in the present context alsohas the structure of a complex analytic space. K must also have complex dimension≥2, which contradicts the classification of the connected locally compact fields

ZARISKI GEOMETRIES5cited in the Introduction. Thus M cannot be ample.

As mentioned in the previoussection, this gives a strong “Abelian” condition; it is shown in [HP] (in a muchmore general context) that if ampleness fails in a given geometry, then every closedirreducible subset of a group A supported by that geometry must be a coset of aclosed subgroup. We apply this to the Albanese variety A of M ([GH, p. 331]).A fiber of the map from M to A must be finite, or we violate the assumptionon infinite analytic subvarieties.

The image of M in A is closed and irreducible,hence a coset of a closed subgroup S of A; “closed” here means a complex analyticsubvariety, hence a subtorus, of A. By translation we obtain a finite holomorphicmap f : M →S.

The branch locus of this map gives an analytic subvariety locallyof dimension dim(M) −1 and hence must be empty. Thus M is a finite covering ofthe complex torus S and hence is itself a complex torus.Theorem 3 resembles Chow’s theorem that a closed analytic subvariety of pro-jective space must be algebraic (see [GH]).

Indeed a somewhat more general resultcan easily be deduced from the theorem.Proposition 2. Let X be a complex algebraic variety.View X as a complexanalytic space (as in [HA, Appendix 1]).

Then any closed analytic subvariety of Xis an algebraic subvariety.This is easily seen by working with constructible sets, defined to be finite Booleancombinations of closed sets. We have two Zariski geometries on X, given by thealgebraic and the analytic structures.

X contains smooth algebraic curves, on whichagain two geometries are induced. By Theorem 1, one obtains the same geometryon each such curve.

It follows easily that the two geometries have the same classof constructible sets, so any closed analytic subvariety V of X is algebraicallyconstructible. Then it is easy to see that V must in fact be Zariski closed.Another application of [HP] yields the following statement:Proposition 3.

Let G = Cn/L be a complex torus. Then either G has closedanalytic subgroups G1 ⊆G2, such that G2/G1 is isomorphic to a nontrivial Abelianvariety, or the only complex analytic submanifolds of G are subtori and finite unionsof their cosets.5.

Method of proofWe begin with a description of the proof of Theorem 2.1.A universal domain. We are given a Zariski geometry X, which we think ofas analogous to the Zariski geometry on a curve over an algebraically closed field,and its powers.

We wish to find an analog to the finer notion of the K-topology(where K is a subfield of an algebraically closed field); in other words, we wishto have a concept of a closed set “defined over a given substructure K”.Thiscannot be usefully done for X itself; we need to embed X in a larger geometryX∗, a “universal domain”. (This is analogous to viewing a number field as asubfield of a larger field of infinite transcendence degree.) This construction of auniversal domain is in fact a standard one in model theory; X∗is obtained via thecompactness theorem of model theory or by using ultrapowers; see [CK] (saturatedmodels) or [FJ] (enlargements).2.A combinatorial geometry.

We assume (1) has been carried out and workdirectly with the universal domain X∗. An element a of X∗is algebraically de-pendent on a tuple of elements b ∈X∗n if there exists a 0-definable closed set

6EHUD HRUSHOVSKI AND BORIS ZILBERC ⊆X∗n × X∗such that C(b) is finite and a ∈C(b). In the case of algebraicallyclosed fields, this coincides with the usual notion.

It can be shown in general thatthis notion of algebraic closure yields a combinatorial pregeometry, i.e., it satis-fies the exchange axiom: If a is algebraically dependent on b1, . .

. , bn, c but not onb1, .

. .

, bn, then c depends on b1, . .

. , bn, a.

Formally, one can define a transcendencebasis, dimension, etc. In particular, the rank of a subset of X∗is the size of anymaximal independent subset thereof.3.Group configurations.

To construct a field, we will need to find its additiveand multiplicative groups. For this purpose we use a general machinery (valid in thestrongly minimal context and in fact considerably beyond it) to recognize groupsfrom the trace that they leave on the combinatorial geometry.

We will apply thismachinery to the affine translation group to obtain the field.Suppose G is a 1-dimensional group interpretable in the geometry X∗. (Assumefor simplicity that the elements of G are points of X∗; the graph of multiplicationis assumed to be locally closed.) Let a1, a2, .

. .

, b1, b2, . .

. be generic points of X,i.e., an independent set of elements of X∗(over some algebraically closed basesubstructure B).

Let cij = ai + bj. Then (cij : i ∈I, j ∈J) forms an array ofelements of the combinatorial pregeometry of B-dependence.

It is easy to see thatthe rank of any m × n-rectangle in this array is m + n −1. Moreover, for anypermutations σ of I and τ of J, the corresponding permutation of the array arisesfrom an automorphism of the geometry.Conversely, suppose (cij : i, j) is an array of elements of the combinatorial pre-geometry enjoying the above symmetry property and in which any m × n-rectanglehas rank m + n −1.

Then one proves the following theorem: There exists a 1-dimensional Abelian group G and independent generic elements ai, bj of G, suchthat cij and ai + bj depend on each other over B. In particular, an infinite Abeliangroup is involved.A similar theory is available for a connected group not necessarily Abelian, ofany dimension.We note in this connection Weil’s theorem on “group chunks”[W].

Weil’s theorem allows the recognition of a group from generic data: A binaryfunction which is generically associative and invertible arises from a definable group.Our result is similar but more general. In particular, it permits the function f to bemultivalued (i.e., the graph of an algebraic correspondence).

(In other words, thehypothesis concerns algebraic rather than rational dependence on certain genericpoints.) It is shown that given a trace of associativity, there exists a group G suchthat f is conjugate (by a multivalued correspondence between the given set and G)to the single-valued function xy−1z of G. (See [EH].

)Applying the theorem to the two-dimensional group of affine transformations ofa field F, we obtain the following higher-dimensional analog:Proposition 4. Let cij (i, j = 1, 2, .

. .) be a symmetric array of elements of thedependence geometry over B.

Suppose every m × n = rectangle of elements of cijhas rank 2m + n −2 (m, n ≥2). Then there exists an algebraically closed field Fdefined over B and generic independent elements ai, bj, gk of F, such that cij andai + bigj depend on each other.We will merely use the existence of F. We note that if cij = ai + bigj, then theelements cij satisfy the relations: (cij −cij′)/(ci′j −ci′j′) = (cij −cij′′)/(ci′j −ci′j′′).4.Tangency.

So far we have used only the strongly minimal set structure, not

ZARISKI GEOMETRIES7the more detailed knowledge of the identity of the closed sets. This comes in viathe notion of a specialization.

A map f from a subset C of X∗to X∗is said to bea specialization (over B) if for every B-closed set F ⊂Cn and all a1, . .

. , an ∈C,if (a1, .

. .

, an) ∈F then (fa1, . .

. , fan) ∈F.

Note that no such notion is availablefor X itself. The properness axiom immediately yields the extension theorem forspecializations.

The dimension theorem gives a more subtle property, one of whoseprincipal consequences is a “preservation of number” principle: If C(a) is a closedset depending continuously on a, a →a′ is a specialization, and C(a), C(a′) areboth finite, then there exists a specialization of {a} ∪C(a) onto {a′} ∪C(a′); inparticular, the number of points in C(a) cannot go up (but remain finite) underspecializations.This allows us to formalize a notion of intersection multiplicity. Let C1, C2 becurves in the “surface” X × X, depending as above on parameters c1, c2 (so Ci =Ci(ci) for some 0-closed set Ci).

Let a1, a2 be distinct points in the intersectionof C1, C2. If (c1, c2, a1, a2) →(c′1, c′2, a′1, a′2) is a specialization and a′1 = a′2, wesay that the specialized curves C′1, C′2 are tangent at a′1.

In general this notion isnot intrinsic to C′1, C′2, a′1, but rather depends on the choice of C1, C2. It does notalways coincide with the usual notion if X is an algebraic curve (and sometimesyields a more fruitful notion, for our purposes).5.The field structure.

From the given two-dimensional family of curves, wemay obtain a one-dimensional family of curves passing through a single point p0 inthe plane. These curves are considered to be multivalued functions from X to X. Assuch they can be composed.

One would like to identify two curves that are tangentat p0; intuitively, an equivalence class of curves corresponds to a slope; one thenwants to show that composition gives a well-defined operation, which correspondsto multiplication on the slopes and at all events gives a group structure. In practice,tangency is not necessarily an equivalence relation, and a number of other technicalproblems arise.

However, one can find curves cij such that c−1ij ci′j and c−1ij′ c′ij′ aretangent for all i < i′ and j < j′. One then shows that this gives an array as in (3),and hence gives an Abelian group structure.Eugenia Rabinovich observed that in some cases the group obtained in this wayis in fact the additive rather than the multiplicative group.One would like to find the field directly, by considering functions from the planeto itself, modulo tangency, thus interpreting the tangent space to the plane andwhat should be GL2 acting on it.

This approach poses severe technical problems.Hence, one first interprets an Abelian group as above, then uses it as a crutch to findthe field configuration described above. We find curves satisfying, up to tangency,the relations noted following Proposition 4, in which addition is interpreted as thegiven Abelian group structure and multiplication is interpreted as composition.

Weshow that this gives the field array.6.Theorem 3. So far we have essentially described the proof of Theorem 2.

ForTheorem 3 one is given a field F and a Zariski geometry on F, refining the usualZariski topology. One must show that they coincide.

If they do not, it can be shownthat there exists a plane curve C not contained in any algebraic curve. From thishypothesis one must obtain a contradiction.We do this by developing an analog of Bezout’s theorem.

One defines the degreeof a curve C to be the number of points of intersection of C with a generic line.

8EHUD HRUSHOVSKI AND BORIS ZILBERBezout’s theorem then states that the number of points of intersection of C withan algebraic curve of degree d is at most d · deg(C). We give a version of a classicalproof of this, “moving” from an arbitrary algebraic curve to a generic one, and fromthere to a special one for which the result is clear (the unions of d lines).

A certainamount of preliminary work is required, showing that the projective plane over Fis sufficiently complete for the purposes of such moves by specializations.Now one considers intersections of C with algebraic curves of high degree d. Thenumber of points of intersection increases linearly with d, but the dimension of thespace of curves of degree d increases quadratically; hence, for large d one can finda curve intersecting C in more than d · deg(C) points. This is only possible if theintersection is infinite and hence contains C, thus contradicting the choice of Cabove.Theorem 1 follows from Theorem 2, Theorem 3, and an analysis of covers inthe Zariski category.

This analysis shows that any ample Zariski geometry X is acover of a canonical algebraic curve C, such that the pullback of a curve on Cn istypically irreducible in Xn. It follows that if X is very ample, then X = C. It maybe in general that the richness of the geometry of X arises entirely from that of C,but we do not know a precise statement.AcknowledgmentThe authors would like to thank Gregory Cherlin, Moshe Jarden, and Mike Friedfor close readings of this manuscript.References[A]M. Artin, Algebraic spaces, Yale University Press, New Haven, CT, 1969.[BL]T.

J. Baldwin and A. Lachlan, On strongly minimal sets, Symbolic Logic 36 (1971), 79–96.[CK]C. C. Chang and H. J. Keisler, Model theory, North-Holland, Amsterdam, 1973.[EH]D.

Evans and E. Hrushovski, Embeddings of matroids in fields of prime characteristic,Proc. London Math.

Soc. (to appear).[FJ]M.

Fried and M. Jarden, Field arithmetic, Springer-Verlag, Berlin, 1986.[GH]P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley-Interscience, New York,1978.[Ha]P.

Hartshorne, Algebraic geometry, Springer-Verlag, New York, 1977.[HL]E. Hrushovski and J. Loveys, Locally modular strongly minimal sets (to appear).[HP]E.

Hrushovski and A. Pillay, Weakly normal groups, Logic Colloquium 85 (Paris), North-Holland, Amsterdam, 1986.[Hr1]E. Hrushovski, A new strongly minimal set, (to appear in Ann.

Pure Appl. Logic).

[Hr2], Strongly minimal expansions of algebraically closed fields, Israel J. Math.

(toappear.).[Mac]A. Macintyre, On aleph-one categorical theories of fields, Fund.

Math. 71 (1971), 1–25.[W]A.

Weil, On algebraic groups of transformations, Amer. J.

Math. 77 (1955), 355–391.[Z]B.

Zilber, The structure of models of uncountably categorical theories, Proc. Internat.Congr.

Math. (Warsaw, 1983), vol.

1, North-Holland, Amsterdam, 1984, pp. 359–368.Massachusetts Institute of Technology and Hebrew University at JerusalemCurrent address: Department of Mathematics, Massachusetts Institute of Technology, Cam-bridge, Massachusetts 02139E-mail address: ehud@math.mit.eduKemerovo University and University of Illinois at ChicagoCurrent address: Department of Mathematics, Kemerovo University, Kemerovo, 650043, Rus-sia

ZARISKI GEOMETRIES9E-mail address: zilber@kemucnit.kemerovo.su

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