A NOTE ON SUBVARIETIES OF POWERS OF OT-MANIFOLDS

RAHIM MOOSA AND MATEI TOMA

Abstract. It is shown that the space of finite-to-finite holomorphic correspondences on an OT-manifold is discrete. When the OT-manifold has no proper infinite complex-analytic subsets, it then follows by known model-theoretic results that its cartesian powers have no interesting complex-analytic families of subvarieties. The methods of proof, which are similar to [Moosa, Moraru, and Toma “An essentially saturated surface not of Kähler-type”, Bull. of the LMS, 40(5):845–854, 2008], require studying finite unramified covers of OTmanifolds.

1. introduction

This note is concerned with complex-analytic families of subvarieties in cartesian powers of the compact complex manifolds introduced by Oeljeklaus and the second author in [7], here referred to as OT-manifolds. These manifolds are higher dimensional analogues of Inoue surfaces of type S_{M}. In [4], we, along with Ruxandra Moraru, showed that if X is an Inoue surface of type S_{M} then X^{n} contains no infinite complex-analytic families of subvarieties, except for the obvious ones such as ({a} × V : a ∈ X^{m}) where V is a fixed subvariety of X^{n−m}. Using model-theoretic techniques we were able to reduce the problem to considering only the case of n = 2. That case amounted to showing that the set of finite-to-finite holomorphic correspondences on X, viewed as subvarieties of X^{2}, is discrete. Here we extend this result to OT-manifolds in general. Actually, it is useful to consider the following higher arity version of correspondences: for any compact complex manifold X, let Corr_{n}(X) denote the set of irreducible complex-analytic S ⊂ X^{n} such that the co-ordinate projections pr_{i} : S → X are surjective and finite for all i = 1, …, n. So Corr_{2}(X) is the set of finite-to-finite holomorphic correspondences.^{1}

Theorem 1. If X is an OT-manifold then Corr_{n}(X) is discrete for all n > 0.

The proof, which we will give in Section 3, follows to some extent what was done for Inoue surfaces of type S_{M} in [4]. But this approach leads naturally to the consideration of finite unramified coverings of OT-manifolds, and the latter are not formally instances of the original construction in [7]. However, we show in Section 2 that a mild generalisation of that construction leads to a class of manifolds which is closed under finite unramified coverings. We call these manifolds also OT-manifolds and the theorem is valid for this larger class.

Date: November 25, 2014.

R. Moosa was partially supported by an NSERC Discovery Grant. M. Toma was partially supported by the ANR project MNGNK, decision N ANR-10-BLAN-0118.

1It may be worth pointing out that the elements of Corr_{n}(X) are simply components of intersections of pull-backs of finite-to-finite holomorphic correspondences. That is, for n > 1, if S ∈ Corr_{n}(X) and π_{i} : X^{n} → X^{2} is the co-ordinate projection (x_{1}, …, x_{n}) ↦ (x_{1}, x_{i}), for i = 2, …, n, then each π_{i}(S) ⊂ X^{2} is a correspondence and S is an irreducible component of ⋂_{i=2}^{n} π_{i}^{−1}(π_{i}(S)). This is an easy dimension calculation, see [5, Lemma 3.2].

The theorem is particularly significant when X has no proper positive dimensional subvarieties, because of the following fact coming from model theory.

Fact 2. Suppose X is a compact complex manifold that is not an algebraic curve, is not a complex torus, and has no proper infinite complex-analytic subsets. Then every irreducible complex-analytic subset of a cartesian power of X is a cartesian product of points and elements of Corr_{n}(X) for various n > 0.

Proof. This is Proposition 5.1 of [9] together with Lemma 3.3(b) of [5]. □

That OT-manifolds without proper positive dimensional subvarieties are ubiquitous in all dimensions follows from work of Ornea and Verbitsky [8] showing that we get examples whenever X is the OT-manifold corresponding to a number field that has precisely two complex embeddings which are not real.

Putting together the Theorem and the Fact, we conclude:

Corollary 3. Suppose X is an OT-manifold that has no proper infinite complexanalytic subsets. Then, for all n > 0, X^{n} has no infinite complex-analytic families of subvarieties that project onto each co-ordinate.

Remark 4. The model theorist should note that for X to have no proper infinite complex-analytic subsets is exactly strong minimality of X as a first-order structure in the language of complex-analytic sets. Strongly minimal OT-manifolds are of trivial acl-geometry by the manifestation of the Zilber trichotomy in this context. By [5, Proposition 3.5], the discreteness of Corr_{2}(X) implies that strongly minimal OT-manifolds are essentially saturated in the sense of [3]. In particular, we obtain in every dimension examples of essentially saturated manifolds that are not of Kähler-type. This was the original motivation for both [4] and the current note.

2. Finite covers of OT-manifolds

We will quickly review the original construction of OT-manifolds from [7] and then describe how to generalise it.

Fix a number field K admitting n = s + 2t distinct embeddings into ℂ, which we will denote by σ_{1}, …, σ_{n} where σ_{1}, …, σ_{s} are real and each σ_{s+i} is complex conjugate to σ_{s+i+t}. Assume that s and t are positive. By Dirichlet’s Theorem the multiplicative group of units O_{K}^{∗} of the ring of integers O_{K} of K has rank s + t − 1. The subgroup

O_{K}^{∗,+} := {a ∈ O_{K}^{∗} : σ_{i}(a) > 0 for all 1 ≤ i ≤ s}

of “positive” units is free abelian of finite index in O_{K}^{∗}. Let U be a rank s subgroup of O_{K}^{∗,+} that is admissible for K in the sense of [7]. With respect to the natural action of U on the additive group O_{K}, consider the semidirect product Γ = U ⋉O_{K}. Let m = s + t and consider the action of Γ on ℂ^{m} given by,

(a, x)(z_{1}, …, z_{m}) := (σ_{1}(ax) + σ_{1}(a)z_{1}, …, σ_{m}(ax) + σ_{m}(a)z_{m}).

As U < O_{K}^{∗,+}, this action leaves ℍ^{s}×ℂ^{t} invariant, and the admissibility condition is equivalent to the action being proper and discontinuous. The original OT-manifold, denoted by X(K, U), is the quotient of ℍ^{s} × ℂ^{t} by this action. In the sequel we will denote these manifolds by X(O_{K}, U) in order to distinguish them from their generalisations.

The above construction is generalised by replacing the role of O_{K} in Γ by any rank n additive subgroup M ≤ O_{K} that is stable under the action of U. We say then that U is admissible for M. Taking Γ = U ⋉ M, we again get a proper and discontinuous action on ℍ^{s} × ℂ^{t}, and the quotient is denoted by X(M, U). We will continue to call these compact complex manifolds OT-manifolds. To avoid confusing them with the previous construction we will occasionally say that they are of type X(M, U) (otherwise of type X(O_{K}, U)). Note that the possibility of generalising the original construction by replacing O_{K} with an order of K is already mentioned in [7]. However only the ℤ-submodule structure of M and the stability under the U-action are necessary to make the construction work.

The universal cover of X(M, U) is ℍ^{s} ×ℂ^{t} and the fundamental group is U ⋉M. As the latter is of finite index in U ⋉O_{K}, we see that X(M, U) is a finite unramified covering of X(O_{K}, U). In fact, all finite unramified covers are of this form:

Lemma 5. The class of OT-manifolds of type X(M, U) is closed under finite unramified coverings.

Proof. Given X(M, U), such a covering would correspond to a finite index subgroup Γ_{1} ≤ U ⋉ M. Taking U_{1} to be the image of Γ_{1} in U, and setting M_{1} := Γ_{1} ∩ M, it is not hard to check that U_{1} is admissible for M_{1} and that the covering is nothing other than X(M_{1}, U_{1}). □

Much of the theory of OT-manifolds developed in [7] goes through in this more general setting. In particular,

Lemma 6. If X = X(M, U) is an OT-manifold then H^{0}(X, T_{X}) = 0.

Proof. For OT-manifolds of type X(O_{K}, U) this is Proposition 2.5 of [7]. Imitating that argument, it suffices to prove for M a rank n additive subgroup of O_{K}, that the image of M in ℝ^{s} under (σ_{1}, …, σ_{s}) is dense. But this is the case because M has finite index in O_{K} and the latter does have dense image (see the proof of Lemma 2.4 of [7]). □

The following remarks serve as further evidence that the above extension of the definition of OT-manifolds is natural.

Remark 7. Any OT-manifold of type X(M, U) admits a finite unramified cover of type X(O_{K}, U).

Indeed, since M is of maximal rank in O_{K}, there exists a positive integer l such that lO_{K} ⊂ M. Thus X(lO_{K}, U) is a finite unramified cover of X(M, U). But the multiplication by l at the level of ℍ^{s} × ℂ^{t} conjugates the actions of U ⋉ O_{K} and of U ⋉ lO_{K} and thus induces an isomorphism between X(O_{K}, U) and X(lO_{K}, U).

Remark 8. When s = t = 1 the class OT-manifolds of type X(M, U) coincides with the class of Inoue surfaces of type S_{M} defined in [2].

Indeed, if one starts with the manifold X(M, U), then choosing a generator a of U with σ_{1}(a) > 1 and a base (α_{1}, α_{2}, α_{3}) of M over ℤ one obtains a matrix A(a) ∈ GL(3, ℤ) which represents the action of a on M with respect to this basis. Applying the embedding σ_{k} to the relation a(α_{1}, α_{2}, α_{3})^{⊤} = A(a)(α_{1}, α_{2}, α_{3})^{⊤} shows that (σ_{k}(α_{1}), σ_{k}(α_{2}), σ_{k}(α_{3}))^{⊤} is an eigenvector of A(a) associated to the eigenvalue σ_{k}(a). In particular this implies A(a) ∈ SL(3, ℤ) since σ_{1}(a) > 0. At this point one sees that X(M, U) coincides with the surface S_{A(a)} as defined in [2].

Conversely, starting with any matrix A ∈ SL(3, ℤ), with one real eigenvalue larger than 1 and two complex non-real eigenvalues, we denote by K the splitting field of the characteristic polynomial χ_{A} of A over ℚ. Then there exists an element a ∈ O_{K}^{∗,+} such that the eigenvalues of A (i.e the roots of χ_{A}) are precisely σ_{1}(a), σ_{2}(a), σ_{3}(a). We find now an eigenvector v ∈ ℤ[σ_{1}(a)]^{3} associated to σ_{1}(A) by solving the system (A − σ_{1}(a)I_{3})v^{⊤} = 0 over K. There exist now elements α_{1}, α_{2}, α_{3} ∈ O_{K} such that v = (σ_{1}(α_{1}), σ_{1}(α_{2}), σ_{1}(α_{3})). Moreover α_{1}, α_{2}, α_{3} are linearly independent over ℚ since a linear relation would entail a linear relation between the components v_{1}, v_{2}, v_{3} of v over ℚ, which combined with the equations (A − σ_{1}(a)I_{3})v^{⊤} = 0 would show that σ_{1}(a) is quadratic over ℚ. Now choosing M to be the ℤ-sumbodule of K generated by α_{1}, α_{2}, α_{3} and U the multiplicative group generated by a we get again X(M, U) = S_{A(a)}.

3. The Proof

As in the case of Inoue surfaces of type S_{M} studied in [4], we will make use of some deformation theory to prove the main theorem. But we will need a bit more than was used in [4]. We say that a holomorphic map f : V → W between compact complex manifolds is rigid over W if there are no nontrivial deformations of f that keep W fixed. More precisely: Whenever V → D is a proper and flat holomorphic map of compact complex varieties with V = V_{d} for some d ∈ D, and F : V → D×W is a holomorphic map over D with F_{d} = f, then there is an open neighbourhood U of d in D and a diagram

\xymatrix{V_{U} \ar[dr] \ar[drr]^{F_{U}} \ar[dd]_{ϕ} \\  & U & U × W \ar[l] \\ U × V \ar[ur] \ar[urr]_{id_{U} ×f}}

where ϕ is a biholomorphism. In particular F_{s}(V_{s}) = f(V) for all s ∈ U.

Fact 9 (Section 3.6 of [6]). Suppuse f : V → W is a holomorphic map between compact complex manifolds such that

• H^{0}(V, f^{∗}T_{W}) = 0, and

• f_{∗} : H^{1}(V, T_{V}) → H^{1}(V, f^{∗}T_{W}) is injective.

Then f is rigid over W.

Lemma 10. Suppose X and Y are compact complex manifolds, H^{0}(Y, T_{Y}) = 0, and f : Y → X^{n} is a holomorphic map such that pr_{i} ∘f : Y → X is a finite unramified cover for each i = 1, …, n. Then f is rigid over X^{n}.

Proof. Note that here pr_{i} : X^{n} → X is the projection onto the ith co-ordinate. Let f_{i} := pr_{i} ∘f : Y → X. As each f_{i} is unramified, we have that

f^{∗}T_{X^{n}} = f^{∗} (⨁_{i=1}^{n} pr_{i}^{∗} T_{X}) = ⨁_{i=1}^{n} f_{i}^{∗}T_{X} = ⨁_{i=1}^{n} T_{Y}

Hence, H^{0}(Y, f^{∗}T_{X^{n}}) = ⨁_{i=1}^{n} H^{0}(Y, T_{Y}) = 0. On the other hand, the isomorphism

(f_{1})_{∗} : H^{1}(Y, T_{Y}) → H^{1}(Y, f_{1}^{∗}T_{X}) factors through f_{∗} : H^{1}(Y, T_{Y}) → H^{1}(Y, f^{∗}T_{X^{n}}), and hence the latter is injective. So f : Y → X^{n} is rigid over X^{n} by Fact 9. □

We can now prove the main theorem.

Proof of Theorem 1. Suppose X is an OT-manifold of type X(M, U). As in [4], in order to show that Corr_{n}(X) is discrete we let S ∈ Corr_{n}(X) be arbitrary, consider the irreducible component D of the Douady space of X^{n} in which S lives, and show that D is zero-dimensional. This suffices as it proves that each element of Corr_{n}(X) is isolated in the Douady space.

Let Z ⊂ D×X^{n} be the restriction of the universal family to D. By the flatness of Z → D, for general d ∈ D, Z_{d} ∈ Corr_{n}(X) also. Let Z̃ → Z be a normalisation and denote by f : Z̃ → D × X^{n} the composition of the normalisation with the inclusion of Z in D × X^{n}. Then for general d ∈ D we have that f_{d} : Z̃_{d} → X^{n} is such that each projection pr_{i} ∘f_{d} : Z̃_{d} → X is a finite surjective map. In [1] it is shown that OT-manifolds of type X(O_{K}, U), and hence also OT-manifolds of type X(M, U), have no divisors. So the purity of branch locus theorem (which applies as Z̃_{d} is normal and X is smooth) implies that pr_{i} ∘f_{d} is a finite unramified covering. In particular, Z̃_{d} is a generalised OT-manifold by Lemma 5, and so H^{0}(Z̃_{d}, TZ̃_{d}) = 0 by Lemma 6. But moreover, by Lemma 10, f_{d} is rigid over X^{n}. It follows that for some open neighbourhood U of d in D, f_{U} : Z̃_{U} → U × X^{n} is biholomorphic over U × X^{n} with id_{U} ×f_{d} : U × Z̃_{d} → U × X^{n}. In particular, for all s ∈ U, Z_{s} = f_{s}(Z̃_{s}) = f_{d}(Z̃_{d}) = Z_{d}. The universality of the Douady space now implies that U = {d}, so that in fact D = {d}, as desired. □

References

[1] L. Battisti and K. Oeljeklaus. Holomorphic line bundles over domains in Cousin groups and the algebraic dimension of OT-manifolds. Preprint, arXiv:1306.3047.

[2] M. Inoue, On surfaces of class VII_{0}, Invent. Math. 24 (1974), 269-310.

[3] R. Moosa. On saturation and the model theory of compact Kähler manifolds. Journal für die reine und angewandte Mathematik, 586:1–20, 2005.

[4] R. Moosa, R. Moraru, and M. Toma. An essentially saturated surface not of Kähler-type. Bulletin of the London Mathematical Society, 40(5):845–854, 2008.

[5] R. Moosa and A. Pillay. ℵ_{0}-categorical strongly minimal compact complex manifolds. Proc. Amer. Math. Soc., 140(5):1785–1801, 2012.

[6] M. Namba. Families of meromorphic functions on compact Riemann surfaces, volume 767 of Lecture Notes in Mathematics. Springer, Berlin, 1979.

[7] K. Oeljeklaus and M. Toma. Non-Kähler compact complex manifolds associated to number fields. Ann. Inst. Fourier (Grenoble), 55(1):161–171, 2005.

[8] L. Ornea and M. Verbitsky. Oeljeklaus-Toma manifolds admitting no complex subvarieties. Math. Res. Lett., 18(4):747–754, 2011.

[9] A. Pillay and T. Scanlon. Meromorphic groups. Transactions of the American Mathematical Society, 355(10):3843–3859, 2003.

Rahim Moosa, University of Waterloo, Department of Pure Mathematics, 200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada

   E-mail address: rmoosa@uwaterloo.ca

Matei Toma, Institut de Mathématiques Elie Cartan, Université de Lorraine, B.P. 70239, 54506 Vandoeuvre-lès-Nancy Cedex, France

   E-mail address: Matei.Toma@univ-lorraine.fr