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Automorphism groups of Moishezon threefolds

We show that automorphism groups of Moishezon threefolds are always Jordan.


Introduction
Jordan property plays an important role in the study of automorphism groups of algebraic varieties and complex manifolds. Following [15,Definition 2.1], we say that a group Γ is Jordan (or has Jordan property) if there is a constant J = J(Γ) such that every finite subgroup G ⊂ Γ contains a normal abelian subgroup A ⊂ G of index at most J.
The groups that enjoy Jordan property include: general linear groups GL n (k), where k is a field of zero characteristic (see for instance [5,Theorem 36.13]); groups of birational selfmaps of rationally connected algebraic varieties (see [17,Theorem 1.8] and [3, Theorem 1.1]); groups of birational selfmaps of non-uniruled algebraic varieties (see [16,Theorem 1.8]); many diffeomorphism groups of smooth compact real manifolds (see for instance [14], [13]). One of the most beautiful results concerning Jordan property for groups of geometric origin is the following theorem due to Sh. Meng and D.-Q. Zhang. Theorem 1 ( [9]). Let X be a projective variety over a field of zero characteristic. Then the group Aut(X) is Jordan.
The paper [7] provides a generalization of Theorem 1 to the case of compact Kähler manifolds.
Note that according to [20] there exist projective surfaces with non-Jordan groups of birational selfmaps; furthermore, there exist smooth compact four-dimensional real manifolds with non-Jordan diffeomorphism groups, see [4]. There is a complete classification of two-and three-dimensional projective varieties with non-Jordan groups of birational selfmaps over algebraically closed fields of zero characteristic, see [15,Theorem 2.32] and [19,Theorem 1.8]. Moreover, it is known that the birational automorphism group of any non-projective compact complex surface is Jordan, see [18], but in higher dimensions there are no significant results in this direction yet. With this in mind, it looks interesting to study automorphism groups of various classes of compact complex manifolds from the point of view of Jordan property.
Recall that a compact complex space X is said to be Moishezon if the transcendence degree of its field of meromorphic functions is maximal, that is, equals the dimension of X. Every proper algebraic variety is a Moishezon compact complex space. Every Moishezon compact complex space is birational to a projective variety (see [10,Theorem 1] or [11,Theorem 3 N ]). While every compact complex curve is projective, there exist two-dimensional Moishezon compact complex spaces that are not projective (see for instance [6,Example VII.6.26]). A smooth Moishezon compact complex space is called a Moishezon manifold. For every Moishezon compact complex space, one can construct a resolution of singularities that is a Moishezon manifold (and even a smooth projective variety), see [11,Theorem 3 N ]. Every two-dimensional Moishezon manifold is projective (see [1,Corollary IV.6.5]). There are well-known examples of three-dimensional non-projective Moishezon manifolds, see for instance [12, §3] and [6, Examples VII.6.20-VII. 6.21].
The purpose of this paper is to prove the following result that is to some extent analogous to Theorem 1.
Theorem 2. Let X be a three-dimensional Moishezon compact complex space. Then the group Aut(X) is Jordan.
It would be interesting to find out if there is a generalization of Theorem 2 to the case of Moishezon compact complex spaces of arbitrary dimension.
We are grateful to Andreas Höring who spotted a gap in the first version of our arguments.

Some projectivity criteria
In this section we collect several assertions on projectivity of certain Moishezon varieties.
Lemma 1. Let X be a Moishezon threefold, and let A be a strongly numerically effective divisor on X. Suppose that for some n > 0 the linear system |nA| has no fixed components. Then A is ample. In particular, the manifold X is projective.
Proof. Follows from Nakai-Moishezon ampleness criterion for Moishezon compact complex spaces, see [10,Theorem 6]. Indeed, by assumption we have A · C > 0 for every curve C. Furthermore, since the divisor A is numerically effective and big, one has A 3 > 0.
Let S ⊂ X be an irreducible surface (that is, a two-dimensional compact complex subspace). Then S is a Moishezon compact complex space by [10,Theorem 3]. In particular, S contains curves. The restriction A S = A| S is an effective divisor on S, and one has A S = 0, because A S has positive intersections with curves on S. Therefore, we see that A 2 · S > 0. Hence A is ample by Nakai-Moishezon criterion.
Lemma 2. Let g :F → F be a surjective morphism of smooth compact complex surfaces. LetÂ be an ample divisor onF . Then the divisor A = g * Â is ample.
Proof. By adjunction formula, we have A · C > 0 for every curve C on F , and also A 2 > 0. Hence A is ample by Nakai-Moishezon criterion. Remark 1. The assertion of Lemma 2 fails in the case when the surface F is singular. However, it still holds if F is a two-dimensional normal compact complex space with Q-factorial singularities.
For a contraction h : Y → Z by Aut(Y ; h) we denote the subgroup of Aut(Y ) that consists of all automorphisms such that h is equivariant with respect to their action. By Aut(Y ; h) ′ we denote the maximal subgroup of Aut(Y ; h) that acts trivially on H * (Y, Z). Note that Aut(Y ; h) ′ is a normal subgroup of Aut(Y ; h), and the quotient Aut(Y ; h)/ Aut(Y ; h) ′ has bounded finite subgroups by the classical theorem of Minkowski. Therefore, the group Aut(Y ; h) has bounded finite subgroups if and only if this holds for Aut(Y ; h) ′ .

Lemma 3. Let Y be a Moishezon threefold.
Let h : Y → Z be a contraction to a non-rational curve. Furthermore, assume that the image of the homomorphism Aut(Y ; h) ′ → Aut(Z) is infinite. Then the threefold Y is projective.
Then the divisor A is big. By projection formula, there is at most a finite number of curves C i ⊂ Y such that A · C i 0 (these curves must be contained in the image of the g-exceptional divisor).
Note that for any δ ∈ Aut(Y ; h) ′ we have Hence the set {C i } of all such curves is invariant under Aut(Y ; h) ′ . By our assumptions Z is an elliptic curve and the image of Aut(Y ; h) ′ → Aut(Z) consists of translations. This implies that no curve C i can be contained in a fiber of h, i.e. all the curves C i dominate Z. Indeed, otherwise the group Aut(Y ; h) ′ would preserve a non-empty finite subset of Z that consists of the images of the curves C i contained in the fibers of h, and thus Aut(Y ; h) ′ would be finite. Therefore, for a sufficiently ample divisor D on Z one has (A + h * D) · C i > 0 for all C i . Thus A ′ = A + h * D is a big and strongly numerically effective divisor. Hence it is ample by Lemma 1, and the threefold Y is projective. Proof. Similarly to the proof of Lemma 3, consider a birational morphism g :Ŷ → Y such thatŶ is projective. LetÂ be an ample divisor onŶ , and set A = g * Â . There is at most a finite number of curves C i ⊂ Y such that A · C i 0. As in the proof of Lemma 3 the set {C i } of all such curves is invariant under Aut(Y ; h) ′ . Suppose that some curve C i is contained in a fiber of h. Let z = h(C i ), let Aut(Z, z) ⊂ Aut(Z) be the stabilizer of z, and let Υ z = Υ ∩ Aut(Z, z). The index [Υ : Υ z ] is finite. Therefore, Υ z has unbounded finite subgroups. By [9, Lemma 2.5] the finite subgroups of Υ z /(Υ z ∩ Aut 0 (Z)) are bounded, where Aut 0 (Z) is the connected component of identity in Aut(Z). On the other hand, since Z is not ruled, Aut 0 (Z) is either trivial or is an abelian variety and so the stabilizer Aut 0 (Z, z) of z in Aut 0 (Z) must be trivial. The contradiction shows that no curve C i can be contained in a fiber of h. Thus for a sufficiently ample divisor D on Z the divisor A ′ = A + h * D is big and strongly numerically effective. Hence A ′ is ample by Lemma 1, and the threefold Y is projective.
Note that the construction of [6, Example VII.6.20] allows one to obtain an example of a non-projective Moishezon threefold such that the base of its maximal rationally connected fibration has arbitrary dimension (from 0 to 3).

Proof of Theorem 2
In this section we prove Theorem 2.
Suppose that the group Aut(X) is not Jordan. Since Aut(X) is a subgroup of the group Bir(X) of birational selfmaps of X, we conclude that Bir(X) is not Jordan either. Note that Bir(X) is isomorphic to the group of birational selfmaps of some projective vari-etyX birational to the compact complex space X. According to [16,Theorem 1.8(ii)], the varietyX is uniruled, and by [17,Theorem 1.8] and [3, Theorem 1.1] it is not rationally connected. Since X is birational toX, we see that X is also uniruled but not rationally connected. There exists the maximal rationally connected fibration f : X V , and one has 0 < dim V < dim X, see [8,Theorem 5.5.4]. The compact complex space V is Moishezon by [10,Theorem 2]. Note that the maximal rationally connected fibration is defined only as a rational map. Thus, resolving the singularities of V , we may assume that V is smooth.
One of our main tools is the following result that is implied by the existence of a canonical resolution of singularities (see [2, §13]).
Theorem 3. Let M be an irreducible compact complex manifold, and let W ⊂ M be its compact complex subspace. Then there exists a sequence of blow ups π :M → M with smooth centers such that the union of the proper transform π −1 W with the exceptional locus E of π is a simple normal crossing divisor. Moreover, the morphism π is canonical in the following sense: every automorphism M → M preserving W can be extended to an automorphismM →M that commutes with π.
Theorem 3 allows us to prove the following result.
Lemma 5. Let X be a Moishezon compact complex space. Let f : X V be the maximal rationally connected fibration, where we choose V to be smooth. Suppose that dim V 2. Let Z be the minimal model of V . Then there is an Aut(X)-equivariant commutative diagram Here Y is a Moishezon manifold, Z is smooth and projective, Y → X is a birational morphism, and h : Y → Z is the maximal rationally connected fibration for Y .
Proof. Since the manifold V is Moishezon and has dimension at most 2, it is projective. Hence its minimal model Z is projective (and smooth) as well.
Recall that the group Aut(X) acts on V by birational maps (possibly non-faithfully). Since Z is a minimal model of V , the group Aut(V ) acts on Z biregularly. The composition σ : X V of f with the contraction V → Z is an Aut(X)-equivariant map. Consider the closureΓ σ of the graph of this map in X × Z. Since the action of Aut(X) on X × Z is biregular, the action of Aut(X) onΓ σ is biregular as well. Finally, let Y be the canonical resolution of singularities ofΓ σ provided by Theorem 3. The action of Aut(X) on Y is again biregular, which gives us the commutative Aut(X)-equivariant diagram (1).
Apply Lemma 5 to our Moishezon compact complex space X. We have an embedding Aut(X) ⊂ Aut(Y ).
Since the map h : Y → Z is the maximal rationally connected fibration for Y , we see that the group Aut(Y ) acts (possibly non-faithfully) on Z, and the map h is Aut(Y )equivariant. Let Aut(Y ) h be the subgroup of Aut(Y ) that consists of all automorphisms whose action is fiberwise with respect to h, and let Υ ∼ = Aut(Y )/ Aut(Y ) h be the image of the group Aut(Y ) in Aut(Z). All finite subgroups of Aut(Y ) h act faithfully on a nonmultiple fiber of h, and thus Aut(Y ) h is Jordan by [18,Theorem 1.5]. This implies that if Υ has bounded finite subgroups, then the group Aut(Y ) is Jordan as well. Therefore, we will assume that the group Υ has unbounded finite subgroups.
Suppose that the dimension of Z equals 1. Then Z is a non-rational curve. In this case Y is projective by Lemma 3. Suppose that the dimension of Z equals 2. Then Z is a non-ruled surface. In this case Y is projective by Lemma 4.
Therefore, we see that under our assumptions the group Aut(X) is contained in the automorphism group of the projective variety Y . Now Theorem 2 follows from Theorem 1.