Automorphism groups of Moishezon threefolds
Yuri Prokhorov, Constantin Shramov

TL;DR
This paper proves that the automorphism groups of Moishezon threefolds are always Jordan, contributing to the understanding of symmetry groups in complex geometry.
Contribution
It establishes that automorphism groups of Moishezon threefolds are always Jordan, a property previously studied in other classes of complex manifolds.
Findings
Automorphism groups of Moishezon threefolds are always Jordan.
Provides a new class of complex manifolds with Jordan automorphism groups.
Abstract
We show that automorphism groups of Moishezon threefolds are always Jordan.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Algebra and Geometry · Finite Group Theory Research
Automorphism groups of Moishezon threefolds
Yu. G. Prokhorov, C. A. Shramov
[email protected], [email protected]
Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina st., Moscow, 119991, Russia
National Research University Higher School of Economics, Laboratory of Algebraic Geometry, 6 Usacheva str., Moscow, 119048, Russia
Abstract.
We study automorphism groups of Moishezon threefolds and show that such groups are always Jordan.
This work is supported by the Russian Science Foundation under grant №18-11-00121.
1. 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 is Jordan (or has Jordan property) if there is a constant such that every finite subgroup contains a normal abelian subgroup of index at most .
The groups that enjoy Jordan property include: general linear groups , where 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 be a projective variety over a field of zero characteristic. Then the group 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 is said to be Moishezon if the transcendence degree of its field of meromorphic functions is maximal, that is, equals the dimension of . 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 ]). 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 ]. 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 be a three-dimensional Moishezon compact complex space. Then the group 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.
2. Some projectivity criteria
In this section we collect several assertions on projectivity of certain Moishezon varieties.
Definition**.**
A divisor is strongly numerically effective, if for every curve .
Lemma 1**.**
Let be a Moishezon threefold, and let be a strongly numerically effective divisor on . Suppose that for some the linear system has no fixed components. Then is ample. In particular, the manifold is projective.
Proof.
Follows from Nakai–Moishezon ampleness criterion for Moishezon compact complex spaces, see [10, Theorem 6]. Indeed, by assumption we have for every curve . Furthermore, since the divisor is numerically effective and big, one has .
Let be an irreducible surface (that is, a two-dimensional compact complex subspace). Then is a Moishezon compact complex space by [10, Theorem 3]. In particular, contains curves. The restriction is an effective divisor on , and one has , because has positive intersections with curves on . Therefore, we see that . Hence is ample by Nakai–Moishezon criterion. ∎
Lemma 2**.**
Let be a surjective morphism of smooth compact complex surfaces. Let be an ample divisor on . Then the divisor is ample.
Proof.
By adjunction formula, we have for every curve on , and also . Hence is ample by Nakai–Moishezon criterion. ∎
Remark 1**.**
The assertion of Lemma 2 fails in the case when the surface is singular. However, it still holds if is a two-dimensional normal compact complex space with -factorial singularities.
For a contraction by we denote the subgroup of that consists of all automorphisms such that is equivariant with respect to their action. By we denote the maximal subgroup of that acts trivially on . Note that is a normal subgroup of , and the quotient has bounded finite subgroups by the classical theorem of Minkowski. Therefore, the group has bounded finite subgroups if and only if this holds for .
Lemma 3**.**
Let be a Moishezon threefold. Let be a contraction to a non-rational curve. Furthermore, assume that the image of the homomorphism is infinite. Then the threefold is projective.
Proof.
Let be a birational morphism such that is projective; such a morphism always exists, see [11, Theorem ]. Let be an ample divisor on . Set . Then the divisor is big. By projection formula, there is at most a finite number of curves such that (these curves must be contained in the image of the -exceptional divisor).
Note that for any we have
[TABLE]
Hence the set of all such curves is invariant under . By our assumptions is an elliptic curve and the image of consists of translations. This implies that no curve can be contained in a fiber of , i.e. all the curves dominate . Indeed, otherwise the group would preserve a non-empty finite subset of that consists of the images of the curves contained in the fibers of , and thus would be finite.
Therefore, for a sufficiently ample divisor on one has for all . Thus is a big and strongly numerically effective divisor. Hence it is ample by Lemma 1, and the threefold is projective. ∎
Lemma 4**.**
Let be a Moishezon threefold, and let be a contraction to a non-ruled surface. Assume that the image of the homomorphism has unbounded finite subgroups. Then the threefold is projective.
Proof.
Similarly to the proof of Lemma 3, consider a birational morphism such that is projective. Let be an ample divisor on , and set . There is at most a finite number of curves such that . As in the proof of Lemma 3 the set of all such curves is invariant under . Suppose that some curve is contained in a fiber of . Let , let be the stabilizer of , and let . The index is finite. Therefore, has unbounded finite subgroups. By [9, Lemma 2.5] the finite subgroups of are bounded, where is the connected component of identity in . On the other hand, since is not ruled, is either trivial or is an abelian variety and so the stabilizer of in must be trivial. The contradiction shows that no curve can be contained in a fiber of . Thus for a sufficiently ample divisor on the divisor is big and strongly numerically effective. Hence is ample by Lemma 1, and the threefold 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 [math] to ).
3. Proof of Theorem 2
In this section we prove Theorem 2.
Suppose that the group is not Jordan. Since is a subgroup of the group of birational selfmaps of , we conclude that is not Jordan either. Note that is isomorphic to the group of birational selfmaps of some projective variety birational to the compact complex space . According to [16, Theorem 1.8(ii)], the variety is uniruled, and by [17, Theorem 1.8] and [3, Theorem 1.1] it is not rationally connected. Since is birational to , we see that is also uniruled but not rationally connected. There exists the maximal rationally connected fibration , and one has , see [8, Theorem 5.5.4]. The compact complex space 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 , we may assume that 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 be an irreducible compact complex manifold, and let be its compact complex subspace. Then there exists a sequence of blow ups with smooth centers such that the union of the proper transform with the exceptional locus of is a simple normal crossing divisor. Moreover, the morphism is canonical in the following sense: every automorphism preserving can be extended to an automorphism that commutes with .
Theorem 3 allows us to prove the following result.
Lemma 5**.**
Let be a Moishezon compact complex space. Let be the maximal rationally connected fibration, where we choose to be smooth. Suppose that . Let be the minimal model of . Then there is an -equivariant commutative diagram
[TABLE]
Here is a Moishezon manifold, is smooth and projective, is a birational morphism, and is the maximal rationally connected fibration for .
Proof.
Since the manifold is Moishezon and has dimension at most , it is projective. Hence its minimal model is projective (and smooth) as well.
Recall that the group acts on by birational maps (possibly non-faithfully). Since is a minimal model of , the group acts on biregularly. The composition of with the contraction is an -equivariant map. Consider the closure of the graph of this map in . Since the action of on is biregular, the action of on is biregular as well. Finally, let be the canonical resolution of singularities of provided by Theorem 3. The action of on is again biregular, which gives us the commutative -equivariant diagram (1). ∎
Apply Lemma 5 to our Moishezon compact complex space . We have an embedding .
Since the map is the maximal rationally connected fibration for , we see that the group acts (possibly non-faithfully) on , and the map is -equivariant. Let be the subgroup of that consists of all automorphisms whose action is fiberwise with respect to , and let be the image of the group in . All finite subgroups of act faithfully on a non-multiple fiber of , and thus is Jordan by [18, Theorem 1.5]. This implies that if has bounded finite subgroups, then the group is Jordan as well. Therefore, we will assume that the group has unbounded finite subgroups.
Suppose that the dimension of equals . Then is a non-rational curve. In this case is projective by Lemma 3.
Suppose that the dimension of equals . Then is a non-ruled surface. In this case is projective by Lemma 4.
Therefore, we see that under our assumptions the group is contained in the automorphism group of the projective variety . Now Theorem 2 follows from Theorem 1.
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