"Конечные группы, действующие на алгебраических и комплексных многообразиях" тема диссертации и автореферата по ВАК РФ 00.00.00, кандидат наук Голота Алексей Сергеевич

• Голота Алексей Сергеевич
• кандидат наук
• 2023

Оглавление диссертации кандидат наук Голота Алексей Сергеевич

Contents

Introduction

1 Non-uniruled Kaehler threefolds

1.1 Preliminaries

1.2 Singular compact Kahler spaces and their minimal models

1.3 Bimeromorphic maps of quasi-minimal models

1.4 The main results

2 Finite abelian subgroups

2.1 Preliminaries

2.1.1 Structure of abelian subgroups

2.1.2 The MRC fibration

2.1.3 Finite group actions

2.2 Main results

2.2.1 Groups of pseudoautomorphisms

2.2.2 Non-uniruled varieties

2.2.3 Rationally connected varieties

2.2.4 The general case

3 Compact parallelizable manifolds

3.1 Preliminaries

3.2 Discrete subgroups of Lie groups

3.3 Compact parallelizable manifolds

3.4 Outer automorphisms of lattices in semisimple Lie groups

3.5 Application of rigidity to deformations of lattices

3.6 Main results

4 Delta-invariants with group actions

4.1 General definitions

4.1.1 Notation and conventions

4.1.2 Log canonical thresholds

4.1.3 The space of valuations

4.1.4 Automorphisms preserving a big divisor class

4.1.5 G-invariant valuations

4.2 Equivariant K-stability

4.2.1 Basic definitions related to K-stability

4.2.2 The main result

4.3 The greatest Ricci lower bound

4.4 Varieties with an action of a torus

4.5 Spherical Fano varieties

4.6 Finite automorphism groups

Conclusion

Bibliography

Введение диссертации (часть автореферата) на тему «"Конечные группы, действующие на алгебраических и комплексных многообразиях"»

Introduction

Group actions are ubiquitous in classical and modern geometry. Since the publication of the famous Erlangen program by F. Klein in 1872, it has become a common practice to study groups via their actions on various geometric objects, that is, to study groups of symmetries. This viewpoint has become extremely influential and productive both for group theory and for many branches of geometry, including algebraic geometry and complex analytic geometry.

A powerful method to study various complicated groups of geometric nature (such as automorphism groups of varieties, Cremona groups, mapping class groups, etc) is to explore their finite subgroups. For example, one can consider groups of biregular automorphisms of projective algebraic varieties over algebraically closed fields of zero characteristic. If X is a nonsingular projective variety, then the group Aut(X) of biregular automorphisms of X fits into an exact sequence

1 ^ Aut0 (X) ^ Aut(X) ^ Aut(X)/ Aut0 ^ 1, (0.0.1)

where Aut0(X) denotes the connected component of the identity and Aut(X)/ Aut0(X) is a discrete group, called the group of connected components of Aut(X). It is well-known that the group Aut0(X) has a structure of a connected algebraic group scheme. On the other hand, groups of connected components are far from being well understood. For instance, there exist projective varieties such that the group Aut(X)/ Aut0(X) is not finitely generated [77]. In general, it is not clear how to characterize groups isomorphic to Aut(X)/ Aut0(X) for some projective variety X.

For a smooth projective variety X the action of the automorphism group on the second Betti cohomology (modulo torsion) gives a representation Aut(X)/Aut0(X) ^ GLN(Z) with finite kernel (see [85, Lemma 2.5]). For finite subgroups in the general linear group over Z (or over Q) there is a well-known theorem of H. Minkowski (see e.g. [112, Theorem 1]).

Theorem 0.0.1. The orders of finite subgroups in G = GLn(Z) are bounded by a natural number M (n) depending on n only.

Despite the fact that automorphism groups of projective varieties are, in general, not algebraic groups, it turns out that finite subgroups in these groups share similar properties with finite linear groups. Recall that a classical theorem of C. Jordan, proved in [65], describes finite subgroups in the general linear group.

Theorem 0.0.2. For every n e N there exists a natural number J(n) such that for any finite subgroup G ^ GLn(C) there exists a finite abelian subgroup A < G of index at most J(n).

In other words, for a fixed n all finite subgroups of the general linear group GLn(C) are obtained as extensions of groups of bounded orders by finite abelian subgroups. V. Popov in [100] suggested the name "Jordan groups" for groups with the above property. For a projective

variety X of any dimension over a field k of char(k) = 0 the Jordan property for Aut(X) was established by S. Meng and D.-Q. Zhang in [85].

Significantly more challenging is the study of finite subgroups in the groups of birational automorphisms. A particularly famous example is the Cremona group

Cr„(C) = Bir(Pg).

For n = 1 the Cremona group Cri(C) is isomorphic to PGL2(C), so the classification of finite subgroups in this group goes back to F. Klein. Already for n = 2 the classification of finite subgroups in Cr2(C) is very complicated. It was completed only in 2009 by I. Dolgachev and V. Iskovskikh [40]. We refer to [36] for a survey of results related to higher-dimensional Cremona groups.

Therefore, a more reasonable goal is to establish a "qualitative" boundedness result, such as the Jordan property, for finite subgroups in Bir(X). The Jordan property for Cr2(C) = Bir(PC) was established by Serre in [112]. For complex projective surfaces V. Popov in [100] proved that the Jordan property holds in all cases except P1 xE where E is an elliptic curve. Later, Yu. Zarhin [137] showed that Bir(P1 xE) is not Jordan and provided similar examples in higher dimensions.

Important results on the Jordan property of Bir(X) for X projective over a field of characteristic 0 were obtained by Yu. Prokhorov and C. Shramov. They applied powerful methods of birational geometry, such that the Minimal Model Program (MMP) and various natural fibrations (such as the MRC fibration or the Iitaka fibration), to the study of finite subgroups of Bir(X). In particular, they proved that the Jordan property holds for Bir(X) for X non-uniruled, that is, not covered by rational curves [102].

Theorem 0.0.3. Let X be a non-uniruled smooth projective variety over a field of zero characteristic. Then the group Bir(X) is Jordan. Moreover, if the irregularity q(X) = h1(X, Ox) = 0 then the group Bir(X) has bounded finite subgroups.

For rationally connected projective varieties of dimension n Yu. Prokhorov and C. Shramov [103] have shown the "uniform" Jordan property. A key component of the proof is a boundedness result for Fano varieties, proved by C. Birkar in [11].

Theorem 0.0.4. For every n e N there exists a natural number R(n) such that the following holds. Let X be a rationally connected variety over an algebraically closed field of char = 0. Then for any finite subgroup G ^ Bir(X) there exists a normal abelian subgroup A < G of index at most R(n).

In particular, the Cremona groups Crn(C) are Jordan for every n.

For compact complex manifolds it is natural to consider the group Bim(X) of bimeromorphic maps from X to itself. Yu. Prokhorov and C. Shramov [106] considered the case of compact complex surfaces. They proved that that automorphism groups of all compact complex surfaces are Jordan. Moreover, they proved that bimeromorphic automorphism groups of all compact complex surfaces except the ones birational to P1 xE are Jordan as well.

In higher dimensions it is more reasonable to study bimeromorphic automorphisms of compact Kaehler manifolds, since these manifolds have many common properties with projective manifolds. The groups of biholomorphic automorphisms of compact Kaehler spaces are Jordan [69]. In a series of papers [104, 105, 107] Yu. Prokhorov and C. Shramov studied groups of bimeromorphic

selfmaps of compact Kahler spaces of dimension 3. In the uniruled case, they proved that Bim(X) is Jordan unless X is bimeromorphic to a space from one of finitely many explicitly described families, see [105, Theorems 1.3 and 1.4].

Theorem 0.0.5. Let X be a uniruled compact Kaehler manifold of dimension 3. Then the group Bim(X) is not Jordan if and only if X is bimeromorphic either to P2 xE, where E is an elliptic curve, or to P1 xS, where S is isomorphic to one of the following surfaces:

• an abelian surface;

• a bielliptic surface;

• a surface of Kodaira dimension 1 such that the Jacobian fibration of its pluricanonical map is locally trivial in the Zariski topology.

As for groups of bimeromorphic automorphisms of non-uniruled Kahler spaces, Yu. Prokhorov and C. Shramov [107] were able to show the Jordan property under an additional assumption. Recall that the Kodaira dimension k(x) of a compact Kahler space X is defined to be the Kodaira dimension of any smooth manifold X1 bimeromorphic to X. Analogously, the irregularity of X is defined to be q(X) = H 1(X1, Ox') for any smooth manifold X1 bimeromorphic to X.

Theorem 0.0.6. [107, Theorem 1.3] Let X be a compact Kahler space of dimension 3. Assume that k(x) ^ 0 and q(X) > 0. Then the group Bim(X) is Jordan.

The first chapter of this thesis is devoted to the proof of the following theorem, which completes the study of the Jordan property for groups of bimeromorphic automorphisms of compact Kaehler threefolds.

Theorem 0.0.7. The group Bim(X) is Jordan for any non-uniruled compact Kahler space X of dimension 3.

To achieve this we study (in sections 1.2 and 1.3) the groups of pseudoautomorphisms of compact Kaehler spaces and generalize the arguments from [42, Corollary 3.3] and [107, Proposition 4.5] to singular Kahler spaces, see Theorems 1.3.5 and 1.4.1, respectively. Together with existence of minimal models for Q-factorial terminal Kahler spaces of dimension 3 from [61] this gives the desired conclusion of Theorem 0.0.7. In higher dimensions, we show that the same result holds assuming existence of a quasi-minimal model of X (Theorem 2.2.6).

A closely related problem is to study "large" finite abelian subgroups in Bir(X) for X projective. More precisely, let G c Bir(X) be a finite abelian subgroup. The most basic invariant of a finite abelian G is the rank r = r(G), that is, the minimal number of elements generating G. In the second chapter of the thesis we study "asymptotic" properties of finite abelian subgroups in the groups of birational automorphisms of projective varieties over a field of zero characteristic. The starting point for us is the following recent theorem by I. Mundet i Riera [92, Theorem 1.15].

Theorem 0.0.8. Let X be a compact Kahler manifold. Suppose that there exists r e N such that for infinitely many positive integers N the group Aut(X) contains a subgroup isomorphic to (Z/NjZ)r. Then Aut(X) contains a subgroup isomorphic to a compact real torus of dimension r. Moreover; one has r ^ 2dim(X), and if r = 2dim(X) then X is biholomorphic to a compact complex torus.

In the same paper, I. Mundet i Riera asked if the same upper bound is valid also for birational automorphism groups of projective varieties (or for bimeromorphic automorphisms of compact Kaehler manifolds).

In fact, finite abelian groups of large orders have appeared in algebraic geometry in various contexts. In [135, Theorem 2.9] J. Xu has shown that "large" finite abelian p-subgroups of birational automorphism group of non-uniruled projective varieties do indeed come from birational actions of abelian varieties.

Theorem 0.0.9. Let X be a non-uniruled variety over an algebraically closed field of characteristic zero. There exists a constant b(X), depending only on the birational equivalence class of X, such that the group Bir(X) contains an element of order greater than b(X) if and only if X is birational to a variety X' which admits an effective action of an abelian variety.

Moreover, J. Xu proved a rationality criterion for rationally connected varieties admitting an action of a "large" elementary p-group (Z/pZ)r of maximal rank r (see [135, Theorem 4.5]).

Theorem 0.0.10. Let X be a rationally connected variety of dimension n over an algebraically closed field of characteristic zero. Then there exists a constant R(n) such that if Bir(X) contains a subgroup isomorphic to (Z/pZ)r for some p > R(n) then X is rational.

Informally speaking, these results suggest that "large" finite abelian subgroups of Bir(X) should come from certain algebraic groups (e. g. semi-abelian varieties) acting on X by birational automorphisms. This "toroidalization principle" has been studied recently by J. Moraga in his works on Kawamata log terminal singularities [87, 88, 89]. In particular, he shows that existence of "large" finite abelian groups of rank n acting on a projective Fano type variety of dimension n implies that X is birational to a log Calabi-Yau toric pair ([87, Theorem 2]). In [89, Theorem 1] a toroidalization principle for finite group actions on klt singularities is proved.

The main result of the second chapter is a generalization of Theorem 0.0.8 to the groups of birational automorphisms.

Theorem 0.0.11. Let X be a projective variety over an algebraically closed field of zero characteristic. Suppose that the group Bir(X) contains subgroups isomorphic to (Z/NiZ)r for some fixed r and arbitrarily large Ni. Then r < 2dim(X) and in case of equality X is birational to an abelian variety.

Combining these ideas with some technical results from Chapter 1, we were able to prove the same result for groups of bimeromorphic selfmaps of compact Kaahler spaces under an additional assumption.

Theorem 0.0.12. Let X be a compact Kahler space. Suppose that the group Bim(X) contains subgroups isomorphic to (Z/NiZ)r for some fixed r and arbitrarily large Ni. Suppose also that the base of the MRC-fibration of X has dimension at most 3. Then the upper bound r < 2dim(X) and in case of equality X is bimeromorphic to a compact complex torus.

Suppose that X is a (not necessarily Kaehler) compact complex manifold. Then the automorphism group Aut(X) fits into an exact sequence (3.4.1) and it is well-known that Aut0(X) is a complex Lie group. However, boundedness of finite subgroups in the group of connected components remains an open problem. For non-Kaahler compact complex manifolds there are only a few known results on the Jordan property for (biregular and bimeromorphic) automorphism groups.