A. S. Golota, “Finite abelian subgroups in the groups of birational and bimeromorphic selfmaps”, Izv. Math., 88:5 (2024), 856

Abstract: Let $X$ be a complex projective variety. Suppose that the group of birational automorphisms of $X$ contains finite subgroups isomorphic to $(\mathbb{Z}/N\mathbb{Z})^r$ for $r$ fixed and $N$ arbitrarily large. We show that $r$ does not exceed $2\dim(X)$. Moreover, the equality holds if and only if $X$ is birational to an abelian variety. We also show that an analogous result holds for groups of bimeromorphic automorphisms of compact Kähler spaces under some additional assumptions.

§ 1. Introduction

In the present paper, we study finite abelian subgroups in the groups of birational automorphisms of projective algebraic varieties (over a field of zero characteristic), or in the groups of bimeromorphic automorphisms of compact Kähler spaces. The starting point for us is the following recent theorem by Mundet i Riera (see Theorem 1.9 in [22]).

Theorem 1. Let $X$ be a connected compact Kähler manifold. Suppose that there exists $r \in \mathbb{N}$ such that, for arbitrarily large positive integers $N$, the group $\operatorname{Aut}(X)$ contains a subgroup isomorphic to $(\mathbb{Z}/N\mathbb{Z})^r$. Then $\operatorname{Aut}(X)$ contains a subgroup isomorphic to a compact real torus of dimension $r$. In addition, $r \leqslant 2\dim(X)$, and if $r = 2\dim(X)$, then $X$ is biholomorphic to a compact complex torus.

The maximal number $r$ satisfying the assumptions of Theorem 1 is called in [22] the (holomorphic) discrete degree of symmetry of $X$. More generally, Mundet i Riera [22] defines and studies this invariant for continuous group actions on topological manifolds. In some cases, the discrete degree of symmetry can be compared to the maximal dimension of a torus acting effectively on a manifold (see Theorem 1.7 in [22]). In connection with Theorem 1, Mundet i Riera also asks whether the same bound on $r$ holds also for birational automorphism groups. In fact, this invariant has implicitly appeared in the study of $p$-subgroups of birational automorphism groups. For instance, Xu (see Theorem 2.9 in [34]) proved the following result for non-uniruled algebraic varieties.

Theorem 2. Let $X$ be a non-uniruled algebraic variety over an algebraically closed field of characteristic zero. There exists a constant $b(X)$ such that the group $\operatorname{Bir}(X)$ 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.

A remarkable result of Prokhorov and Shramov (Theorem 1.10 in [25]), together with Birkar’s solution of the BAB conjecture (Theorem 1.1 in [3]), provides a stronger bound for rationally connected varieties.

Theorem 3. Let $X$ be a rationally connected algebraic variety of dimension $n$ over an algebraically closed field of characteristic zero. There exists a constant $L = L(n)$ such that, for any prime number $p > L(n)$, each finite $p$-subgroup $G \subset \operatorname{Bir}(X)$ is isomorphic to $(\mathbb{Z}/p\mathbb{Z})^r$ for some $r \leqslant n$.

By a result of Xu [35], the constant $L$ in the above theorem can be taken to be $n\,{+}\, 1$. Moreover, Xu proved a rationality criterion for rationally connected varieties admitting an action of $(\mathbb{Z}/p\mathbb{Z})^r$ in terms of $r$ and $p$ (see Theorem 4.5 in [34]).

Theorem 4. Let $X$ be a rationally connected algebraic variety of dimension $n$ over an algebraically closed field of characteristic zero. Then there exists a constant $R(n)$ such that if $\operatorname{Bir}(X)$ contains a subgroup isomorphic to $(\mathbb{Z}/p\mathbb{Z})^n$ for some $p > R(n)$, then $X$ is rational.

Informally speaking, these results suggest that existence of finite abelian subgroups in $\operatorname{Bir}(X)$ of unbounded orders should imply existence of algebraic groups (of positive dimension depending on $r$) acting on $X$ by birational automorphisms, at least if the ranks $r$ of the finite abelian groups are close to maximal. For smaller values of $r$ the relation between finite abelian and algebraic subgroups of $\operatorname{Bir}(X)$ is more delicate. For instance, there exists a sequence of finite cyclic subgroups of $\mathrm{Cr}_2(\mathbb{C}) = \operatorname{Bir}(\mathbb{P}^2_{\mathbb{C}})$ which generate a subgroup isomorphic to $\mathbb{Q}/\mathbb{Z}$ but are not contained in any torus in the Cremona group [33]. Existence of finite abelian subgroups of unbounded orders in the group $\operatorname{Bir}(X)$, where $X$ is a non-rational rationally connected threefold, is a difficult open problem (see Question 4.8 in [26]). Another related open problem (cf. Conjecture 1.7 in [34]) is a conjectural description of projective varieties with non-Jordan groups of birational automorphisms. The first examples of such varieties were constructed in [36]; a complete description exists in dimension 3 by Theorem 1.8 in [26] and Theorem 1.6 in [34].

We should also mention a “toroidalization principle” recently studied by Moraga in his works on Kawamata log terminal singularities [19]–[21]. In particular, he showed 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 (see Theorem 2 in [19]). In [21], Theorem 1, a general result on toroidalization for finite group actions on klt singularities is proved. The case of cyclic group actions on Fano type surfaces is studied in [20].

The aim of this paper is to initiate a systematic study of an invariant similar to the discrete degree of symmetry for groups of birational (and bimeromorphic) automorphisms. Our main result is a generalization of Theorem 1 to groups of birational automorphisms.

Theorem 5. Let $X$ be a projective algebraic variety over an algebraically closed field of zero characteristic. Suppose that there exists an unbounded sequence $\{N_i\}_{i \in \mathbb{N}}$ of positive integers such that the group $\operatorname{Bir}(X)$ contains subgroups isomorphic to $(\mathbb{Z}/N_i\mathbb{Z})^r$ for some fixed $r$. Then $r \leqslant 2\dim(X)$, and, in case of equality, $X$ is birational to an abelian variety.

Compared to Theorem 2, we consider also uniruled varieties; moreover, we do not assume that the orders $N_i$ of generators of the finite groups are prime. The main idea of the proof is to consider the action of $\operatorname{Bir}(X)$ on the maximal rationally connected (MRC) fibration of $X$ (see Definition 4 below); this idea is already present in Xu’s work (see Proposition 2.12 in [34]). Combining it with some technical results from our paper [10], we prove an analogous result for groups of bimeromorphic selfmaps of compact Kähler spaces. We have to assume the existence of quasi-minimal models (see Definition 6 and Proposition 8 below) for the (non-uniruled) base of the MRC fibration of $X$. This is the case if the base has dimension at most 3 by Theorem 1.1 in [14], and is expected to be true in any dimension.

Theorem 6. Let $X$ be a compact Kähler space. Assume that the base $B$ of the MRC fibration of $X$ admits a quasi-minimal model. Suppose that there exists an unbounded sequence $\{N_i\}_{i \in \mathbb{N}}$ of positive integers such that the group $\operatorname{Bim}(X)$ contains subgroups isomorphic to $(\mathbb{Z}/N_i\mathbb{Z})^r$ for some fixed $r$. Then $r \leqslant 2\dim(X)$, and, in case of equality, $X$ is bimeromorphic to a compact complex torus.

Let us outline the structure of the paper. In § 2, we gather some technical results. Section 3 is devoted to the proof of our main theorem. First, in § 3.1, we use techniques from our previous paper [10] to generalize Theorem 1 to pseudoautomorphisms of compact Kähler spaces with rational singularities (see Theorem 11). Then, in § 3.2, we prove the main theorem for non-uniruled projective varieties (Theorem 13), following the ideas from § 2 in [34]. In § 3.3, we use the results of Prokhorov and Shramov from [25] to derive the bound on $r$ for abelian groups acting on rationally connected varieties. Finally, in § 3.4, we derive Theorem 5 from Theorems 13 and 14 using the maximal rationally connected fibration of $X$. We also prove Theorem 6 in this section.

Acknowledgements

The author thanks C. A. Shramov for suggesting this problem; he also thanks I. Mundet i Riera and the anonymous referee for valuable remarks.

§ 2. Preliminaries

2.1. Conventions and terminology

An algebraic variety (or just a variety) is an integral separated scheme of finite type over a field $k$. Unless explicitly stated otherwise, the base field $k$ is always assumed to be algebraically closed and of characteristic zero.

In what follows we consider irreducible and reduced compact complex spaces, see Chap. 1 in [12] for a general reference on complex analytic spaces. A complex manifold is a nonsingular complex space. We consider only compact Kähler manifolds. For the definition of a singular compact Kähler space see Definition 5; this definition follows the one in [14].

2.2. Structure of abelian subgroups

In this subsection, we collect a few technical statements on subgroups of finite abelian groups.

Lemma 1. Let $G \simeq (\mathbb{Z}/N\mathbb{Z})^r$ be a finite abelian group. Let $H \subset G$ be a subgroup of index

$$ \begin{equation*} I_H \leqslant N-1. \end{equation*} \notag $$

Then there exists an elementary subgroup $H' \subset H$ such that $H' \simeq (\mathbb{Z}/N'\mathbb{Z})^r$ for some $N' \geqslant N/I_H$.

Proof. Let $H \subseteq G$ be a subgroup of index $I_H \leqslant N-1$. Then $H$ is a finite abelian group of order at least $N(r-1) + 1$. The orders of generators of $H$ do not exceed $N$, so the rank of $H$ is at least $r$. By the structure theorem for finite abelian groups we have

$$ \begin{equation*} H \simeq \bigoplus_{1 \leqslant i \leqslant r}\mathbb{Z}/N_i\mathbb{Z}, \end{equation*} \notag $$

where $N_i | N_{i+1}$ for all $i \in \{1, \ldots, r-1\}$. Next, from the equality

$$ \begin{equation*} |H| = N^r/I_H = N_1 \cdots N_r \end{equation*} \notag $$

we have $N_1 \geqslant N/I_H$. Now it suffices to take the elementary subgroup $ H' = (\mathbb{Z}/N_1\mathbb{Z})^r \subseteq H$. This proves Lemma 1.

Lemma 2. Let $G \simeq (\mathbb{Z}/N\mathbb{Z})^r$ be a finite abelian group and let $H \subset G$ be a subgroup. There exist a set of generators $\{b_1, \dots, b_r\}$ for $H$ and a set of generators $\{a'_1, \dots, a'_r\}$ for $G$ such that the embedding $H \to G$ can be written as the direct sum of homomorphisms

$$ \begin{equation*} \mathbb{Z}/N_i\mathbb{Z} \to \mathbb{Z}/N\mathbb{Z}. \end{equation*} \notag $$

Proof. We choose a set of generators $a_1, \dots, a_r$ for $\mathbb{Z}^r$ such that their images under the projection

$$ \begin{equation*} \mathbb{Z}^r \to \mathbb{Z}^r/(N\mathbb{Z})^r \end{equation*} \notag $$

give an isomorphism $(\mathbb{Z}/N\mathbb{Z})^r \simeq G$. Let us denote by $\widetilde{H}$ the preimage of $H$ under the map $\mathbb{Z}^r \to G$. Then $\widetilde{H}$ is a free abelian group of rank $r$ containing the subgroup $(\mathbb{Z}/N\mathbb{Z})^r$. Let $A$ be a presentation matrix for $\widetilde{H}$. Using, for example, Theorem (4.3) in [2], we find there exists the Smith normal form

$$ \begin{equation*} A' = QAP^{-1}, \end{equation*} \notag $$

where $Q, P \in \mathrm{GL}_r(\mathbb{Z})$ and the matrix $A'$ is diagonal with entries $N_1, \dots, N_r$ such that $N_i$ divides $N_{i+1}$ for any $i \in \{1, \dots, r-1\}$. So, there exists a basis $a'_1, \dots, a'_r$ of $\mathbb{Z}^r$ such that $\widetilde{H}$ is generated by

$$ \begin{equation*} b_1 = a'_1N_1,\quad \dots,\quad b_r = a'_rN_r. \end{equation*} \notag $$

Since multiplication by invertible matrices preserves the sublattice $(N\mathbb{Z})^r \subset \mathbb{Z}^r$, we can take the images of $b_1, \dots, b_r$ under the projection $\mathbb{Z}^r \to \mathbb{Z}^r/(n\mathbb{Z})^r \simeq G$ as generators for $H$. This proves the lemma.

For the discussion that follows, it will be convenient to introduce the following definition.

Definition 2. Let $\{G_i\}_{i \in \mathbb{N}}$ be a sequence of finite groups. We define the asymptotic rank of the sequence $\{G_i\}$ to be the minimal number $r$ such that the following condition is satisfied. There exists a constant $L$ such that, for infinitely many indices $i \in \mathbb{N}$, we can find an abelian subgroup $H_i \subset G_i$ such that

$\bullet$ $H_i$ is generated by $r$ elements;

$\bullet$ the orders of the subgroups $H_i$ are unbounded as $i$ tends to infinity;

$\bullet$ the index of $H_i$ in $G_i$ does not exceed $L$.

Example 1. If the orders of the groups $G_i$ are bounded by a constant then the asymptotic rank of the sequence $\{G_i\}$ is equal to zero. The asymptotic rank of the sequence $G_i = (\mathbb{Z}/i\mathbb{Z})^{r} \times (\mathbb{Z}/2\mathbb{Z})^{10}$ is equal to $r$. The asymptotic rank of the sequence $G_r = (\mathbb{Z}/r\mathbb{Z})^r$ is infinite.

Remark 1. The motivation for the above definition comes from the study of Jordan groups. Recall (see Definition 2.1 in [23]) that a group $G$ is Jordan if there exists a constant $J(G) \in \mathbb{N}$ such that, for every finite subgroup $H \subset G$, there exists a normal abelian subgroup $A \lhd H$ of index at most $J(G)$. Suppose that the group $G$ is Jordan and that the orders of finite subgroups of $G$ are unbounded. Then there exists a sequence $\{G_i\}_{i \in \mathbb{N}}$ of finite subgroups of $G$ satisfying the assumptions in Definition 2 for some $r$ and $L = J(G)$ (the Jordan constant of $G$). The maximum value of $r$ over all such sequences of finite subgroups of $G$ is a natural invariant of the group $G$.

Remark 2. The most basic example of a Jordan group is a linear algebraic group $G$ over an algebraically closed field $k$ of characteristic zero. In this case, by Lemma 3.7 in [34], there exists a constant $B(n)$ such that, for every connected linear algebraic group $G$ over $k$ of rank at most $n$ and for every finite subgroup $H \subset G$, there exists a finite subgroup $N \subset H$ of index at most $B(n)$ such that $N$ is contained in a maximal torus of $G$. Thus, the asymptotic rank of any sequence of finite subgroups of $G$ is bounded from above by the rank of $G$.

Remark 3. Let $\{G_i\}$ be a sequence of finite groups of asymptotic rank $r$. Then we can take a sequence of abelian subgroups $H_i \subset G_i$ as in Definition 2; then the asymptotic rank of $\{H_i\}$ is equal to the asymptotic rank of $\{G_i\}$. More generally, if $\{G'_i \subset G_i\}$ is a sequence of subgroups of uniformly bounded index, then the asymptotic ranks of the sequences $\{G'_i\}$ and $\{G_i\}$ are equal. Therefore, by Lemma 1, it suffices to consider sequences of elementary abelian groups.

For a sequence of finite abelian groups , the asymptotic rank can be computed using the direct sum decomposition provided by the structure theorem.

Proposition 1. Let $\{G_i\}_{i \in \mathbb{N}}$ be a sequence of finite abelian groups. Suppose that for every $i \in \mathbb{N}$ the rank of $G_i$ is $r$. Consider the decomposition

$$ \begin{equation*} G_i \simeq \bigoplus_{1 \leqslant k \leqslant r}\mathbb{Z}/N_{i,k}\mathbb{Z}, \end{equation*} \notag $$

where $N_{i,k} | N_{i, k+1}$ for every $k \in \{1, \dots, r-1\}$. Then the asymptotic rank of the sequence $\{G_i\}$ is equal to

$$ \begin{equation*} r - \max\{k \mid \textit{the sequence }\{N_{i, k}\},\, i \in \mathbb{N}, \textit{ is bounded as }i \to \infty\}. \end{equation*} \notag $$

Proof. We set

$$ \begin{equation*} k_{\max} = \max\{k \mid \text{the sequence }\{N_{i, k}\},\, i \in \mathbb{N}, \text{ is bounded as }i \to \infty\}. \end{equation*} \notag $$

Considering the sequence of subgroups

$$ \begin{equation*} H_i = \bigoplus_{k_{\max}+1 \leqslant k \leqslant r}\mathbb{Z}/N_{i,k}\mathbb{Z} \subset G_i, \end{equation*} \notag $$

we find that the asymptotic rank of the sequence $\{G_i\}$ is at most $r-k_{\max}$. We denote by $L$ a constant such that $|G_i|/|H_i| \leqslant L$ for all $i \in \mathbb{N}$.

Suppose that the asymptotic rank $r'$ of $\{G_i\}$ is smaller than $r - k_{\max}$. Then there exists a sequence of subgroups $\{H'_i \subset G_i\}$ such that, for every $i \in \mathbb{N}$, the group $H'_i$ is generated by $r' < r - k_{\max}$ elements, and the indices $|G_i|/|H'_i|$ are bounded as $i \to \infty$. We have

$$ \begin{equation*} |H'_i| = |H'_i/(H_i \cap H'_i)|\cdot |H_i \cap H'_i| \leqslant L \cdot|H_i \cap H'_i|. \end{equation*} \notag $$

Hence

$$ \begin{equation*} \frac{|G_i|}{|H'_i|} \geqslant \frac{|G_i|}{L\cdot|H_i\cap H'_i|} = \frac{|G_i|}{L|H_i|}\cdot\frac{|H_i|}{|H_i\cap H'_i|}. \end{equation*} \notag $$

On the other hand, since the subgroup $H_i \cap H'_i$ is generated by $r' < r - k_{\max}$ elements, the indices $|H_i|/|H_i \cap H'_i|$ are unbounded as $i \to \infty$. This contradiction shows that the asymptotic rank of $\{G_i\}$ is equal to $r - k_{\max}$. This proves the proposition.

Another convenient way to express the asymptotic rank of a sequence of finite abelian groups is given in the corollary below.

Corollary 1. Let $\{G_i\}$ be a sequence of finite abelian groups. Suppose that for every $i \in \mathbb{N}$, the group $G_i$ can be generated by $r$ elements. Then the asymptotic rank of $\{G_i\}$ is equal to

$$ \begin{equation*} \max\{r \mid G_i \supset (\mathbb{Z}/M_i\mathbb{Z})^r\textit{ for an infinite number of }i \in \mathbb{N}\textit{ as some }M_i \to \infty\}. \end{equation*} \notag $$

Proof. By Proposition 1, the asymptotic rank of the sequence $\{G_i\}$ is $r - k_{\max}$, where

$$ \begin{equation*} k_{\max} = \max\{k \mid \text{the sequence }\{N_{i, k}\},\, i \in \mathbb{N}, \text{ is bounded as }i \to \infty\}. \end{equation*} \notag $$

Consider the sequence of subgroups

$$ \begin{equation*} H_i = \bigoplus_{k_{\max}+1 \leqslant k \leqslant r}\mathbb{Z}/N_{i,k}\mathbb{Z} \subset G_i. \end{equation*} \notag $$

Then each $H_i$ contains a subgroup isomorphic to $(\mathbb{Z}/M_i\mathbb{Z})^{r-k_{\max}}$, where $M_i = N_{i, k_{\max} + 1}$. Suppose that, for infinitely many $i \in \mathbb{N}$, we can find subgroups $H'_i \subset G_i$ such that $H'_i \simeq (\mathbb{Z}/M'_i\mathbb{Z})^s$ for $M_i \to \infty$. Consider the images of $H'_i$ under the quotient homomorphisms $G_i \to G_i/H_i$. Since the indices $|G_i|/|H_i|$ are bounded by a constant $L$ independent of $i \in \mathbb{N}$, the number $s$ does not exceed $r - k_{\max}$. This proves the corollary.

Lemma 3. Let $\{G_i\}$ be a sequence of abelian groups. Consider a sequence of subgroups $G'_i \subset G_i$ for $i \in \mathbb{N}$, and denote the quotient groups by $G''_i$. Suppose that the asymptotic rank of the sequence $\{G'_i\}$ is at most $r'$ and that the asymptotic rank of the sequence $\{G''_i\}$ is at most $r''$. Then the asymptotic rank $r$ of the sequence $\{G_i\}$ is at most $r'+r''$.

Proof. By Remark 3, we may assume that $G_i \simeq (\mathbb{Z}/N_i\mathbb{Z})^r$ are elementary abelian subgroups for some $N_i$ that tend to infinity. By Lemma 2, we may choose compatible systems of generators in $G'_i$ and $G_i$ for every $i \in \mathbb{N}$. Let us denote by

$$ \begin{equation*} N'_{i, 1} \,|\, N'_{i, 2} \,|\, \cdots \,|\, N'_{i, r} \end{equation*} \notag $$

the divisors in the decomposition of $G'_i$ given by the structure theorem. Then the quotient groups $G''_i$ are isomorphic to

$$ \begin{equation*} \bigoplus_{1 \leqslant j \leqslant r}\mathbb{Z}/\frac{N_i}{N'_{i,j}}\mathbb{Z}. \end{equation*} \notag $$

By the assumption, the asymptotic rank of the sequence $\{G'_i\}$ is at most $r'$. By Proposition 1,

$$ \begin{equation*} r - \max\{j \mid \text{the sequence }\{N_{i, j}\}\text{ is bounded as }i \to \infty\} \leqslant r'. \end{equation*} \notag $$

Similarly, since the asymptotic rank of $\{G''_i\}$ is $\leqslant r''$, we have, by Proposition 1,

$$ \begin{equation*} r - \min\biggl\{j \biggm| \text{the sequence }\biggl\{\frac{N_i}{N'_{i, j}}\biggr\} \text{ is bounded as }i \to \infty\biggr\} \leqslant r''. \end{equation*} \notag $$

However, since $N_i$ tend to infinity, the sequences $\{N'_{i,j}\}$ and $\{N_i/N'_{i,j}\}$, for a fixed $j \in \{1, \dots, r\}$, cannot be bounded simultaneously. Therefore, adding the above inequalities, we obtain

$$ \begin{equation*} r \leqslant r' + r'', \end{equation*} \notag $$

the result required. This proves the lemma.

2.3. The MRC fibration

In this section, we briefly recall the construction of the maximal rationally connected (MRC) fibration for a compact Kähler manifold $X$. In this generality, the existence of the MRC fibration was established in [8]. We refer to [8] for details, including the definition of the cycle space (Barlet space) $\mathcal{C}(X)$ for a compact complex space $X$. For a purely algebraic proof of this result in the case of projective algebraic varieties, see [7], Théorème 2.3, or [17].

Definition 3. A covering family of cycles on a complex space $X$ is a complex subspace $S \subset \mathcal{C}(X)$ such that

$\bullet$ $S$ is a countable union of compact irreducible complex subspaces;

$\bullet$ for $s \in S_i$ a general point the cycle $Z_s$ is irreducible and reduced;

$\bullet$ $X$ is a union of $\mathrm{Supp}(Z_s)$ for $s \in S$.

A covering family of cycles induces an equivalence relation $R(S)$ on points of $X$. Namely, two points $x, y \in X$ are equivalent if and only if they are contained in a connected union of a finite number of cycles parameterized by $S$. The following theorem (see Theorem 1.1 in [8] for the proof) shows the existence of meromorphic reduction maps for covering families of cycles. Recall that a fibration is a dominant meromorphic map of normal complex spaces with connected fibers. A typical fiber of a fibration is a fiber over a point in the complement to a proper analytic subset in the base.

Theorem 7. Let $X$ be a normal compact connected complex space. Let $S \subset \mathcal{C}(X)$ be a covering family of cycles on $X$. Denote by $R(S)$ the equivalence relation on $X$ induced by $S$. Then there exists a meromorphic fibration $q_S \colon X \dashrightarrow B_S$ such that a typical fiber of $q_S$ is an equivalence class for $R(S)$.

An important result, proved independently in [9] and [18] is the compactness of the irreducible components of $\mathcal{C}(X)$ in the Kähler case.

As a consequence of Theorem 8, for any compact Kähler manifold, one can define the following natural meromorphic fibration.

Obviously, if $X$ is not covered by rational curves then $f$ is birational. A crucial property of the MRC fibration is that its base $B$ is not covered by rational curves. This statement was shown in Corollary 1.4 of [11] for $X$ an algebraic variety. The same argument generalizes to the Kähler case (see, for instance, Remark 3.2 in [13] or Proposition 3.8 in [27]).

Moreover, the smooth fibers of the MRC fibration of a compact Kähler manifold $X$ are in fact projective, see [27], Theorem 3.9, for a proof.

2.4. Finite group actions

We will need a well-known result (see for example, Lemma 3 in [24] ) on existence of regularizations of birational actions of finite groups.

Proposition 3. Let $X$ be a normal projective variety and let $G \subset \operatorname{Bir}(X)$ be a finite group. Then there exists a smooth projective variety $\overline{X}$ with a regular action of $G$ and a $G$-equivariant birational map

$$ \begin{equation*} \varphi \colon \overline{X} \dashrightarrow X. \end{equation*} \notag $$

Proof. Replacing $X$ by an affine open subset, we may assume that the action of $G$ on $X$ is regular. Then, by Theorem 3 in [31], there exists a $G$-equivariant projective completion

$$ \begin{equation*} \varphi \colon \overline{X} \dashrightarrow X. \end{equation*} \notag $$

Replacing $\overline{X}$ by a $G$-equivariant resolution of singularities of $\overline{X}$ (see for example, [4]) we may assume $\overline{X}$ to be smooth.

The following proposition shows that actions of finite groups by automorphisms can be linearized in the fixed points. For the proof in the complex analytic setup we refer to [1], p. 38.

§ 3. Main results

3.1. Groups of pseudoautomorphisms

In this section, we extend Theorem 1 to automorphism groups of singular compact Kähler spaces. For a complex space $X$, we denote the subsets of its singular and non-singular points by $X_{\mathrm{sing}}$ and $X_{\mathrm{ns}}$, respectively.

Definition 5. Let $X$ be an irreducible and reduced complex space. A Kähler form on $X$ is a closed positive real $(1,1)$-form $\omega$ on $X_{\mathrm{ns}}$ satisfying the following condition: for any $x \in X_{\mathrm{sing}}$ there exists an open neighbourhood $x \in U \subset X$ with closed embedding $i_U \colon U \subset V$ into an open subset $V \subset \mathbb{C}^N$ such that

$$ \begin{equation*} \omega|_{U \cap X_{\mathrm{ns}}} = i\,\partial\,\overline\partial f|_{U \cap X_{\mathrm{ns}}} \end{equation*} \notag $$

for a smooth strictly plurisubharmonic function $f \colon V \to \mathbb{C}$. An irreducible and reduced complex space $X$ is Kähler if there exists a Kähler form on $X$.

Remark 5. If $X$ is a singular Kähler space, one can always find a resolution of singularities $\varphi \colon X' \to X$ where $X'$ is a compact Kähler manifold (see Remark 2.3 in [14]). The MRC fibration for $X$ can be defined as the MRC fibration of (any) compact Kähler manifold $X'$ bimeromorphic to $X$.

Lemma 4. Let $X$ be a normal compact Kähler space. Suppose that there exists a bimeromorphic morphism

$$ \begin{equation*} \varphi \colon T \to X, \end{equation*} \notag $$

where $T$ is a compact complex torus. Then $\varphi$ is an isomorphism.

Proof. Let $E$ be an irreducible component of the exceptional locus of $\varphi$ of dimension $c > 0$. Consider a Kähler class $\omega$ on $X$. By the projection formula,

$$ \begin{equation*} (\varphi^*\omega)^c \cdot E = \omega^c \cdot (\varphi_*E) = 0. \end{equation*} \notag $$

On the other hand, we can choose a general translation $\tau \colon T \to T$ such that the image of $\tau^*(E)$ under the map $\varphi$ is not contained in the singular locus of $X$. Therefore,

$$ \begin{equation*} (\varphi^*\omega)^c \cdot \tau^*E = \omega^c \cdot (\varphi_*\tau^*E) > 0. \end{equation*} \notag $$

However, since $\tau$ is an automorphism of $T$, we have $(\varphi^*\omega)^c \cdot E = (\varphi^*\omega)^c \cdot (\tau^*E)$. This contradiction shows that the exceptional locus of $\varphi$ is empty, so $\varphi$ is an isomorphism. This proves the lemma.

We also state another result by Mundet i Riera (see Theorem 1.10 in [22]). Theorem 1 is immediate from this result.

Theorem 10. Let $G$ be a Lie group with finitely many connected components. For every natural number $r$, the following properties are equivalent:

$\bullet$ the group $G$ contains subgroups of the form $(\mathbb{Z}/N\mathbb{Z})^r$ for arbitrarily large positive integers $N$;

$\bullet$ the group $G$ contains a subgroup isomorphic to a compact real torus $(S^1)^{r}$ of real dimension $r$.

We can deduce the following corollary from Theorem 10 by an argument similar to the proof of Theorem 1.9 in [22].

Corollary 2. Let $X$ be a (possibly singular) normal compact Kähler space. For every natural number $r$, the following properties are equivalent:

$\bullet$ the group $\operatorname{Aut}(X)$ contains finite abelian subgroups of the form $(\mathbb{Z}/N\mathbb{Z})^r$ for arbitrarily large positive integers $N$;

$\bullet$ the group $\operatorname{Aut}(X)$ contains a subgroup isomorphic to $(S^1)^r$.

In addition, $r \leqslant 2\dim(X)$, and if $r = 2\dim(X)$, then $X$ is biholomorphic to a compact complex torus.

Proof. The group of connected components $\operatorname{Aut}(X)/\operatorname{Aut}^0(X)$ has bounded finite subgroups (see Lemma 3.1 in [16]). So, by Lemma 1, we may assume that the finite abelian subgroups in question lie in $\operatorname{Aut}^0(X)$. By a well-known theorem of S. Bochner and H. Montgomery, the group $\operatorname{Aut}^0(X)$ is a connected complex Lie group acting holomorphically on $X$ (see for example, the theorem on p. 40 in [1] for a modern proof). Now the first statement of the corollary follows from Theorem 10.

Suppose now that there is an effective action of $(S^1)^r$ on $X$ by holomorphic automorphisms. Then by the results of [4], we can take $(S^1)^r$-equivariant resolution of singularities $\varphi \colon X' \to X$, where $X'$ is a compact Kähler manifold. Applying Theorem 1 to $X'$, we get the estimate

$$ \begin{equation*} r \leqslant 2\dim(X') = 2\dim(X). \end{equation*} \notag $$

Now, if $r = 2\dim(X')$, then $X'$ is biholomorphic to a compact complex torus. By Lemma 4, $\varphi$ is an isomorphism, and so $X$ is nonsingular and biholomorphic to a compact complex torus. This proves the corollary.

The next step is to extend the above result to groups of pseudoautomorphisms of singular compact Kähler spaces. Recall that a bimeromorphic map $f \colon X \dashrightarrow X$ is a pseudoautomorphism if both $f$ and $f^{-1}$ do not contract divisors. The group of pseudoautomorphisms of $X$ is denoted by $\operatorname{Psaut}(X)$.

For convenience of the reader, we reproduce here the following result (see Corollary 4.6 in [10]).

Proposition 5. Let $X$ be a normal compact Kähler space with rational singularities. Let $f \colon X \dashrightarrow X$ be a pseudoautomorphism. Suppose that there exists a Kähler class $\omega$ such that $f_*\omega$ is also a Kähler class. Then $f$ is a biholomorphic automorphism of $X$.

Now using this proposition we can easily generalize Corollary 2 to the group $\operatorname{Psaut}(X)$.

Theorem 11. Let $X$ be a normal compact Kähler space with rational singularities. Suppose that there exists $r \in \mathbb{N}$ such that the group $\operatorname{Psaut}(X)$ contains finite abelian subgroups isomorphic to $(\mathbb{Z}/N\mathbb{Z})^r$ for arbitrarily large $N$. Then $r \leqslant 2\dim(X)$ and $\operatorname{Psaut}(X)$ contains a subgroup isomorphic to a compact real torus $(S^1)^r$. In addition, if $r = 2\dim(X)$, then $X$ is biholomorphic to a compact complex torus.

Proof. As in the proof of Theorem 4.5 in [10], we consider the action of $\operatorname{Psaut}(X)$ on $H^{2}(X, \mathbb{Q})$ by pushforward. We have the exact sequence of groups

$$ \begin{equation*} 1 \to \operatorname{Psaut}(X)_{\tau} \to \operatorname{Psaut}(X) \to \operatorname{Psaut}(X)/\operatorname{Psaut}(X)_{\tau} \to 1, \end{equation*} \notag $$

where we set

$$ \begin{equation*} \operatorname{Psaut}(X)_{\tau} = \{f \in \operatorname{Psaut}(X) \mid f_*|_{H^{2}(X, \mathbb{Q})} = \mathrm{Id}\}. \end{equation*} \notag $$

Note that the quotient group $\operatorname{Psaut}(X)/\operatorname{Psaut}(X)_{\tau}$ embeds into $\mathrm{GL}(H^{2}(X, \mathbb{Q}))$, therefore, by Minkowski’s theorem (see, for example, Theorem 1 in [30]), the orders of finite subgroups of $\operatorname{Psaut}(X)/\operatorname{Psaut}(X)_{\tau}$ are bounded by a constant $M(X)$ depending on $h^{2}(X, \mathbb{Q})$ only. Hence the group $\operatorname{Psaut}(X)_{\tau}$ contains a sequence of finite abelian subgroups of asymptotic rank $r$; in addition, by Lemma 1, we may assume that these subgroups are of the form $(\mathbb{Z}/N_i\mathbb{Z})^r$, where $N_i$ tend to infinity. The group $\operatorname{Psaut}(X)_{\tau}$ acts trivially on $H^{2}(X, \mathbb{R}) = H^{2}(X, \mathbb{Q}) \otimes_{\mathbb{Q}} \mathbb{R}$, and, in particular, it preserves each Kähler class on $X$. Thus by Proposition 5, the group $\operatorname{Psaut}(X)_{\tau}$ is contained in $\operatorname{Aut}(X)$. The theorem now follows from Corollary 2.

3.2. Non-uniruled varieties and complex spaces

In this subsection we use Theorem 11 to derive a slightly more general version of Theorem 2 from the introduction.

To define minimal and quasi-minimal models of compact Kähler spaces, we need to introduce notions of nefness and modified nefness in the non-projective context (see [5] and [14] for more details).

Definition 6. Let $X$ be a normal compact Kähler space with rational singularities. We say that a class $\alpha \in H^{1,1}(X, \mathbb{R})$ is

$\bullet$ nef if it belongs to the closure of the cone of Kähler classes;

$\bullet$ modified nef if it belongs to the closure of the cone generated by classes of the form $\mu_*\omega$ where $\mu \colon Y \to X$ is an arbitrary bimeromorphic morphism from a smooth compact Kähler manifold $Y$ and $\omega$ is a Kähler class on $Y$.

Definition 7. A compact Kähler space (or a projective variety) $X$ with terminal $\mathbb{Q}$-factorial singularities is called

$\bullet$ minimal (or a minimal model) if the canonical class $K_X$ is nef;

$\bullet$ quasi-minimal (or a quasi-minimal model) if the canonical class $K_X$ is a modified nef.

Note that a minimal model is also quasi-minimal. Existence of quasi-minimal models for non-uniruled projective varieties was shown in Lemma 4.4 in [24].

In the case of non-uniruled compact Kähler spaces of dimension 3, minimal models exist by Theorem 1.1 in [14].

The reason to consider quasi-minimal models is the following description of their bimeromorphic (or birational) automorphisms. The case when $X$ is a projective variety was settled in Corollary 4.7 in [24]; for the general case of compact Kähler spaces see Proposition 4.2 in [10].

Now we can prove Theorem 5 for a non-uniruled projective variety $X$ over an algebraically closed field $k$ of zero characteristic. Without loss of generality we may assume that $k = \mathbb{C}$.

Theorem 13. Let $X$ be a non-uniruled projective variety over the field of complex numbers. Suppose that there exists $r \in \mathbb{N}$ such that the group $\operatorname{Bir}(X)$ contains finite abelian subgroups isomorphic to $(\mathbb{Z}/N\mathbb{Z})^r$ for arbitrarily large positive integers $N$. Then

$$ \begin{equation*} r \leqslant 2\dim(X), \end{equation*} \notag $$

and the group $\operatorname{Bir}(X)$ contains a subgroup isomorphic to an abelian variety of dimension $\lceil r/2\rceil$. In the case, $r = 2\dim(X)$ the variety $X$ is birational to an abelian variety.

Proof. Since $X$ is non-uniruled, by Proposition 6 there exists a quasi-minimal projective variety $X'$ birational to $X$. By Proposition 7, $\operatorname{Bir}(X) \simeq \operatorname{Bir}(X') = \operatorname{Psaut}(X')$. The upper bound $r \leqslant 2\dim(X)$ now follows from Theorem 11. Since $X$ is not covered by rational curves, the compact real torus $(S^1)^r$ in the connected component $\operatorname{Aut}^0(X)$ can only be contained in an abelian variety of complex dimension at least $\lceil r/2\rceil$. This proves the theorem.

An analogous result holds for a compact Kähler space $X$ under the assumption that a quasi-minimal model of $X$ exists. By Theorem 12, this condition holds if $\dim(X) \leqslant 3$.

Proposition 8. Let $X$ be a non-uniruled compact Kähler space admitting a quasi-minimal model. Let, for some $r \in \mathbb{N}$, the group $\operatorname{Bim}(X)$ contain finite abelian subgroups isomorphic to $(\mathbb{Z}/N\mathbb{Z})^r$ for arbitrarily large $N$. Then $r \leqslant 2\dim(X)$, and the group $\operatorname{Bim}(X)$ contains a subgroup isomorphic to a compact complex torus of dimension $\lceil r/2\rceil$. In addition, if $r = 2\dim(X)$, then $X$ is bimeromorphic to a compact complex torus.

3.3. Rationally connected varieties

We recall an important result on boundedness for finite groups acting on rationally connected algebraic varieties.

Proposition 9. Let $X$ be a rationally connected algebraic variety of dimension $n$ over an algebraically closed field $k$ of zero characteristic. Then there exists a constant $J(n)$, depending on $n$ only, such that, for any finite subgroup $G \subseteq \operatorname{Aut}(X)$ there exists a subgroup $H \subseteq G$ of index at most $J(n)$ acting on $X$ with a fixed point.

As a result, we have the following upper bound for ranks of finite abelian subgroups in the group $\operatorname{Bir}(X)$, where $X$ is a geometrically rationally connected algebraic variety over any field of zero characteristic.

Corollary 3. Let $X$ be a geometrically integral and geometrically rationally connected algebraic variety of dimension $n$ over an arbitrary field $k$ of zero characteristic. There exists a constant $M = M(n)$ such that, for any finite subgroup $G \subset \operatorname{Bir}(X)$, there exists an abelian subgroup $H \subset G$ of index at most $M(n)$ and such that the rank of $H$ does not exceed $n$.

Proof. We pass to an algebraic closure $\overline{k}$ of $k$ and replace $X$ by $X \times_{k}\overline{k}$. Let $\varphi \colon X' \dashrightarrow X$ be a smooth birational regularization of the action of $G$, which exists by Proposition 3. Note that $X'$ is rationally connected as well. Therefore by Proposition 9 there exists a constant $J'(n)$ such that $G$ contains a subgroup $H$ of index at most $J'(n)$ acting on $X'$ with a fixed point. By Proposition 4 the group $H$ embeds into $\mathrm{GL}_n(\overline{k})$. Therefore by Jordan’s theorem (see [15] or [29]), there exists a constant $J''(n)$ such that the group $H$ contains an abelian subgroup $A \subset H$ of index at most $J''(n)$. Then $A \subset G$ is a subgroup of index at most

$$ \begin{equation*} M(n) = J'(n)\cdot J''(n), \end{equation*} \notag $$

moreover, since $A$ is linear, it is generated by at most $n$ elements. This proves the corollary.

Let us now show that any sequence of finite abelian subgroups in $\operatorname{Bir}(X)$ has asymptotic rank at most $n$ (see Corollary 1).

Theorem 14. Let $X$ be a geometrically integral and geometrically rationally connected algebraic variety of dimension $n$ over an arbitrary field $k$ of zero characteristic. Suppose that there exists an unbounded sequence $\{N_i\}_{i \in \mathbb{N}}$ of positive integers such that the group $\operatorname{Bir}(X)$ contains a subgroup $G_i \simeq (\mathbb{Z}/N_i\mathbb{Z})^r$ for some fixed $r \in \mathbb{N}$. Then $r \leqslant n$.

Proof. By Proposition 3, there exists a constant $M(n)$ such that, for each $i \in \mathbb{N}$, there exists an abelian subgroup $H_i \subset G_i$ of index $\leqslant M(n)$. Hence the asymptotic rank of $\{G_i\}$ is equal to that of $\{H_i\}$, which is at most $n$, because all finite abelian groups $H_i$ are of rank $\leqslant n$. This proves the theorem.

3.4. The general case

Proof of Theorem 5. Passing to a resolution of singularities, we may assume that $X$ is smooth. If $X$ is not uniruled then the result follows from Theorem 13. Suppose that $X$ is uniruled and consider its MRC fibration $f \colon X \dashrightarrow B$, where $\dim(B) < \dim(X)$. Now, for every $i \in \mathbb{N}$, we have the exact sequence

$$ \begin{equation*} 1 \to G'_i \to G_i \to G''_i \to 1, \end{equation*} \notag $$

where the action of $G'_i$ is fiberwise with respect to $f$ (that is, every element $g \in G'_i$ maps a point in a fiber of $f$ where $g$ is defined to a point in the same fiber) and $G''_i$ acts faithfully on the base $B$. Let $X_{\eta}$ be the scheme-theoretic generic fiber of $f$. Then for every $i \in \mathbb{N}$ we have $G'_i \subset \operatorname{Bir}(X_{\eta})$. We denote $n' = \dim(X_{\eta})$. Then by Theorem 14 the asymptotic rank of the sequence $\{G'_i\}$ is at most $n'$. Since $B$ is not uniruled by Proposition 9, we apply Theorem 13 and obtain that the asymptotic rank of the sequence $\{G''_i\}$ is at most $2\dim(B)$. Therefore by Lemma 3, the asymptotic rank $r$ of the sequence $\{G_i\}$ is at most $n' + 2\dim(B)$. In particular,

$$ \begin{equation*} r \leqslant 2\dim(B) + n' = \dim(X) + \dim(B) < 2n, \end{equation*} \notag $$

the result required. This proves the theorem.

It is also possible to describe projective varieties such that $\operatorname{Bir}(X)$ contains a sequence of finite abelian groups of submaximal asymptotic rank.

Corollary 4. Let $X$ be a projective variety over an algebraically closed field of zero characteristic. Suppose that there exists an unbounded sequence $\{N_i\}_{i \in \mathbb{N}}$ of positive integers such that the group $\operatorname{Bir}(X)$ contains subgroups $G_i$ isomorphic to $(\mathbb{Z}/N_i\mathbb{Z})^r$ for $r = 2\dim(X) - 1$. Then $X$ is birational either to

$\bullet$ an abelian variety $A$;

$\bullet$ the product $\mathbb{P}^1 \times A$, where $A$ is an abelian variety of dimension $\dim(X)-1$.

Proof. For $\dim(X) = 1$ the result is obvious, since $X$ is then isomorphic to a rational or elliptic curve. Suppose from now on that $\dim(X) > 1$. If $X$ is not uniruled, then, by Theorem 13, there exists a birational model $X'$ of $X$ and a faithful action of an abelian variety of dimension

$$ \begin{equation*} \biggl\lceil \frac{2\dim(X') - 1}{2} \biggr\rceil = \dim(X') = \dim(X) \end{equation*} \notag $$

on $X'$, so that, by Theorem 5, $X$ is birational to an abelian variety.

Suppose now that $X$ is uniruled and consider the MRC fibration $f \colon X \dashrightarrow B$. Since $\dim(X) > 1$, we have $2\dim(X) - 1 > \dim(X)$ and, therefore, by Theorem 14, $X$ cannot be rationally connected, that is, $\dim(B) > 0$. Let $X_{\eta}$ be the general fiber of $f$. Let also $\{G'_i\}$ be the sequence of subgroups acting fiberwise with respect to $f$, and denote by $\{G''_i\}$ the sequence of quotient groups. By Theorem 14, the asymptotic rank of $\{G'_i\}$ does not exceed $\dim(X_{\eta}) \leqslant \dim(X)$. Therefore, the asymptotic rank of $\{G_i''\}$ is at least

$$ \begin{equation*} 2\dim(X) - 1 - \dim(X_{\eta}) \geqslant 2\dim(B) > 0. \end{equation*} \notag $$

Now by Theorem 13, the non-uniruled variety $B$ is birational to an abelian variety $A$; moreover, $A$ has maximal possible dimension, equal to $\dim(X) - 1$. Since the asymptotic rank of $\{G_i'\}$ is equal to 1, it follows by Theorem 4.14 in [6] that $X_\eta \simeq \mathbb{P}^1_{k(B)}$, and so $X$ is birational to a product $\mathbb{P}^1\times A$. This proves the corollary.

Before proceeding with the proof of Theorem 6, we need the following technical lemma (see Lemma 3.1 in [28] for the proof). Recall that a very typical fiber of a dominant meromorphic map $\alpha \colon X \dashrightarrow Y$ is a fiber $X_t = \alpha^{-1}(t)$ over a point $t \in Y$ in the complement to at most countable union of proper analytic subspaces of $Y$. By $\operatorname{Bim}(X)_{\alpha}$, we denote the subgroup of elements of $\operatorname{Bim}(X)$ acting fiberwise with respect to $\alpha$.

Lemma 5. Let $\alpha \colon X \dashrightarrow Y$ be a dominant meromorphic map of compact complex manifolds. Then there exist a constant $I = I(\alpha)$ with the following property. Let $\{G_i\}_{i \in \mathbb{N}}$ be a sequence of finite subgroups of $\operatorname{Bim}(X)_{\alpha}$. Then there exists a reduced fiber $F$ of $\alpha$ and its irreducible component $F'$ of dimension $\dim(X) - \dim(Y)$ such that, for every $i \in \mathbb{N}$ the group $G_i$ contains a subgroup of index at most $I$, which is isomorphic to a subgroup of $\operatorname{Bim}(F')$. Moreover, if $\dim(Y) > 0$ the fiber $F$ can be chosen to be very typical.

We can proceed as in the case of compact Kähler spaces with the help of Lemma 5 when dealing with the MRC fibration of $X$.

Proof of Theorem 6. Passing to a resolution of singularities, we may assume that $X$ is smooth. If $X$ is not uniruled then the result follows from Proposition 8.

Suppose that $X$ is uniruled. Then we consider the MRC fibration $f \colon X \dashrightarrow B$ with $B$ non-uniruled and $\dim(B) < \dim(X)$. If $\dim(B) = 0$, then $X$ is rationally connected and hence projective by Proposition 2; this case follows from Theorem 14. Assume from now on that $\dim(B) > 0$. Then, for every $i \in \mathbb{N}$, there exists an exact sequence of groups

$$ \begin{equation*} 1 \to G'_i \to G_i \to G''_i \to 1, \end{equation*} \notag $$

where the action of $G'_i$ is fiberwise with respect to $f$ and $G''_i$ acts faithfully on $B$. Since the set of finite groups $\{G_i\}_{i \in \mathbb{N}}$ is countable, by Lemma 5, we may assume that

$$ \begin{equation*} G'_i \subset \operatorname{Bim}(X_t), \end{equation*} \notag $$

where $X_t$ is a very typical (in particular, smooth) fiber of $f$. Note that by Proposition 2 smooth fibers of $f$ are projective. Now by Theorem 14, the asymptotic rank of $\{G'_i\}$ is at most $\dim(X_t)$. Moreover, by the assumptions on $B$ and by Theorem 13 the asymptotic rank of the sequence $\{G''_i\}$ is at most $2\dim(B)$. By Lemma 3, the asymptotic rank $r$ of the sequence $\{G_i\}$ is at most $2\dim(B) + \dim(X_t)$; in particular,

$$ \begin{equation*} r \leqslant \dim(X_t) + 2\dim(B) < 2\dim(X), \end{equation*} \notag $$

as desired. This proves the theorem.

Remark 6. To prove Theorem 6 in full generality, it suffices to prove that “large” finite abelian subgroups of $\operatorname{Bim}(X)$ can be pseudo-regularized on a compact Kähler manifold $X'$ bimeromorphic to $X$. By considering the algebraic reduction (see Definition 3.3 in [32]) of a compact Kähler space $X$, it suffices to resolve the above problem in the case when $X$ has algebraic dimension 0. In particular, if $X$ has no divisors (like a general compact complex torus) this statement is clear, since $\operatorname{Bim}(X) = \operatorname{Psaut}(X)$ in this case.

Citation in format AMSBIB

\Bibitem{Gol24}

\by A.~S.~Golota

\paper Finite abelian subgroups in the groups of birational and bimeromorphic selfmaps

\jour Izv. Math.

\yr 2024

\vol 88

\issue 5

\pages 856--872

\mathnet{http://mi.mathnet.ru//eng/im9568}

\crossref{https://doi.org/10.4213/im9568e}

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