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This article is cited in 3 scientific papers (total in 3 papers) Jordan property for groups of bimeromorphic automorphisms of compact Kähler threefolds A. S. Golotaab a Laboratory of Algebraic Geometry and Its Applications, National Research University Higher School of Economics, Moscow, Russia b Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
Russian version:
Matematicheskii Sbornik, 2023, Volume 214, Number 1, DOI: https://doi.org/10.4213/sm9743 
Bibliographic databases:
    § 1. IntroductionA fruitful way to study various groups of geometric nature is to explore their finite subgroups. For example, we can consider groups of automorphisms of algebraic (for instance, projective) varieties. More challenging are groups of birational automorphisms, the case of Cremona group $\mathrm{Cr}_n(\mathbb{C}) = \mathrm{Bir}(\mathbb{P}^n_{\mathbb{C}})$ being particularly famous (see, for example, [1] and the references there for more details). A well-known theorem of Jordan proved in [2] describes finite subgroups of the general linear group. It says that there exists a function $J \colon \mathbb{N} \to \mathbb{N}$ such that all finite subgroups of $\mathrm{GL}_n(\mathbb{C})$ can be obtained as extensions of groups of order at most $J(n)$ by finite abelian groups. Popov [3] proposed the name of Jordan groups for groups with the above property. In the case when $X$ is a projective variety of any dimension over a field $\Bbbk$ with $\mathrm{char}(\Bbbk) = 0$ the Jordan property for $\operatorname{Aut}(X)$ was established by Meng and Zhang [4]. This result was extended to automorphism groups of compact Kähler spaces by Kim [5] and to the automorphism groups of compact complex spaces in the Fujiki class $\mathcal{C}$ by Meng, Perroni and Zhang [6]. Also, Popov [7] proved that real Lie groups have the Jordan property; in particular, for any compact complex space $X$ the neutral component of $\operatorname{Aut}(X)$ is Jordan. As concerns groups of birational automorphisms, the Jordan property of $\mathrm{Cr}_2(\mathbb{C}) = \mathrm{Bir}(\mathbb{P}^2_{\mathbb{C}})$ was established by Serre in [8]. For complex projective surfaces Popov [3] verified the Jordan property in all cases except $\mathbb{P}^1 \times E$ where $E$ is an elliptic curve. Subsequently, Zarhin [9] showed that $\mathrm{Bir}(\mathbb{P}^1 \times E)$ is not Jordan and provided similar examples in higher dimensions. Important results on the Jordan property of $\mathrm{Bir}(X)$ for $X$ projective over a field of characteristic 0 were obtained by Prokhorov and Shramov. They proved that the Jordan property holds for $\mathrm{Bir}(X)$ for nonuniruled [10] or rationally connected $X$ [11]. For compact complex manifolds it is natural to consider the group $\operatorname{Bim}(X)$ of bimeromorphic maps from $X$ to itself. Prokhorov and Shramov [12] settled the case of compact complex surfaces. They proved that the automorphism groups of all compact complex surfaces are Jordan and that the bimeromorphic automorphism groups of all compact complex surfaces except for those bimeromorphic to $\mathbb{P}^1 \times E$ are Jordan too. In the series of papers [13]–[15] Prokhorov and Shramov investigated groups of bimeromorphic selfmaps of compact Kähler spaces of dimension 3. In the uniruled case they proved that $\operatorname{Bim}(X)$ is Jordan unless $X$ is bimeromorphic to a space in one of finitely many explicitly described families (see [14], Theorems 1.3 and 1.4). For groups of bimeromorphic automorphisms of nonuniruled Kähler spaces they were able to show the Jordan property under an additional assumption. Recall that the Kodaira dimension $\kappa(X)$ of a compact Kähler space $X$ is defined to be the Kodaira dimension of any smooth manifold $X'$ bimeromorphic to $X$. Analogously, the irregularity of $X$ is defined by $q(X) = H^1(X', \mathscr{O}_{X'})$ for any smooth manifold $X'$ bimeromorphic to $X$. Theorem 1.1 (see [15], Theorem 1.3). Let $X$ be a compact Kähler space of dimension 3. Assume that $\kappa(X) \geqslant 0$ and $q(X) > 0$. Then the group $\operatorname{Bim}(X)$ is Jordan. The purpose of this note is to prove the following theorem. Theorem 1.2. The group $\operatorname{Bim}(X)$ is Jordan for any nonuniruled compact Kähler space $X$ of dimension 3. To achieve this result we generalize the arguments presented in Corollary 3.3 in [16] and Proposition 4.5 in [15] to singular Kähler spaces (see Theorems 4.5 and 5.1, respectively). Together with the existence of minimal models for $\mathbb{Q}$-factorial terminal Kähler spaces of dimension 3 established in [17], this gives us the required conclusion of Theorem 1.2. In higher dimensions we show that the same result holds under the assumption that a quasi-minimal model of $X$ exists (Theorem 5.2). AcknowledgementsThe author thanks Constantin Shramov for suggesting this problem to him and Yuri Prokhorov for valuable discussions. § 2. Generalities on the Jordan propertyThe following definition was proposed by Popov [3]. Definition 2.1. Let $G$ be a group. We say that $G$ is Jordan (or has the Jordan property) if there is a constant $J(G) \in \mathbb{N}$ such that for any finite subgroup $H \subset G$ there is a normal abelian subgroup $A \subset H$ of index at most $J(G)$. A classical result of Jordan says that this property holds for $G = \mathrm{GL}_n(\mathbb{C})$ (for a proof, see, for example, Theorem 36.13 in [18]). Clearly, a subgroup of a Jordan group is again Jordan; therefore, all linear algebraic groups over $\mathbb{C}$ are Jordan. On the other hand, quotient groups and extensions of Jordan groups are not necessarily Jordan. To overcome this difficulty we also consider the more restrictive class of groups with bounded finite subgroups. Recall that a group $G$ has bounded finite subgroups if there is a number $B(G)$ such that for each finite subgroup $H \subset G$ we have $|H| \leqslant B(G)$ (see [3], Definition 2.7). An easy but important result below (see [3], Lemma 2.9) says that an extension of a group with bounded finite subgroups by a Jordan group remains Jordan. Proposition 2.2. Consider an exact sequence of groups Assume that $G_1$ is Jordan and $G_3$ has bounded finite subgroups. Then $G_2$ is Jordan. Examples of groups with bounded finite subgroups are given by a classical theorem of Minkowski (see Theorem 1 in [8], for example). Theorem 2.3. The orders of finite subgroups of $G = \mathrm{GL}_n(\mathbb{Q})$ are bounded by a natural number $M(n)$ depending on $n$ only. We need the following result due to Kim (see [5], Theorem 1.1); for the notion of a singular Kähler space, see Definition 3.1 below. Theorem 2.4. Let $X$ be a normal compact Kähler space. Then the group $\operatorname{Aut}(X)$ of biholomorphic automorphisms of $X$ is Jordan. § 3. Singular compact Kähler spaces and their minimal modelsWe recall basic definitions related to singular compact Kähler spaces, mostly following [17]. Definition 3.1. Let $X$ be an irreducible reduced complex space; denote the subsets of its singular and nonsingular points by $X_{\mathrm{sing}}$ and $X_{\mathrm{ns}}$, respectively. A Kähler form on $X$ is a closed positive real $(1,1)$-form $\omega$ on $X_{\mathrm{ns}}$ satisfying the following condition of existence of local potentials: for any $x \in X_{\mathrm{sing}}$ there exists an open neighbourhood $x \in U \subset X$ with a closed embedding $i_U \colon U \subset V$ into an open subset $V \subset \mathbb{C}^N$ such that for a smooth strictly plurisubharmonic function $f \colon V \to \mathbb{C}$. A Kähler form $\omega$ defines a $\partial\overline\partial$-cohomology class $[\omega] \in H^{1,1}_{\mathrm{BC}}(X, \mathbb{R})$. An irreducible reduced complex space $X$ is Kähler if there exists a Kähler form on $X$. Remark 3.2. We restrict ourselves to singular Kähler spaces that are normal and have rational singularities. The latter condition ensures that the space $N^1(X)=H^{1,1}_{\mathrm{BC}}(X, \mathbb{R})$ defined via the $\partial\overline\partial$-cohomology embeds into $H^2(X, \mathbb{R})$ (see [17], Remark 3.7). The definition of terminal singularities for singular Kähler spaces is identical to the projective case. Every terminal singularity is rational by [19] (also see [20], Theorem 5.22). To define minimal and quasiminimal models of compact Kähler spaces we need to introduce the notions of nefness and modified nefness (‘nefness in codimension 1’) in the nonprojective context (see [21] and [17] for more details). Definition 3.3. 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 nef if it belongs to the closure of the cone of Kähler classes. Also, a class $\alpha$ is called semipositive if there exists a smooth semipositive form representing $\alpha$. Definition 3.4. 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 modified nef if it belongs to the closure of the cone generated by the classes of the forms $\mu_*\omega$ where $\mu \colon Y \to X$ is an arbitrary bimeromorphic morphism from a smooth Kähler manifold $Y$ and $\omega$ is a Kähler class on $Y$. Recall that a class $\alpha \in H^{1,1}(X, \mathbb{R})$ is pseudoeffective if it can be represented by a closed positive $(1,1)$-current with local potentials. For any pseudoeffective class on a smooth manifold $X$ Boucksom defined a so-called divisorial Zariski decomposition (see [21], Definition 3.7). Proposition 3.5. Let $X$ be a compact Kähler manifold, and let $\alpha \in H^{1,1}(X, \mathbb{R})$ be a pseudoeffective class. Then there exists a divisorial Zariski decomposition such that Remark 3.6. For a normal compact Kähler space $X$ with rational singularities we can still define the divisorial Zariski decomposition of a pseudoeffective class $\alpha \in H^{1,1}(X, \mathbb{R})$. To do this we take a resolution of singularities $\mu \colon Y \to X$ and apply Proposition 3.5 to $\mu^*\alpha$, to obtain a decomposition Then we can define $P(\alpha) = \mu_*P(\mu^*\alpha)$ and $N(\alpha) = \mu_*N(\mu^*\alpha)$. Since $X$ is nonsingular in codimension $1$, the decomposition $\alpha = P(\alpha) + N(\alpha)$ satisfies the conditions in Proposition 3.5. Moreover, it does not depend on the choice of $\mu \colon Y \to X$ since any two resolutions of $X$ can be dominated by a common resolution. We define minimal models of singular compact Kähler spaces following the conventions in [17]. Note that the definition of $\mathbb{Q}$-factorial singularities in [17] is as follows: every Weil divisor is $\mathbb{Q}$-Cartier and some reflexive power of the dualizing sheaf $K_X$ is a line bundle. Definition 3.7. A compact Kähler space $X$ with terminal $\mathbb{Q}$-factorial singularities is called Let $X$ be a compact Kähler space of dimension $3$ with terminal $\mathbb{Q}$-factorial singularities. Assume that $X$ is not uniruled; then the canonical class $K_X$ is pseudoeffective. This follows from Theorem 2.6 in [22] for $X$ that is projective in any dimension and from [23] for $X$ that is not projective in dimension $3$. In [17], Theorem 1.1, Höring and Peternell proved the existence of minimal models for compact Kähler spaces of dimension $3$ with terminal $\mathbb{Q}$-factorial singularities and pseudoeffective $K_X$. Thus, any nonuniruled compact Kähler space of dimension $3$ has a minimal model. Theorem 3.8. Let $X$ be a terminal $\mathbb{Q}$-factorial compact Kähler space of dimension $3$. Assume that $X$ is not uniruled. Then there exists a bimeromorphic map $\varphi \colon X \dashrightarrow X'$ such that $X'$ is a minimal model. Note that any minimal model is also a quasi-minimal model. When $X$ is projective, our definition of quasi-minimal models is equivalent to Definition 4.2 in [10]. The existence of quasi-minimal models for projective non-uniruled varieties with terminal $\mathbb{Q}$-factorial singularities was shown in [10], Lemma 4.4. § 4. Bimeromorphic maps of quasi-minimal modelsIn this section we consider bimeromorphic maps of normal compact Kähler spaces with rational singularities and discuss sufficient conditions for such maps to be holomorphic. Recall that for any bimeromorphic map $f \colon X \dashrightarrow X'$ of reduced complex spaces there exists a resolution of indeterminacies Here $Y$ is a smooth complex manifold and $p$ and $q$ are bimeromorphic morphisms.The following fact is well known to experts; we include a proof for the reader’s convenience. Lemma 4.1. Let $f \colon X \dashrightarrow X'$ be a bimeromorphic map of reduced complex spaces such that $X'$ is normal. Assume that there exists a resolution (4.1) such that every fibre of $q$ is mapped to a point by $p$. Then the inverse map $f^{-1}$ is holomorphic. Proof. Suppose that there is a point $x$ in the indeterminacy set $\mathrm{Ind}(f^{-1})$. By assumption we can extend the map $f^{-1}$ to a continuous (but a priori not holomorphic) map $X' \to X$. There is an open subset $U \subset X$ containing $f^{-1}(x)$ such that $U$ embeds into $\mathbb{C}^N$ as an open subset. In other words, in some neighbourhood $V \subset X'$ the map $f^{-1}$ is given by an $N$-tuple of functions which are holomorphic outside a subset $V \cap \mathrm{Ind}(f^{-1})$ of codimension at least 2 in $V$. Since $X'$ (and thus $V$) is normal, by the extension theorem (see [24], p. 144) $f^{-1}$ extends holomorphically to $x$.
The lemma is proved. The following proposition is due to Hanamura (see [25], Lemma 3.4) and Kollár (see [26], Lemma 4.3) in the case of minimal algebraic varieties. We prove a slightly more general version using divisorial Zariski decompositions. Proposition 4.2. Let $f \colon X \dashrightarrow X'$ be a bimeromorphic map between compact Kähler spaces with terminal singularities. Assume that $K_X$ and $K_{X'}$ are modified nef. Then $f$ is an isomorphism in codimension $1$. Proof. We consider a resolution of indeterminacies (4.1). We denote by $E_i \subset Y$ (respectively, $F_j \subset Y$) irreducible $p$-exceptional (respectively, $q$-exceptional) divisors. We have
$$
\begin{equation*}
p^*K_{X} + \sum_ia_iE_i=K_Y=q^*K_{X'} + \sum_jb_jF_j,
\end{equation*}
\notag
$$
where all coefficients $a_i$ and $b_j$ are strictly positive because $X$ and $X'$ are terminal. In particular, $K_Y$ is pseudoeffective, and therefore by Proposition 3.5 it has a divisorial Zariski decomposition: Since $p_*K_Y = K_X$ and $q_*K_Y = K_{X'}$ are modified nef, any irreducible component of the negative part $N(K_Y)$ is both $p$- and $q$-exceptional. On the other hand the exceptional divisors $E = \sum_ia_iE_i$ and $F = \sum_jb_jF_j$ are also exceptional in the sense of Proposition 3.5, and thus for all $i$ and $j$ $E_i$ and $F_j$ are components of the negative part $N(K_Y)$. Therefore, the map $f$ establishes an isomorphism between the open subsets $X\setminus p\bigl(\bigcup_{i,j}E_i \cup F_j\bigr)$ and $X'\setminus q\bigl(\bigcup_{i,j}E_i \cup F_j\bigr)$ whose complements have codimension at least 2.
The proposition is proved. Let $f \colon X \dashrightarrow X'$ be a bimeromorphic map, and let $\alpha \in H^{1,1}(X, \mathbb{R})$ be a class on $X$. To define the pushforward $f_*\alpha \in H^{1,1}(X', \mathbb{R})$ we take a resolution (4.1) and set $f_*\alpha = q_*p^*\alpha$. A standard argument shows that the class $f_*\alpha$ does not depend on the choice of the resolution. If $\alpha$ is pseudoeffective, then $f_*\alpha$ is pseudoeffective too. Recall that the singular locus of a closed positive $(1,1)$-current $T$ is a subset defined locally as the set of points where a local plurisubharmonic potential of $T$ is not bounded below. Definition 4.3. Let $\alpha \in H^{1,1}(X, \mathbb{R})$ be a pseudoeffective class. The singular locus $S(\alpha)$ is defined to be the intersection of the singular loci of all closed positive $(1,1)$-currents $T \in \alpha$. If the class $\alpha$ is semipositive (if $\alpha$ is a Kähler class, for example) then, clearly, $S(\alpha) = \varnothing$. We need the following lemma, which was proved in [16], Lemma 2.4, for smooth manifolds $Y$ and $X$. However, the proof also works when $X$ is singular; we reproduce it here to be self-contained. Recall that a closed subset $S \subset X$ is thin in $X$ if for every point $x \in X$ there exists an open neighbourhood $U$ of $x$ such that $S \cap U$ is contained in a nowhere dense analytic subset of $U$. Lemma 4.4. Let $f \colon Y \to X$ be a bimeromorphic morphism from a smooth Kähler manifold $Y$ to a normal compact Kähler space $X$ with exceptional set $E$. Let $\alpha \in H^{1,1}(Y, \mathbb{R})$ be a pseudoeffective class. If $E \cap S(\alpha)$ is a thin subset of $E$, then where the $E_i$ are irreducible components of $E$ and the $r_i$ are some nonnegative real numbers. Proof. We choose a closed positive current $T$ representing the class $\alpha$. The problem is local on $X$, so we can replace $X$ by an open neighbourhood $U \subset \mathbb{C}^N$ such that $f_*T = i\partial\overline\partial\varphi$ for a plurisubharmonic function $\varphi$ on $U$. Then the class $f^*f_*\alpha$ is represented by a closed positive current $i\partial\overline\partial (\varphi \circ f)$. Let $y \in Y$ be an arbitrary point. We can choose an open neighbourhood $V$ of $y$ such that $T = i\partial\overline\partial \psi$ on $V$ for a plurisubharmonic function $\psi$. Then the function $\varphi \circ f - \psi$ is pluriharmonic on $V \setminus (V \cap E)$. Since $Y$ is smooth and $S(\alpha) \cap E$ is thin in $E$ by assumption, we can extend the function $\varphi \circ f - \psi$ to a plurisubharmonic function on $V$ (see [16], Lemma 2.6). We use Proposition 3.1.3 in [27] (also see [16], pp. 744-745) to obtain the following equality of currents on $V$:
$$
\begin{equation}
i\partial\overline\partial(\varphi \circ f) - i\partial\overline\partial\psi=\sum_ir_i(V \cap E_i),
\end{equation}
\tag{4.2}
$$
where the $r_i$ are nonnegative real numbers. We cover $Y$ by open neighbourhoods in which equalities of the form (4.2) hold. Since the image of a pluriharmonic function under a holomorphic change of coordinates is pluriharmonic, we obtain the required equality of $(1,1)$-classes.
The lemma is proved. The next step of our proof is a generalization of a result for smooth manifolds due to Fujiki ([16], Corollary 3.3). Theorem 4.5. Let $f \colon X \dashrightarrow X'$ be a bimeromorphic map of normal compact Kähler spaces with rational singularities. Assume that $f$ does not contract divisors. If there exists a Kähler class $\alpha \in H^{1,1}(X, \mathbb{R})$ such that the class $\alpha' = f_*\alpha$ is semipositive, then the inverse map $f^{-1}$ is holomorphic. Proof. Again, we consider a resolution of indeterminacies (4.1). We may assume that $q$ is projective. Moreover, we may assume that $\mathrm{Exc}(p) = \sum_iE_i$ and $\mathrm{Exc}(q) = \sum_jF_j$ are divisors. By the definition of a pushforward we have $\alpha' = f_*\alpha = q_*p^*\alpha$. Consider the class
$$
\begin{equation*}
q^*q_*p^*\alpha - p^*\alpha \in H^{1,1}(Y, \mathbb{R}).
\end{equation*}
\notag
$$
Since the class $p^*\alpha$ is represented by a smooth semipositive form, we have $S(\alpha) = \varnothing$ so we can use Lemma 4.4 and obtain
$$
\begin{equation}
q^*q_*p^*\alpha - p^*\alpha=q^*\alpha' - p^*\alpha=\sum_jr_j[F_j]
\end{equation}
\tag{4.3}
$$
for some $r_j \geqslant 0$. Since $\alpha'$ is semipositive by assumption, the same is true for the class $q^*\alpha'$, and therefore $S(q^*\alpha') = \varnothing$. Note that $f$ does not contract divisors by assumption, so every $q$-exceptional divisor is also $p$-exceptional. Therefore, from (4.3) we obtain
$$
\begin{equation*}
p_*q^*\alpha'=p_*\biggl(p^*\alpha + \sum_jr_j[F_j]\biggr)=\alpha + \sum_jr_jp_*[F_j]=\alpha.
\end{equation*}
\notag
$$
We apply Lemma 4.4 to the map $p$ and obtain the equality
$$
\begin{equation}
p^*p_*q^*\alpha' - q^*\alpha'=p^*\alpha - q^*\alpha'=\sum_is_i[E_i]
\end{equation}
\tag{4.4}
$$
for some $s_i \geqslant 0$. Adding the expressions (4.3) and (4.4) we finally obtain
If $q$ is an isomorphism, then $f^{-1}$ is clearly holomorphic. Suppose that $q$ is not an isomorphism. As $q$ is projective, every fibre is projective and therefore covered by curves. If every irreducible curve in every fibre of $q$ is contracted to a point by $p$, then $f^{-1}$ is holomorphic by Lemma 4.1. Thus we may assume that there exists an irreducible curve $\widetilde{C} \subset Y$ contracted to a point by $q$ and such that $p(\widetilde{C})$ is a curve. We set $p_*\widetilde{C} = mC$ for some $m > 0$. By the projection formula and equality (4.5) we have
$$
\begin{equation*}
0 < \alpha \cdot mC=\alpha \cdot p_*\widetilde{C}=p^*\alpha \cdot \widetilde{C}= q^*\alpha'\cdot \widetilde{C}=\alpha' \cdot q_*\widetilde{C}=0,
\end{equation*}
\notag
$$
which is a contradiction. Thus, the set $\mathrm{Ind}(f^{-1})$ is empty and $f^{-1}$ is holomorphic.
The theorem is proved. Theorem 4.5 gives a sufficient condition for an isomorphism in codimension $1$ between compact singular Kähler spaces to be biholomorphic. Corollary 4.6. Let $f \colon X \dashrightarrow X'$ be a bimeromorphic map of normal compact Kähler spaces with rational singularities. Suppose that $f$ is an isomorphism in codimension $1$. If there exists a Kähler class $\alpha \in H^{1,1}(X, \mathbb{R})$ such that $\alpha' = f_*\alpha$ is also Kähler, then $f$ is biholomorphic. Remark 4.7. Jia and Meng proved that the conclusion of Theorem 4.5 holds if the class $f_*\alpha$ is merely assumed to be nef (see [28], Proposition 2.1). In this case our proof of (4.5) does not work since we cannot guarantee that $S(q^*\alpha') \cap \mathrm{Exc}(p)$ is thin in $\mathrm{Exc}(p)$. Jia and Meng overcame this difficulty by using the negativity lemma (see [28], Claim 2.2). § 5. The main resultsNow we can state and prove our main results. The first result generalizes Proposition 4.5 in [15] to the singular case. We denote by $\operatorname{PsAut}(X)$ the group of pseudoautomorphisms of $X$, that is, bimeromorphic maps from $X$ to itself which are isomorphisms in codimension $1$. This group acts on $H^2(X, \mathbb{Q})$ by pushforward $f \to f_* = (f^{-1})^*$; we also set
$$
\begin{equation*}
\operatorname{PsAut}(X)_{\tau}=\bigl\{f \in \operatorname{PsAut}(X) \mid f_*|_{H^{2}(X, \mathbb{Q})}= \mathrm{id} \bigr\}.
\end{equation*}
\notag
$$
Theorem 5.1. Let $X$ be a normal compact Kähler space with rational singularities. Then the group $\operatorname{PsAut}(X)$ is Jordan. Proof. The action of $\mathrm{PsAut}(X)$ on $H^{2}(X, \mathbb{Q})$ gives an exact sequence of groups
Any element $f \in \mathrm{PsAut}(X)_{\tau}$ acts trivially on the space $H^{2}(X, \mathbb{R}) = {H^{2}(X, \mathbb{Q}) \otimes_{\mathbb{Q}} \mathbb{R}}$; in particular, it preserves every Kähler class. By Corollary 4.6 the group $\operatorname{PsAut}(X)_{\tau}$ is a subgroup of $\operatorname{Aut}(X)$. Therefore, it is Jordan by Theorem 2.4. Note that the quotient group $\operatorname{PsAut}(X)/\operatorname{PsAut}(X)_{\tau}$ embeds into the linear group $\mathrm{GL}(H^{2}(X, \mathbb{Q}))$, and therefore it has bounded finite subgroups by Theorem 2.3. Hence the group $\operatorname{PsAut}(X)$ is also Jordan by Proposition 2.2. The theorem is proved. Assuming that $X$ admits a quasi-minimal model, we can deduce the same conclusion for $\operatorname{Bim}(X)$. In particular, this gives us another proof of Theorem 1.8, (ii), in [10]. Theorem 5.2. Let $X$ be a compact Kähler space admitting a quasi-minimal model. Then the group $\mathrm{Bim}(X)$ is Jordan. Proof. By assumption there exists a bimeromorphic map $\varphi \colon X \dashrightarrow X'$ such that $X'$ is a compact Kähler space with terminal $\mathbb{Q}$-factorial singularities and $K_{X'}$ is modified nef. By Proposition 4.2 we have
$$
\begin{equation*}
\operatorname{Bim}(X) \simeq \mathrm{Bim}(X')= \mathrm{PsAut}(X'),
\end{equation*}
\notag
$$
and the latter group is Jordan by Theorem 5.1.
The theorem is proved. In particular, we can prove Theorem 1.2. Proof of Theorem 1.2. We replace $X$ by a resolution of singularities of $X$; then the statement follows immediately from Theorem 5.2 and Theorem 3.8. Remark 5.3. We can also prove Theorem 1.2 using the fact that any bimeromorphic map between minimal models in dimension $3$ can be decomposed as a sequence of analytic flops (see [26], Theorem 4.9). Remark 5.4. Conjecturally, any compact Kähler space $X$ that is not uniruled should admit a minimal (or at least quasi-minimal) model. By Theorem 5.2 this would mean the Jordan property of $\operatorname{Bim}(X)$ in higher dimensions. In [29] Cao and Höring made some progress in this direction.
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