Jordan groups and geometric properties of manifolds

We will use the standard notation \(\BN , \ \BZ , \ \BQ , \ \BC \) for the set of positive integers, the ring of integers, the fields of rational and complex numbers, respectively. If \(q\) is a prime (or a prime power) then we write \(\BF _q\) for the (finite) \(q\)-element field. In this note we consider the following groups.

• \(\Bir (X)\) of all birational self-maps of an irreducible complex algebraic variety \(X;\)
• \(\Bim (X)\) of all bimeromorphic self-maps of a connected complex manifold \(X;\)
• \(\Diff (X)\) of all diffeormorphisms of a smooth real manifold \(X;\)
• \(\Aut _{an}(X)\) and \(\Aut (X)\) of all automorphisms of complex or algebraic variety, respectively.

Sometimes these groups are finite; for example, \(\Bim (X)\) is finite if \(X\) is a compact connected complex manifold of general type (i.e., it has maximal possible Kodaira dimension \(\varkappa (X)= \dim X\) ( [KO] ). However, in general, the groups \(\Bim (X)\) may be infinite and non-algebraic. One of the most interesting and important examples of such groups in birational geometry is the Cremona group \(\Cr _n=\Bir (\BP ^n)\) where \(\BP ^n\) is the \(n-\)dimensional complex projective space. If \(n\ge 2\) then \(\Cr _n\) is a huge non-abelian non-algebraic group. To understand the structure of such groups one is tempted to consider their less complicated subgroups: finite, abelian or their combinations. This is where the Jordan properties come in.

The study of Jordan properties was inspired by the following fundamental results of Jordan and Serre (see [Jor] , [Se16] Theorem 9.9, and [Se09] Theorem 5.3, respectively).

(Later the exact value \(J_{\Cr _2}=7200\) was found by E. Yasinsky [Ya] .)

In his paper [Po11] V.L. Popov asked whether for any algebraic variety \(X\) the groups \(\Aut (X)\) and \(\Bir (X)\) are Jordan. This question stimulated an intensive and fruitful activity, see Section 2 below.

The following “Jordan properties” of groups are also very useful.

In this section we sketch certain facts, methods and tools related to the study of the Jordan properties of groups arising from complex geometry.

As of today (June 2024), there are no examples of complex algebraic varieties (compact or non-compact) with non-Jordan \(\Aut (X)\). If \(X\) is a compact complex connected manifold, then \(\Aut (X)\) carries the natural structure of a (not necessarily connected) complex Lie group [BM] . The identity component \(\Aut _0(X)\) of \(\Aut (X)\) is Jordan for every compact complex space \(X\) [Po18] Theorems 5 and 7.

The group \(\Aut (X)/\Aut _0(X)\) of connected components of \(\Aut (X)\) is bounded if \(X\) is Kähler [BZ20] Proposition 1.4.

It is known that the group \(\Aut (X)\) is Jordan if

• \(X\) is projective (Sh. Meng and D.-Q. Zheng [MZ] );
• \(X\) is a compact complex Kähler manifold (Kim [Kim] );
• \(X\) is a compact complex space in Fujiki's Class \(\mathcal C\) (Sh. Meng, F. Peroni, D.-Q. Zheng, [MPZ] ; see also [PS19] for Moishezon threefolds).

Moreover, \(\Aut (X)\) is very Jordan if the Kodaira dimension \(\varkappa (X)\) of \(X\) is non-negative, or if \(X\) is a \(\BP ^1-\)bundle over a certain non-uniruled complex manifold [BZ20] , [BZ22] , [BZ22a] .

The structure of the groups \(\Bir (X)\) and \(\Bim (X) \) of birational and bimeromorphic selfmaps, respectively, is more complicated. It appears that uniruled varieties play a special role with respect to Jordan properties.

There are examples of

• a projective variety \(X_{pr}\) with non-Jordan group \(\Bir (X_{pr}),\) namely

  \[X_{pr}:=E\times \BP ^1\]

  where \(E \) is any elliptic curve [Zar14] ;

• a non-algebraic connected compact complex manifold \(X_{c}\) with non-Jordan group \(\Bim (X_c):\)

  \[X_{c}:=T\times \BP ^1,\]

  where \(T\) is any non-algebraic complex torus of positive algebraic dimension [Zar19] ;

• a smooth compact real manifold \(M\) with non-Jordan group \(\Diff (M)\) with \(M\) being the direct product of 2-dimensional real torus by 2-dimensional sphere (B. Csikós, L. Pyber, E.Szabó, [CPS] ).

  Note that \(\BP ^1\) is a 2-dimensional sphere as a real manifold.

All these examples are essentially the same. Let us note their main features: all those objects are

• uniruled (covered by rational curves);
• direct products with a torus \(T\);
• a torus \(T\) carries no rational curves and the group \(T\) is an algebraic, commutative, not bounded group.

It seems that the Jordan property (or rather its absence) of the groups \(\Bir (X),\) or \(\Bim (X)\) for a complex manifold (or projective varietiy) \(X\) correlate with such geometric features as being uniruled over a non-uniruled positive dimensional base or being a direct product.

Let us illustrate it in the case of surfaces by the following assertion.

Let us sketch the ideas involved in the proof. They are basic for this theory and, in a more sophisticated form, are widely used.

We will restrict ourselves to the smooth situation. Recall that a smooth surface \(X\) has a minimal model \(X_m\) (that is smooth and contains no (-1) curves, see, e.g., [Sha] ).

Case 1. \(\varkappa (X) \ge 0\). Then \(\Bir (X)=\Bir (X_m) = \Aut (X_m).\)

Every automorphism \(f\in \Aut (X_m)\) induces the automorphism \(\psi (f)\) of the Néron-Severi group \(\NS (X_m)\) (the group of connected components of \(\Pic (X).\)) Let \(G_i:=\ker (\psi )\). This is a complex Lie group that may be included into the exact sequence:

It is known that

• \(G_i\) has finitely many connected components;
• the identity component \(G_i^{0}\) of \(G_i\) is a connected algebraic group;
• Being a connected algebraic group, \(G_i^{0}\) is Jordan;
• The Néron-Severi group \(\NS (X)\) is a finitely generated abelian group; in particular, its torsion subgroup \(F\) is finite and the quotient \(NS(X))/F\) is isomorphic to the free abelian group \(\BZ ^{\rho }\) of finite (positive) rank \(\rho \) where \(\rho \) is the Picard number of \(X.\) This implies that the kernel of the natural homomorphism

  \[\Aut (\NS (X)) \to \Aut (NS(X)/F) \cong \GL (\rho ,\BZ )\]

  is finite. By the theorem of Minkowski, \(\GL (\rho ,\BZ )\) is bounded. This implies that \(\Aut (\NS (X))\) is bounded as well.

Now Equation ( 3 ) implies that \(\Bir (X)= \Aut (X_m)\) is Jordan .

Case 2. \(\varkappa (X) = -\infty \)

As was already mentioned, the case of \(\Cr _2(\BC )=\Bir (\BP ^2)\) is due to Serre (see Theorem 4 above).

If the surface is birational to a direct product \(X_m:=B\times \BP ^1\) of a curve \(B\) of genus \(g\ge 1\) and the projective line then every birational automorphism \(f\in \Bir (X_m)\cong \Bir (X)\) is fiberwise. It means that it can be included into the following commutative diagram:

Here \(\pi :X\to B\) is the natural projection and \(\tau (f)\in \Aut (B).\)

The subgroup \(G_0=\{f\in \Aut (X_m) | \tau (f)=\mathrm {id}\}\subset \PSL (2, K),\) where \(K=\BC (B)\) is the field of rational functions on \(B\), is Jordan.

Once more we have an exact sequence

where \(G_B= \psi (\Aut (X_m))\subset \Aut (B)\) is finite if genus \(g>1.\)

Thus if the genus \(g(B)>1\) then Equation ( 5 ) implies that \(\Bir (X_m)\cong \Bir (X)\) is Jordan.

The special case: \(X\) is birational to \(E\times \BP ^1\) where \(E\) is an elliptic curve, is left.

The proof of this Theorem is done in two steps. First, for every \(N\in \BN \) a certain group \(\mathfrak {G}_{N}\) is constructed and its Jordan number is shown to be \(N.\) Then for every \(N\in \BN \) a surface \(S_N\) is built such that

• \(S_N\) is birational to \(E\times \BP ^1;\)
• \(\Aut (S_N) \) contains a group \(G_N\cong \mathfrak {G}_{N}.\)

Step 1: Analogues of the Heisenberg groups that were used by D. Mumford [Mum66] . Let

• \(\mathbf {K}\) be a finite commutative group of order \(N>1;\)
• \(\mu _N \subset \BC ^{*}\) be the multiplicative group of \(N\)th roots of unity;
• \(\hat {\mathbf {K}}=\mathrm {Hom}(\mathbf {K},\mu _N)\) - the dual of \(\mathbf {K}.\)

The Mumford theta group \(\mathfrak {G}_{\mathbf {K}}\) for \(\mathbf {K}\) is the group of matrices of the type

\[\begin {pmatrix} 1 & \alpha &\gamma \\ 0 & 1 & \beta \\ 0 & 0 & 1 \end {pmatrix}\]

where \(\alpha \in \hat {\mathbf {K}},\) \(\gamma \in \BC ^{*},\) and \(\beta \in \mathbf {K}.\) The product \(\alpha (\beta ) \in \BC ^{*}\) of \(\alpha \in \hat {\mathbf {K}} \) and \(\beta \in \mathbf {K}\) is used in order to define a certain natural non-degenerate alternating bilinear form \(e_{\mathbf {K}}\) on \(\mathbf {H}_{\mathbf {K}}=\mathbf {K}\times \hat {\mathbf {K}}\) with values in \(\BC ^{*}\) [Zar14] , p. 302. This group may be included into a short exact sequence

\[1 \to \BC ^{*} \to \mathfrak {G}_{\mathbf {K}} \to \mathbf {H}_{\mathbf {K}} \to 1\]

where the image of \(\BC ^*\) is the center of \(\mathfrak {G}_{\mathbf {K}}.\) These groups are Jordan and

\[J_{\mathfrak {G}_{\mathbf {K}}}=\sqrt {\#(\mathbf {H}_{\mathbf {K}})}=N=\#(\mathbf {K}).\]

In particular, let us put \(\mathfrak {G}_{N}:=\mathfrak {G}_{\BZ /N\BZ }, \) i.e., \(K=\BZ /N\BZ .\) Then \(J_{\mathfrak {G}_{N}}=N.\)

Step 2: Constructing surfaces \(S_N\).

Fix a point \(P\in E\) and denote by \([P]\) the corresponding divisor on \(E\). Choose an integer \(N>1\) and consider the divisor \(N[P]\) on \(E\). Let \(L_{N[P]}\) be the holomorphic line bundle on \(E\) that corresponds to \(N[P]\). Let \(\mathcal {L}_N\) be the total space of the line bundle \(L_{N[P]}.\) Let \(S_N=\ov {\mathcal {L}_N}\) be its projective closure/compactification , i.e., \(S_N=\mathcal {L}_N\cup \mathcal {T}_{\infty },\) where \(\mathcal {T}_{\infty }\) is the “infinite" section of \(L_{N[P]}\). Actually, \(\ov {\mathcal {L}_N}\) is the \(\BP ^1\)-bundle over \(E\) that is the projectivization of the rank two vector bundle \(L_N\oplus \mathbf {1}_E, \) where \(\mathbf {1}_E=E \times \BC \) is the trivial line bundle over \(E.\) Thus, \(S_N\) is a ruled surface birational to \(E\times \BP ^1.\)

Let \(G(N)\) be the subgroup of all those \(f\in \Aut (S_N) \) that may be included into the following commutative diagram:

\[ \begin {CD} \ov {L_N} @>{f}>>\ov {L_N}\\ @V p VV @V p VV \\ E @>{T_Q}>> E \end {CD} \]

Here \(p:S_N\to E\) is the natural projection, \(E(N)\) stands for the subgroup of points in \(E\) of order dividing \(N,\) point \(Q\in E[N]\) is a point of order dividing \(N,\) and \(T_Q:E\to E\) is the translation map \(e\to e+Q.\) Moreover, \(f\) induces \(\BC -\)linear isomorphisms between the fibers of \(p\) over \(e\) and \(e+Q.\)

On \(E\times \BP ^1\) elements of the group \(G(N)\) induce birational maps and form a subgroup \(G_N\subset \Bir (E\times \BP ^1)\) that may be described as follows.

\(G(N)=\{(Q,f), Q\in E(N), f\in \BC (E)^* \text { such that } (f)=N[P+Q]-N[P] \}\) is acting as

\[(y, t)\in E\times \BP ^{1}\longrightarrow (Q,f)(y, t)=(Q+y, f(y)t).\]

Here \((f)\) is the divisor of a rational function \(f\).

By a result of D. Mumford [Mum66] , Sect. 1, Corollary of Theorem 1, that the group \(G_N\) is isomorphic to \(\mathfrak {G}_{N}\); hence \(J_{G_N}=N.\) Thus, \(J_{\Bir (E\times \BP ^{1})}\ge J_{G_N}=N\) for all \(N\), i.e., \(\Bir (E\times \BP ^{1})\) is not Jordan.

[1, § 1, Corollary of Theorem 1] Based on the proof of the non-Jordanness of \(\Bir (E\times \BP ^1)\) [Zar14] , B. Csikós, L. Pyber, E.Szabó [CPS] constructed a counterexample to

Conjecture of E. Ghys (1997) If \(M\) is a connected compact smooth real manifold then \(\Diff (M)\) is Jordan.

Let us describe their counterexample. From the real point of view, \(\BP ^1\) is the two-dimensional sphere \(\mathbb {S}^2,\) \(E\) is the two-dimensonal real torus \(\mathbb {T}^2,\) and \(S_N\) is an oriented \(\mathbb {S}^2\)-bundle over \(\mathbb {T}^2\).

As a smooth manifold, \(S_N\) is diffeomorphic to the product \(\mathbb {T}^2\times \mathbb {S}^2\) if and only if \(N\) is even . Therefore for each even \(N\) we have

\[G_N\hookrightarrow \Diff (\mathbb {T}^2\times \mathbb {S}^2).\]

Since the set of \(J_{G_N}\) for positive even integers \(N\) is unbounded, the group \(\Diff (\mathbb {T}^2\times \mathbb {S}^2)\) is not Jordan.

For complex projective varieties Yu. Prokhorov and C. Shramov, and C. Birkar proved the following

Here \(q(X)=\dim _{\BC }H^1(X,\mathcal {O}_X)\) is the irregularity of \(X.\) In particular, the Cremona group \(\Cr _n\) of any rank \(n\) is Jordan ( [PS16] , [Bi] ).

The group \(\Diff (M)\) of all diffeomorphisms of a smooth manifold \(M\) also appeared to be Jordan for certain classes of manifolds.

Namely, B. Zimmerman [Zim] proved that if \(M\) is compact and \(\dim (M)\le 3\) then \(\Diff (M)\) is Jordan. The Jordan property of \(\Diff (M)\) was studied by I. Mundet i Riera. In particular, he proved [MR18] that \(\Diff (M)\) is Jordan if \(M\) is one of the following:

(1) open acyclic manifolds,

(2) compact manifolds (possibly with boundary) with nonzero Euler characteristic,

(3) homology spheres.

So, in high dimensions the situation is very similar: the group \(\Bim (X)\) or \(\Bir (X)\) is mostly Jordan, and the worst case from the Jordan properties point of view is the following: a uniruled variety \(X\) with \(q(X)>0\) (or fibered over a non-uniruled base) that has many sections (such as a direct product). A typical example of such a variety \(X\) is a \(\BP ^1\)-bundle over a complex torus \(T\) of positive dimension.

The need of “many sections” may be demonstrated by the case of projective non-trivial conic bundles.

Recall that the generic fiber of \(f\) is an irreducible smooth projective curve \(\mathcal {X}_f\) over the field \(K:=\BC (Y)\) such that its field of rational functions \(K(\mathcal {X}_f)\) coincides with \(\BC (X).\) Notice that \(K\)-points in \(\mathcal {X}_f\) correspond to a rational sections of the conic bundle \(f: X\to Y.\) If such a \(K\)-point exists, then \(\mathcal {X}_f\) is isomorphic over \(K\) to the projective line \(\BP ^1_{K}\) and \(X\) is birational to \(Y\times \BP ^1\) (over \(\BC \)).

Let us sketch the proof.

If \(f: X\to Y\) is a conic bundle and \(Y\) is non-uniruled, then every \(\phi \in \Bir (X)\) is fiberwise (see ( 4 )).

It follows that there is an exact sequence of groups:

Since \(Y\) is non-uniruled, the group \(\Bir (Y)\) is Jordan, thanks to Theorem  20 . Moreover, it is strongly Jordan (see [BZ17] , Cor. 3.8 and its proof). Let us compute \(\Bir _{K}(\mathcal {X}_f)\) (recall that \(K=\BC (Y)\)). We have

• 1. \(\Bir ( \mathcal {X}_f)=\Aut ( \mathcal {X}_f)\), since \(\dim (\mathcal {X}_f)=1.\)

• 2. Since \(X\) is not birational to \(Y\times \BP ^1,\) the genus \(0\) curve \(\mathcal {X}_f\) has no \(K\)-points and therefore there exists a ternary quadratic form

  \[q(T)=a_1 T_1^2 +a_2 T_2^2 +a_3 T_3^2\]

  over \(K\) such that

  — all \(a_i\) are nonzero elements of \(K;\)

  — \(q(T)=0\) if and only if \(T=(0,0,0))\) (this means that \(q\) is anisotropic );

  — \(\mathcal {X}_f\) is biregular over \(K\) to the plane projective quadric

  \[\mathbf {X}_q:=\{(T_1:T_2:T_3)\mid q(T)=0\}\subset \BP ^2_K.\]

• 3. \(K\) is a field of characteristic zero that contains all roots of unity.

Now we can use the following fact that was proven in [BZ17] ).

Thus if \(G\) is a nontrivial finite subgroup of \(\Aut ( \mathcal {X}_f)\) then either \(G\cong \BZ /2\BZ \) or \(G\cong (\BZ /2\BZ )^2\).

Applying Equation ( 6 ), we get that \(\Bir (X)\) is Jordan.

We summarize now what we know about the Jordan properties when \(X\) is a \(\BP ^1\)-bundle over a complex torus \(T\) of positive dimension \(n\). First, let us recall basic facts about complex tori [BL] .

For a complex torus \(T\) there exists its algebraic model \(T_0\) such that:

• \(T_0\) is an abelian variety;
• there is a holomorphic surjective homomorphism \(p:T\to T_0\) with connected kernel that is universal in a sense that every homomorphism from \(T\) to any abelian variety factors uniquely through \(p\);
• the field \(\BC (T)\) of meromorphic functions on \(T\) coincides with \(p^*(\BC (T_0)),\) i.e., every meromorphic function on \(T\) is the lift of a rational function on \(T_0\);
• by definition, the algebraic dimension \(a(T)\) is \(\dim _{\BC } T_0.\)

Now we are ready to state our

Summary.

1. We may consider \(T\) as a real manifold \(T_r.\) It follows from the counterexample to the Ghys Conjecture that

if \(\dim _{\BR }(T_r)\ge 2\) and \(X=\mathbb {S}^2\times T_r\) then \(\Diff (X)\) is not Jordan.

2. Since \(T\) is a complex torus, it is a connected compact Kähler manifold.

2.1 Suppose that \(a(T)=\dim (T)=n.\) This means that \(T\) is algebraic, i.e., is an abelian variety. If \(X=\BP ^1\times T\) then \(\Bir (X)\) is not Jordan (see Theorem  18 ). If \(X\) is not birational to \(\BP ^1\times T\) then \(\Bir (X)\) is Jordan (see Theorem  23 ).

2.2 Suppose that \(0<a(T)<n\). Then \(T\) is a non-algebraic torus and \(n>1\). (In dimension \(1\) all complex tori are algebraic - they are the famous elliptic curves.) If \(X=\BP ^1\times T\) (or has at least three sections) then \(\Bim (X)\) is not Jordan [Zar19] .

2.3 Suppose that \(a(T)=0\). Then \(n \ge 2\) and \(T\) is non-algebraic. This is a “very general” case: in a “versal” family [BL] of all complex tori of a given dimension \(n \ge 2\) the subset of tori with algebraic dimension zero is dense. (See [BZ20] for explicit examples of such tori in all dimensions \(n\ge 2\).) If \(a(T)=0\) then any \(\BP ^1-\)bundle \(X\) over \(T\) that is not biholomorphic to the direct product \(\BP ^1\times T\) has at most two sections and \(\Bim (X)=\Aut (X)\) is Jordan [BZ20] .

Let us mention some open problems. Fix a positive integer \(n\).

Varieties with non-Jordan group \(\Bir (X).\) Let \(\mathcal V_n \) and \(\mathcal X_n\) be the class of connected complex projective varieties \(V\) (respectively, complex compact manifolds \(X\)) of dimension \(n\) such that the group \(\Bir (V)\) (respectively, \(\Bim (X)\) is not Jordan. For \(n\le 3\) these classes are well described (see [Po11] , [Zar14] , [PS14] , [PS18] , [PS19] , [PS20] , [PS20b] ). It is known that \(A\times \BP ^n\in \mathcal V_{n+k} \) if \(A\) is an abelian variety of positive dimension \(k\), and \(T\times \BP ^n\in \mathcal X_{n+k} \) if \(T\) is a complex torus of dimension \(k\) and positive algebraic dimension.

Question 1. Assume that \(V\) is a non-uniruled smooth projective variety and \(Y=V\times \BP ^n. \) Is \(\Bir (Y)\) non-Jordan? More generally, how to describe \(\mathcal V_n \) and \(\mathcal X_n?\)

Quasiprojective varieties Assume that \(W\) is a smooth quasiprojective variety that is an open subset of a smooth projective variety \(X.\) Then

\[\Aut (W)\subset \Bir ( X).\]

If \(\Bir ( X)\) is not Jordan, then, a priori , the same may be true for \(\Aut (W).\) However, to the best of our knowledge there is no example of a complex algebraic variety \(W\) with non-Jordan \(\Aut (W).\) It is known that \(\Aut (W)\) is Jordan if either

• \(\dim W\le 3\) and \(W\) is not birational to \(E\times \BP ^2,\) where \(E\) is an elliptic curve ( [BZ15] ) or
• \(W\) is quasiprojective and birational to a product \(A\times \BP ^n,\) where \(A\) is a smooth irreducible positive-dimensional projective variety that contains no rational curves. (See [BZ18] .)

Question 2. Does there exist a complex algebraic variety \(W\) with non-Jordan group \(\Aut (W)?\)

Line bundles over tori of positive algebraic dimension The statement of Summary 2.2 remains true if the direct product \(X=\BP ^1\times T\) is replaced by the “natural compactification” \(X_{L}\) of the total space of a holomorphic line bundle \(L=p^*(L_0)\) on \(X\) where \(L_0\) is any holomorphic line bundle on the algebraic model \(T_0\) of \(T\) and \(p: T \to T_0\) the universal homomorphism. Here by natural compactification \(X_{L}\) we mean the projectivization of the total space of the rank \(2\) holomorphic vector bundle \(L\oplus \mathbf {1}_T\) where \(\mathbf {1}_T=T \times \BC \) is the trivial holomorphic line bundle. (Summary 2.2 still remains true even if just the Chern class of \(L\) coincides with the Chern class of \(p^*(L_0)\) for some holomorphic line bundle on \(T_0\).) See [Zar19] .

Question 3. Does Summary 2.2 remains true for \(X_{L}\) for an arbitrary holomorphic line bundle \(L\) on \(T\)?

Poor manifolds. The statement of Summary 2.3 remains true if the torus \(T\) is replaced by any poor manifold [BZ20] .

Any complex torus \(T\) with \(\dim (T)\ge 2\) and \(a(T)=0\) is poor. There are examples of poor \(K3\) surfaces.

Question 4. Find a classification of poor manifolds.

Acknowledgements. The authors are most grateful to the referee for valuable comments and for suggestion to add a list of open problems (in particular, Question 2).

Conflict of Interest statement Not Applicable