JORDAN GROUPS AND AUTOMORPHISM GROUPS OF ALGEBRAIC VARIETIES

VLADIMIR L. POPOV^{∗}

Steklov Mathematical Institute, Russian Academy of Sciences Gubkina 8, Moscow 119991, Russia and National Research University Higher School of Economics 20, Myasnitskaya Ulitsa, Moscow 101000, Russia popovvl@mi.ras.ru

Abstract. The first section of this paper is focused on Jordan groups in abstract setting, the second on that in the settings of automorphisms groups and groups of birational self-maps of algebraic varieties. The appendix contains formulations of some open problems and the relevant comments. MSC 2010: 20E07, 14E07 Key words: Jordan, Cremona, automorphism, birational map

This is the expanded version of my talk, based on [Po2010, Sect. 2], at the workshop Groups of Automorphisms in Birational and Affine Geometry, October 29–November 3, 2012, Levico Terme, Italy. The appendix is the expanded version of my notes on open problems posted on the site of this workshop [Po2012_{2}].

Below k is an algebraically closed field of characteristic zero. Variety means algebraic variety over k in the sense of Serre (so algebraic group means algebraic group over k). We use without explanation standard notation and conventions of [Bo1991] and [Sp1998]. In particular, k(X) denotes the field of rational functions of an irreducible variety X. Bir(X) denotes the group of birational self-maps of an irreducible variety X, and Cr_{n} denotes the Cremona group over k of rank n, cf. [Po2011], [Po2012_{1}].

1. Jordan groups

1.1. Main definition. The notion of Jordan group was introduced in [Po2010]:

Definition 1 ([Po2010, Def. 2.1]). A group G is called a Jordan group if there exists a positive integer d, depending only on G, such that every finite subgroup K of G contains a normal abelian subgroup whose index in K is at most d. The minimal such d is called the Jordan constant of G and is denoted by J_{G}.

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Informally, this means that all finite subgroups of G are “almost” abelian in the sense that they are extensions of abelian groups by finite groups taken from a finite list.

Actually, one obtains the same class of groups if the assumption of normality in Definition 1 is dropped. Indeed, for any group P containing a subgroup Q of finite index, there is a normal subgroup N of P such that [P : N] ⩽ [P : Q]! and N ⊆ Q (see, e.g., [La1965, Exer. 12 to Chap. I]).

1.2. Examples.

1.2.1. Jordan’s Theorem. The first example that led to Definition 1 justifies the coined name. It is given by the classical Jordan’s theorem [Jo1878] (see, e.g., [CR1962, §36] for a modern exposition). In terms of Definition 1 the latter can be reformulated as follows:

Theorem 1 (C. Jordan, 1878). The group GL_{n}(k) is Jordan for every n.

Since the symmetric group Sym_{n+1} admits a faithful n-dimensional representation and the alternating group Alt_{n+1} is the only non-identity proper normal subgroup of Sym_{n+1} for n ⩾ 2, n ≠ 3, Definition 1 yields the lower bound

(n + 1)! ⩽ J_{GL_{n}(k)} for n ⩾ 4.  (1)

Frobenius, Schur, and Blichfeldt initiated exploration of the upper bounds for J_{GL_{n}(k)}. In 2007, using the classification of finite simple groups, M. J. Collins [Co2007] gave optimal upper bounds and thereby found the precise values of J_{GL_{n}(k)} for all n. In particular, in [Co2007] is proved that

(i) the equality in (1) holds for all n ⩾ 71 and n = 63, 65, 67, 69; (ii) J_{GL_{n}(k)} = 60^{r}r! if n = 2r or 2r + 1 and either 20 ⩽ n ⩽ 62 or

n = 64, 66, 68, 70; (iii) J_{GL_{n}(k)} = 60, 360, 25920, 25920, 6531840 resp., for n = 2, 3, 4, 5, 6.

The values of J_{GL_{n}(k)} for 7 ⩽ n ⩽ 19 see in [Co2007].

1.2.2. Affine algebraic groups. Since any subgroup of a Jordan groups is Jordan, Theorem 1 yields

Corollary 1. Every linear group is Jordan.

Since every affine algebraic group is linear [Sp1998, 2.3.7], this, in turn, yields the following generalization of Theorem 1:

Theorem 2. Every affine algebraic group is Jordan.

1.2.3. Nonlinear Jordan groups. Are there nonlinear Jordan groups? The next example, together with Theorem 1, convinced me that Definition 1 singles out an interesting class of groups and therefore deserves to be introduced.

Example 1. By [Se2009_{1}, Thm. 5.3], [Se2008, Thm. 3.1], the planar Cremona group Cr_{2} is Jordan. On the other hand, by [CD2009, Prop. 5.1] (see also [Co2007, Prop. 2.2]), Cr_{2} is not linear. Note that in [Se2009_{1}, Thm. 5.3] one also finds a “multiplicative” upper bound for J_{Cr_{2}}: as is specified there, a crude computation shows that every finite subgroup G of Cr_{2} contains a normal abelian subgroup A of rank ⩽ 2 with [G : A] dividing 2^{10} ·3^{4} ·5^{2} ·7 (it is also mentioned that the exponents of 2 and 3 can be somewhat lowered, but those of 5 and 7 cannot). □

Example 2. Let F_{d} be a free group with d free generators and let F_{d}^{n} be its normal subgroup generated by the nth powers of all elements. As is known (see, e.g., [Ad2011, Thm. 2]), the group B(d, n) := F_{d}/F_{d}^{n} is infinite for d ⩾ 2 and odd n ⩾ 665 (recently S. Adian announced in [Ad2013] that 665 may be replaced by 100). On the other hand, by I. Schur, finitely generated linear torsion groups are finite (see, e.g., [CR1962, Thm. 36.2]). Hence infinite B(d, n) is nonlinear. On the other hand, for d ⩾ 2 and odd n ⩾ 665, every finite subgroup in B(d, n) is cyclic (see [Ad2011, Thm. 8]); hence B(d, n) is Jordan and J_{B(d,n)} = 1. □

1.2.4. Diffeomorphism groups of smooth manifolds. Let M be a compact connected n-dimensional smooth manifold. Assume that M admits an unramified covering M̃ → M such that H^{1}(M̃, Z) contains the cohomology classes α_{1}, …, α_{n} satisfying α_{1} ∪ ⋯ ∪ α_{n} ≠ 0. Then, by [MiR2010, Thm. 1.4(1)], the group Diff(M) is Jordan. This result is applicable to T^{n}, the product of n circles, and, more generally, to the connected sum of N♯T^{n}, where N is any compact connected orientable smooth manifold. (I thank I. Mundet i Riera who informed me about his paper [MiR2010], from which I also learned about [Fi2011] and [Pu2007].)

1.2.5. Non-Jordan groups. Are there non-Jordan groups?

Example 3. The group Sym_{∞} of all permutations of Z contains the alternating group Alt_{n} for every n. Hence Sym_{∞} is non-Jordan because Alt_{n} is simple for n ⩾ 5 and |Alt_{n}| = n!/2. □

Using Example 3 one obtains a finitely generated non-Jordan group:

Example 4. Let N be the subgroup of Sym_{∞} generated by the transposition σ := (1, 2) and the “translation” δ defined by the condition

δ(i) = i + 1 for every i ∈ Z.

Then δ^{m}σδ^{−m} is the transposition (m + 1, m + 2) for every m. Since the set of transpositions (1, 2), (2, 3), …, (n − 1, n) generates the symmetric group Sym_{n}, this shows that N contains Alt_{n} for every n; whence N is non-Jordan. □

1.3. General properties.

1.3.1. Subgroups, quotient groups, and products. Exploring whether a group is Jordan or not leads to the questions on the connections between Jordaness of a group, its subgroup, and its quotient group.

Theorem 3 ([Po2010, Lemmas 2.6, 2.7, 2.8]).

(1) Let H be a subgroup of a group G.

(i) If G is Jordan, then H is Jordan and J_{H} ⩽ J_{G}.

(ii) If G is Jordan and H is normal in G, then G/H is Jordan and J_{G/H} ⩽ J_{G} in either of the cases:

(a) H is finite;

(b) the extension 1 → H → G → G/H → 1 splits. (iii) If H is torsion-free, normal in G, and G/H is Jordan, then G

is Jordan and J_{G} ⩽ J_{G/H}.

(2) Let G_{1} and G_{2} be two groups. Then G_{1} × G_{2} is Jordan if and only if G_{1} and G_{2} are. In this case, J_{G_{i}} ⩽ J_{G_{1}×G_{2}} ⩽ J_{G_{1}}J_{G_{2}} for every i.

Proof. (1)(i). This follows from Definition 1.

If H is normal in G, let π: G → G/H be the natural projection.

(1)(ii)(a). Let F be a finite subgroup of G/H. Since H is finite, π^{−1}(F) is finite. Since G is Jordan, π^{−1}(F) contains a normal abelian subgroup A whose index is at most J_{G}. Hence π(A) is a normal abelian subgroup of F whose index in F is at most J_{G}.

(1)(ii)(b). By the condition, there is a subgroup S in G such that π|_{S} : S → G/H is an isomorphism; whence the claim by (1)(i).

(1)(iii). Let F be a finite subgroup of G. Since H is torsion free, F ∩ H = {1}; whence π|_{F} : S → π(F) is an isomorphism. Therefore, as G/H is Jordan, F contains a normal abelian subgroup whose index in F is at most J_{G/H}.

(2) If G := G_{1} × G_{2} is Jordan, then (1)(i) implies that G_{1} and G_{2} are Jordan and J_{G_{i}} ⩽ J_{G} for every i. Conversely, let G_{1} and G_{2} be Jordan. Let π_{i}: G → G_{i} be the natural projection. Take a finite subgroup F of G. Then F_{i} := π_{i}(F) contains an abelian normal subgroup A_{i} such that

[F_{i} : A_{i}] ⩽ J_{G_{i}}.  (2)

The subgroup Ã_{i} := π_{i}^{−1}(A_{i}) ∩ F is normal in F and F/Ã_{i} is isomorphic to F_{i}/A_{i}. From (2) we then conclude that

[F : Ã_{i}] ⩽ J_{G_{i}}.  (3)

Since A := Ã_{1} ∩ Ã_{2} is the kernel of the diagonal homomorphism

F ⟶ F/Ã_{1} × F/Ã_{2}

determined by the canonical projection F → F/Ã_{i}, we infer from (3) that

[F : A] = |F/A| ⩽ |F/Ã_{1} × F/Ã_{2}| = |F_{1}/A_{1}||F_{2}/A_{2}| ⩽ J_{G_{1}}J_{G_{2}}. (4)

By construction, A ⊆ A_{1} × A_{2} and A_{i} is abelian. Hence A is abelian as well. Since A is normal in F, the claim then follows from (4). □

1.3.2. Counterexample. For a normal subgroup H of G, it is not true, in general, that G is Jordan if H and G/H are.

Example 5. For every integer n>0 fix a finite group G_{n} with the properties: (i) G_{n} has an abelian normal subgroup H_{n} such that G_{n}/H_{n} is abelian; (ii) there is a subgroup Q_{n} of G_{n} such that the index in Q_{n} of every

abelian subgroup of Q_{n} is greater or equal than n. Such a G_{n} exists, see below. Now take G := ∏_{n}G_{n} and H := ∏_{n} H_{n}. Then H and G/H are abelian by (i), hence Jordan, but G is not Jordan by (ii).

The following construction from [Za2010, Sect. 3] proves the existence of such a G_{n}. Let K be a finite commutative group of order n written additively and let K̂ := Hom(K, k^{∗}) be the group of characters of K written multiplicatively. The formula

(α, g, ℓ)(α′, g′, ℓ′) := (αα′ℓ′(g), g + g′, ℓℓ′)  (5)

endows the set k^{∗} × K × K̂ with the group structure. Denote by G_{K} the obtained group. It is embedded in the exact sequence of groups

{1} → k^{∗} ⟶^{ι} G_{K} ⟶^{π} K × K̂ → {(0, 1)},

where ι(α) := (α, 0, 1) and π((α, g, ℓ)) := (g, ℓ).

Thus, if one takes G_{n} := G_{K} and H_{n} := ι(k^{∗}), then property (i) holds. Let μ_{n} be the subgroup of all nth roots of unity in k^{∗}. From (5) and |K| = n we infer that the subset Q_{K} := μ_{n} × K × K̂ is a subgroup of G_{K}. In [Za2010, Sect. 3] is proved that for Q_{n} = Q_{K} property (ii) holds. □

1.3.3. Bounded groups. However, under certain conditions, G is Jordan if and only if H and G/H are. An example of such a condition is given in Theorem 4 below; it is based on Definition 2 below introduced in [Po2010].

Given a group G, put

b_{G} := sup_{F} |F |,

where F runs over all finite subgroups of G.

Definition 2 ([Po2010, Def. 2.9]). A group G is called bounded if b_{G} ≠ ∞.

Example 6. Finite groups and torsion free groups are bounded. □

Example 7. It is immediate from Definition 2 that every extension of a bounded group by bounded is bounded. □

Example 8. By the classical Minkowski’s theorem GL_{n}(Z) is bounded (see, e.g., [Hu1998, Thm. 39.4]). Since every finite subgroup of GL_{n}(Q) is conjugate to a subgroup of GL_{n}(Z) (see, e.g., [CR1962, Thm. 73.5]), this implies that GL_{n}(Q) is bounded and b_{GL_{n}(Q)} = b_{GL_{n}(Z)}. H. Minkowski and I. Schur obtained the following upper bound for b_{GL_{n}(Z)}, see, e.g., [Hu1998, §39]. Let P(n) be the set of all primes p ∈ N such that [n/(p−1)] > 0. Then

b_{GL_{n}(Z)} ⩽ ∏_{p∈P(n)}p^{d_{p}}, where d_{p} = ∑_{i=0}^{∞}[\frac{n}{p^{i}(p − 1)}].  (6)

In particular, the right-hand side of the inequality in (6) is

2, 24, 48, 5760, 11520, 2903040 resp., for n = 1, 2, 3, 4, 5, 6. □

Example 9. Maintain the notation and assumption of Subsection 1.2.4. If χ(M) ≠ 0, then by [MiR2010, Thm. 1.4(2)], the group Diff(M) is bounded. Further information on smooth manifolds with bounded diffeomorphism groups is contained in [Pu2007]. □

Example 10. Every bounded group G is Jordan with J_{G} ⩽ b_{G}, and there are non-bounded Jordan groups (e.g., GL_{n}(k)). □

Theorem 4 ([Po2010, Lemma 2.11]). Let H be a normal subgroup of a group G such that G/H is bounded. Then G is Jordan if and only if H is Jordan, and in this case

J_{G} ⩽ b_{G/H}J_{H}^{b_{G/H}} .

Proof. A proof is needed only for the sufficiency. So let H be Jordan and let F be a finite subgroup of G. By Definition 1

L := F ∩ H  (7)

contains a normal abelian subgroup A such that

[L : A] ⩽ J_{H}.  (8)

Let g be an element of F. Since L is a normal subgroup of F, we infer that gAg^{−1} is a normal abelian subgroup of L and

[L : A] = [L : gAg^{−1}].  (9)

The abelian subgroup

M := ⋂_{g∈F} gAg^{−1}.  (10)

is normal in F. We intend to prove that [F : M] is upper bounded by a constant not depending on F. To this end, fix the representatives g_{1}, …, g_{|F/L|} of all cosets of L in F. Then (10) and normality of A in L imply that

M = ⋂_{i=1}^{|F/L|} g_{i}Ag_{i}^{−1}.  (11)

From (11) we deduce that M is the kernel of the diagonal homomorphism

L ⟶ ∏_{i=1}^{|F/L|} L/g_{i}Ag_{i}^{−1}

determined by the canonical projections L → L/g_{i}Ag_{i}^{−1}. This, (9), and (8) yield

[L : M] ⩽ [L : A]^{|F/L|} ⩽ J_{H}^{|F/L|}.  (12)

Let π: G → G/H be the canonical projection. By (7) the finite subgroup π(F) of G/H is isomorphic to F/L. Since G/H is bounded, this yields |F/L| ⩽ b_{G/H}. We then deduce from (12) and [F : M] = [F : L][L : M] that

[F : M] ⩽ b_{G/H}J_{H}^{b_{G/H}};

whence the claim. □

The following corollary should be compared with statement (1)(ii)(a) of Theorem 3:

Corollary 2. Let H be a finite normal subgroup of a group G such that the center of H is trivial. If G/H is Jordan, then G is Jordan and

J_{G} ⩽ |Aut(H)|J_{G/H}^{|Aut(H)|}.

Proof. Let φ: G → Aut(H) be the homomorphism determined by the conjugating action of G on H. Triviality of the center of H yields H ∩ker φ = {1}. Hence the restriction of the natural projection G → G/H to ker φ is an embedding ker φ ↪ G/H. Therefore, ker φ is Jordan since G/H is. But G/ker φ is finite since it is isomorphic to a subgroup of Aut(H) for the finite group H. By Theorem 4 this implies the claim. □

2. When are Aut(X) and Bir(X) Jordan?

2.1. Problems A and B. In [Po2010, Sect. 2] were posed the following two problems:

Problem A. Describe algebraic varieties X for which Aut(X) is Jordan. Problem B. The same with Aut(X) replaced by Bir(X).

Note that for rational varieties X Problem B means finding n such that the Cremona group Cr_{n} is Jordan; in this case, it was essentially posed in [Se2009_{1}, 6.1].

Describing finite subgroups of the groups Aut(X) and Bir(X) for various varieties X is a classical research direction, currently flourishing. Understanding which of these groups are Jordan sheds a light on the structure of these subgroups.

2.2. Groups Aut(X). In this subsection we shall consider Problem A.

Lemma 1. Let X_{1}, …, X_{n} be all the irreducible components of a variety X. If every Aut(X_{i}) is Jordan, then Aut(X) is Jordan.

Proof. Define the homomorphism π: Aut(X) → Sym_{n} by g · X_{i} = X_{π(g)} for g ∈ Aut(X). Then g · X_{i} = X_{i} for every g ∈ Ker(π) and i, so the homomorphism π_{i}: Ker(π) → Aut(X_{i}), g ↦ g|_{X_{i}}, arises. The definition implies that π_{1} × … × π_{n}: Ker(π) → ∏_{i=1}^{n} Aut(X_{i}) is an injection; whence Ker(π) is Jordan by Theorem 3(2). Therefore, Aut(X) is Jordan by Theorem 4. □

At this writing (July 2013), not a single variety X with non-Jordan Aut(X) is known (to me).

Question 1 ([Po2010, Quest. 2.30 and 2.14]). Is there an irreducible variety X such that Aut(X) is non-Jordan? Is there an irreducible affine variety X with this property?

Remark 1. This may be compared with the question originally posed by É. Ghys, see [Fi2011, Quest. 13.1]. Using Definition 1, it is reformulated as follows: Is the diffemorphism group of any compact smooth manifold Jordan?

On the other hand, in many cases it can be proven that Aut(X) is Jordan. Below are described several extensive classes of X with this property.

2.2.1. Toral varieties. First, consider the wide class of affine varieties singled out by the following

Definition 3 ([Po2010, Def. 1.13]). A variety is called toral if it is isomorphic to a closed subvariety of some A^{n} \⋃_{i=1}^{n} H_{i}, where H_{i} is the set of zeros of the ith standard coordinate function x_{i} on A^{n}.

Remark 2. A^{n} \ ⋃_{i=1}^{n} H_{i} is the group variety of the n-dimensional affine torus; whence the terminology. Warning: “toral” does not imply “affine toric” in the sense of [Fu1993].

The class of toral varieties is closed with respect to taking products and closed subvarieties.

Lemma 2 ([Po2010, Lemma 1.14(a)]). The following properties of an affine variety X are equivalent:

(i) X is toral; (ii) k[X] is generated by k[X]^{∗}, the group of units of k[X].

Proof. If X is closed in A^{n} \ ⋃_{i=1}^{n} H_{i}, then the restriction of functions is an epimorphism k[A^{n} \ ⋃_{i=1}^{n} H_{i}] → k[X]. Since k[A^{n} \ ⋃_{i=1}^{n} H_{i}] = k[x_{1}, …, x_{n}, 1/x_{1}, … 1/x_{n}], this proves (i) ⇒ (ii).

Conversely, assume that (ii) holds and let

k[X] = k[f_{1}, …, f_{n}]  (13)

for some f_{1}, …, f_{n} ∈ k[X]^{∗}. Since X is affine, (13) implies that ι: X → A^{n}, x ↦ (f_{1}(x), …, f_{n}(x)), is a closed embedding. The standard coordinate functions on A^{n} do not vanish on ι(X) since every f_{i} does not vanish on X. Hence ι(X) ⊆ A^{n} \ ⋃_{i=1}^{n} H_{i}. This proves (ii)⇒(i). □

Lemma 3. Any quasiprojective variety X endowed with a finite automorphism group G is covered by G-stable toral open subsets.

Proof. First, any point x ∈ X is contained in a G-stable affine open subset of X. Indeed, since the orbit G · x is finite and X is quasiprojective, there is an affine open subset U of X containing G · x. Hence V := ⋂_{g∈G} g · U is a G-stable open subset containing x, and, since every g · U is affine, V is affine as well, see, e.g., [Sp1998, Prop. 1.6.12(i)].

Thus, the problem is reduced to the case where X is affine. Assume then that X is affine, and let k[X] = k[h_{1}, …, h_{s}]. Replacing h_{i} by h_{i} + α_{i} for an appropriate α_{i} ∈ k, we may (and shall) assume that every h_{i} vanishes nowhere on the G · x. Expanding the set {h_{1}, …, h_{s}} by including g · h_{i} for every i and g ∈ G, we may (and shall) assume that {h_{1}, …, h_{s}} is Gstable. Then h := h_{1} ⋯ h_{s} ∈ k[X]^{G}. Hence the affine open set X_{h} := {z ∈ X | h(z) ≠ 0} is G-stable and contains G·x. Since k[X_{h}] = k[h_{1}, …, h_{s}, 1/h] and h_{1}, …, h_{s}, 1/h ∈ k[X_{h}]^{∗}, the variety X_{h} is toral by Lemma 2. □

Remark 3. Lemma 3 and its proof remain true for any variety X such that every G-orbit is contained in an affine open subset; whence the following Corollary 3. Every variety is covered by open toral subsets.

For irreducible toral varieties the following was proved in [Po2010, Thm. 2.16].

Theorem 5. The automorphism group of every toral variety is Jordan.

Proof. By Theorem 1 it suffices to prove this for irreducible toral varieties. By [Ro1957], for any irreducible variety X,

Γ := k[X]^{∗}/k^{∗}

is a free abelian group of finite rank. Let X be toral and let H be the kernel of the natural action of Aut(X) on Γ. We claim that H is abelian. Indeed, for every function f ∈ k[X]^{∗}, the line in k[X] spanned over k by f is H-stable. Since GL_{1} is abelian, this yields that

h_{1}h_{2}·f = h_{2}h_{1}·f for any elements h_{1}, h_{2} ∈ H.  (14)

As X is toral, k[X]^{∗} generates the k-algebra k[X] by Lemma 2. Hence (14) holds for every f ∈ k[X]. Since X is affine, the automorphisms of X coincide if and only if they induce the same automorphisms of k[X]. Therefore, H is abelian, as claimed.

Let n be the rank of Γ. Then Aut(Γ) is isomorphic to GL_{n}(Z). By the definition of H, the natural action of Aut(X) on Γ induces an embedding of Aut(X)/H into Aut(Γ). Hence Aut(X)/H is isomorphic to a subgroup of GL_{n}(Z) and therefore is bounded by Example 6(2). Thus, Aut(X) is an extension of a bounded group by an abelian group, hence Jordan by Theorem 4. This completes the proof. □

Remark 4. Maintain the notation of the proof of Theorem 5 and assume that X is irreducible. Let f_{1}, … f_{n} be a basis of Γ. There are the homomorphisms λ_{i}: H → k^{∗}, i = 1, …, n, such that g · f_{i} = λ(g)f_{i} for every g ∈ H and i. Since k[X]^{∗} generates k[X], the diagonal map H → (k^{∗})^{n}, h ↦ (λ_{1}(g), …, λ_{n}(g)), is injective. This and the proof of Theorem 5 show that for any irreducible toral variety X with rk k[X]^{∗}/k^{∗} = n, there is an exact sequence

{1} → D → Aut(X) → B → {1},

where D is a subgroup of the torus (k^{∗})^{n} and B is a subgroup of GL_{n}(Z).

Combining Theorem 5 with Corollary of Lemma 3, we get the following:

Theorem 6. Any point of any variety has an open neighborhood U such that Aut(U) is Jordan.

2.2.2. Affine spaces. Next, consider the fundamental objects of algebraic geometry, the affine spaces A^{n}. The group Aut(A^{n}) is the “affine Cremona group of rank n”.

Since Aut(A^{1}) is the affine algebraic group Aff_{1}, it is Jordan by Theorem 2. Since Aut(A^{2}) is the subgroup of Cr_{2}, it is Jordan by Example 1. Another

proof: By [Ig1977] every finite subgroup of Aut(A^{2}) is conjugate to a subgroup of GL_{2}(k), so the claim follows from Theorem 1.

The group Aut(A^{3}) is Jordan being the subgroup of Cr_{3} that is Jordan by Corollary 13 below.

At this writing (July 2013) is unknown whether Aut(A^{n}) is Jordan for n ⩾ 4 or not. By Theorem 14, if the so-called BAB Conjecture (see Subsection 2.3.5 below) holds true in dimension n, then Cr_{n} is Jordan, hence Aut(A^{n}) is Jordan as well.

2.2.3. Fixed points and Jordaness. The following method of proving Jordaness of Aut(X) was suggested in [Po2010, Sect. 2] and yields extensive classes of X with Jordan Aut(X). It is based on the use of the following known fact:

Lemma 4. Let X be an irreducible variety, let G be a finite subgroup of Aut(X), and let x ∈ X be a fixed point of G. Then the natural action of G on T_{x,X}, the tangent space of X at x, is faithful.

Proof. Let 𝔪_{x,X} be the maximal ideal of O_{x,X}, the local ring of X at x. Being finite, G is reductive. Since char k = 0, this implies that 𝔪_{x,X} = L ⊕ 𝔪_{x,X}^{2} for some submodule L of the G-module 𝔪_{x,X}. Let K be the kernel of the action of G on L and let L^{d} be the k-linear span in 𝔪_{x,X} of the dth powers of all the elements of L. By the Nakayama’s Lemma, the restriction to L^{d} of the natural projection 𝔪_{x,X} → 𝔪_{x,X}/𝔪_{x,X}^{d+1} is surjective. Hence K acts trivially on 𝔪_{x,X}/𝔪_{x,X}^{d+1} for every d.

Take an element f ∈ 𝔪_{x,X}. Since G is finite, the k-linear span ⟨K · f⟩ of the K-orbit of f in 𝔪_{x,X} is finite-dimensional. This and ⋂_{s} 𝔪_{x,X}^{s} = {0} (see, e.g., [AM1969, Cor. 10.18]) implies that ⟨K · f⟩ ∩ 𝔪_{x,X}^{d+1} = {0} for some d. Since f −g ·f ∈ 𝔪_{x,X}^{d+1} for every element g ∈ K, we conclude that f = g ·f, i.e., f is K-invariant. Thus, K acts trivially on 𝔪_{x,X}, hence on O_{x,X} as well. Since k(X) is the field of fractions of O_{x,X}, K acts trivially on k(X), and therefore, on X. But K acts on X faithfully because K ⊆ Aut(X). This proves that K is trivial. Since L is the dual of the G-module T_{x,X}, this completes the proof. □

The idea of the method is to use the fact that if a finite subgroup G of Aut(X) has a fixed point x ∈ X, then, by Lemma 4 and Theorem 1, there is a normal abelian subgroup of G whose index in G is at most J_{GL_{n}(k)} for n = dim T_{x,X}.

This yields the following:

Theorem 7. Let X be an irreducible variety and let G be a finite subgroup of Aut(X). If G has a fixed point in X, then there is a normal abelian subgroup of G whose index in G is at most J_{GL_{m}(k)}, where

m = max_{x} dim T_{x,X}.  (15)

Corollary 4. If every finite automorphism group of an irreducible variety X has a fixed point in X, then Aut(X) is Jordan and

J_{Aut(X)} ⩽ J_{GL_{m}(k)},

where m is defined by (15).

Corollary 5. Let p be a prime number. Then every finite p-subgroup G of Aut(A^{n}) contains an abelian normal subgroup whose index in G is at most J_{GL_{n}(k)}.

Proof. This follows from Theorem 7 since in this case (A^{n})^{G} ≠ ∅, see [Se2009_{2}, Thm. 1.2]. □

Remark 5. At this writing (July 2013), it is unknown whether or not (A^{n})^{G} ≠ ∅ for every finite subgroup G of Aut(A^{n}). By Theorem 7 the affirmative answer would imply that Aut(A^{n}) is Jordan (cf. Subsection (2.2.2)).

Remark 6. The statement of Corollary 5 remains true if A^{n} is replaced by any p-acyclic variety X, and n in J_{GL_{n}(k)} is replaced by m (see (15)). This is because in this case X^{G} ≠ ∅ for every finite p-subgroup G of Aut(X), see [Se2009_{2}, Sect. 7–8].

The following applications are obtained by combining the above idea with Theorem 4.

Theorem 8. Let X be an irreducible variety. Consider an Aut(X)-stable equivalence relation ∼ on the set its points. If there is a finite equivalence class C of ∼, then Aut(X) is Jordan and

J_{Aut(X)} ⩽ |C|!J_{GL_{m}(k)}^{|C|!},

where m is defined by (15).

Proof. By the assumption, every equivalence class of ∼ is Aut(X)-stable. The kernel K of the action of Aut(X) on C is a normal subgroup of Aut(X) and, since the elements of Aut(X) induce permutations of C,

[Aut(X):K] ⩽ |C|!.  (16)

By Theorem 4, Jordaness of Aut(X) follows from that of K. To prove that the latter holds, take a point of x ∈ C. Since x is fixed by every finite subgroup of K, Theorem 7 implies that K is Jordan and J_{K} ⩽ J_{GL_{m}(k)}. By Theorem 4, this and (16) imply the claim. □

Example 11. Below are several examples of Aut(X)-stable equivalence relations on an irreducible variety X:

(i) x ∼ y ⟺ O_{x,X} and O_{y,X} are k-isomorphic; (ii) x ∼ y ⟺ dim T_{x,X} = dim T_{y,X}; (iii) x ∼ y ⟺ the tangent cones of X at x and y are isomorphic. □

Corollary 6. If an irreducible variety X has a point x such that the set

{y ∈ X | O_{x,X} and O_{y,X} are k-isomorphic}

is finite, then Aut(X) is Jordan.

Call a point x ∈ X a vertex of X if

dim T_{x,X} ⩾ dim T_{y,X} for every point y ∈ X.

Thus every point of X is a vertex of X if and only if X is smooth.

Corollary 7. The automorphism group of every irreducible variety with only finitely many vertices is Jordan.

Corollary 8. The automorphism group of every nonsmooth irreducible variety with only finitely many singular points is Jordan.

Corollary 9. Let X ⊂ A^{n} be the affine cone of a smooth closed proper irreducible subvariety Z of P^{n−1} that does not lie in any hyperplane. Then Aut(X) is Jordan.

Proof. The assumptions imply that the singular locus of X consists of a single point, the origin; whence the claim by Corollary 8. □

Corollary 10. If an irreducible variety X has a point x such that there are only finitely many points y ∈ X for which the tangent cones of X at x and at y are isomorphic, then Aut(X) is Jordan.

Remark 7. Smoothness in Corollary 9 may be replaced by the assumption that Z is not a cone. Indeed, in this case the origin constitutes a single equivalence class of equivalence relation (iii) in Example 11; whence the claim by Corollary 10.

2.2.4. The Koras–Russell threefolds. Let X = X_{d,s,l} be the so-called Koras– Russell threefold of the first kind [M-J2011], i.e., the smooth hypersurface in A^{4} defined by the equation

x_{1}^{d}x_{2} + x_{3}^{s} + x_{4}^{l} + x_{1} = 0,

where d ⩾ 2 and 2 ⩽ s ⩽ l with s and l relatively prime; the case d = s = 2 and l = 3 is the famous Koras–Russell cubic. According to [M-J2011, Cor. 6.1], every element of Aut(X) fixes the origin (0, 0, 0, 0) ∈ X. By Corollary 4 and item (iii) of Subsection 1.2.1 this implies that Aut(X) is Jordan and

J_{Aut(X)} ⩽ 360.

Actually, during the conference I learned from L. Moser-Jauslin that X contains a line ℓ passing through the origin, stable with respect to Aut(X), and such that every element of Aut(X) fixing ℓ pointwise has infinite order. This implies that every finite subgroup of Aut(X) is cyclic and hence

J_{Aut(X)} = 1.

2.2.5. Small dimensions. Since Aut(X) is a subgroup of Bir(X), Jordaness of Bir(X) implies that of Aut(X). This and Theorem 12 below yield the following

Theorem 9. Let X be an irreducible variety of dimension ⩽ 2 not birationally isomorphic to P^{1} × E, where E is an elliptic curve. Then Aut(X) is Jordan.

Note that if E is an elliptic curve and X = P^{1} × E, then Aut(X) = PGL_{2}(k) × Aut(E), see [Ma1971, pp. 98–99]. Fixing a point of E, endow E with a structure of abelian variety E_{ab}. Since Aut(E) is an extension of the finite group Aut(E_{ab}) by the abelian group E_{ab}, Theorems 2, 4, and 3(2) imply that Aut(P^{1} × E) is Jordan.

Note also that all irreducible curves (not necessarily smooth and projective) whose automorphism group is infinite are classified in [Po1978].

2.2.6. Non-uniruled varieties. Again, using that Jordaness of Bir(X) implies that of Aut(X), we deduce from recent Theorem 15(i)(a) below the following

Theorem 10. Aut(X) is Jordan for any irreducible non-uniruled variety X.

2.3. Groups Bir(X). Now we shall consider Problem B (see Subsection 2.1). Exploring Bir(X), one may, maintaining this group, replace X by any variety birationally isomorphic to X. Note that by Theorem 6 one can always attain that after such a replacement Aut(X) becomes Jordan.

The counterpart of Question 1 is

Question 2 ([Po2010, Quest. 2.31]). Is there an irreducible variety X such that Bir(X) is non-Jordan?

In contrast to the case of Question 1, at present we know the answer to Question 2: motivated by my question, Yu. Zarhin proved in [Za2010] the following

Theorem 11 ([Za2010, Cor. 1.3]). Let X be an abelian variety of positive dimension and let Z be a rational variety of positive dimension. Then Bir(X × Z) is non-Jordan.

Sketch of proof. By Theorem 3(1)(i), it suffices to prove that Bir(X×A^{1}) is non-Jordan. Consider an ample divisor D on X and the sheaf L:= O_{X}(D). For a positive integer n, consider the following group Θ(L^{n}). Its elements are all pairs (x, [f]) where x∈X is such that L^{n} ≅T_{x}^{∗}(L^{n}) for the translation T_{x}: X → X, z ↦ z + x, and [f] is the automorphism of the additive group of k(X) induced by the multiplication by f ∈ k(X)^{∗}. The group structure of Θ(L^{n}) is defined by (x, [f])(y, [h]) = (x + y, [T_{x}^{∗}h · f]). One proves that Θ(L^{n}) enjoys the properties: (i) φ: Θ(L^{n}) → Bir(X ×A^{1}), φ(x, [f])(y, t) = (x+y, f(y)t), is a group embedding; (ii) Θ(L^{n}) is isomorphic to a group G_{K} from Example 5 with |K|⩾n. This implies the claim (see Example 5). □

Below Problem B is solved for varieties of small dimensions (⩽ 2).

2.3.1. Curves. If X is a curve, then the answer to Question 2 is negative.

Proving this, we may assume that X is smooth and projective; whence Bir(X)=Aut(X).

If g(X), the genus of X, is 0, then X = P^{1}, so Aut(X) = PGL_{2}(k). Hence Aut(X) is Jordan by Theorem 2.

If g(X) = 1, then X is an elliptic curve, hence Aut(X) is Jordan (see the penultimate paragraph in Subsection 2.2.5).

If g(X) ⩾ 2, then, being finite, Aut(X) is Jordan.

2.3.2. Surfaces. Answering Question 2 for surfaces X, we may assume that X is a smooth projective minimal model.

If X is of general type, then by Matsumura’s theorem Bir(X) is finite, hence Jordan.

If X is rational, then Bir(X) is Cr_{2}, hence Jordan, see Example 1.

If X is a nonrational ruled surface, it is birationally isomorphic to P^{1} ×B where B is a smooth projective curve such that g(B) > 0; we may then take X = P^{1} × B. Since g(B) > 0, there are no dominant rational maps P^{1} −−→ B; whence the elements of Bir(X) permute fibers of the natural projection P^{1}×B → B. The set of elements inducing trivial permutation is a normal subgroup Bir_{B}(X) of Bir(X). The definition implies that Bir_{B}(X) = PGL_{2}(k(B)), hence Bir_{B}(X) is Jordan by Theorem 2. Identifying Aut(B) with the subgroup of Bir(X) in the natural way, we get the decomposition

Bir(X) = Bir_{B}(X) ⋊ Aut(B).  (17)

If g(B) ⩾ 2, then Aut(B) is finite; whence Bir(X) is Jordan by virtue of (17) and Theorem 4. If g(B)=1, then Bir(X) is non-Jordan by Theorem 11.

The canonical class of all the other surfaces X is numerically effective, so, for them, Bir(X) = Aut(X), cf. [IS1996, Sect. 7.1, Thm. 1 and Sect. 7.3, Thm. 2].

Let X be such a surface. The group Aut(X) has a structure of a locally algebraic group with finite or countably many components, see [Ma1958], i.e., there is a normal subgroup Aut(X)^{0} in Aut(X) such that

(i) Aut(X)^{0} is a connected algebraic group, (ii) Aut(X)/Aut(X)^{0} is either a finite or a countable group,

By (i) and the structure theorem on algebraic groups [Ba1955], [Ro1956] there is a normal connected affine algebraic subgroup L of Aut(X)^{0} such that Aut(X)^{0}/L is an abelian variety. By [Ma1963, Cor. 1] nontriviality of L would imply that X is ruled. Since we assumed that X is not ruled, this means that Aut(X)^{0} is an abelian variety. Hence Aut(X)^{0} is abelian and, a fortiori, Jordan.

By (i) the group Aut(X)^{0} is contained in the kernel of the natural action of Aut(X) on H^{2}(X, Q) (we may assume that k = C). Therefore, this action defines a homomorphism Aut(X)/Aut(X)^{0} → GL(H^{2}(X, Q)). The kernel of this homomorphism is finite by [Do1986, Prop. 1], and the image is bounded by Example 8. By Examples 6, 7 this yields that Aut(X)/Aut(X)^{0} is bounded. In turn, since Aut(X)^{0} is Jordan, by Theorem 4 this implies that Aut(X) is Jordan.

2.3.3. The upshot. The upshot of the last two subsections is

Theorem 12 ([Po2010, Thm. 2.32]). Let X be an irreducible variety of dimension ⩽ 2. Then the following two properties are equivalent:

(a) the group Bir(X) is Jordan;

(b) the variety X is not birationally isomorphic to P^{1} × B, where B is an elliptic curve.

2.3.4. Finite and connected algebraic subgroups of Bir(X) and Aut(X). Recall that the notions of algebraic subgroup of Bir(X) and Aut(X) make sense, and every algebraic subgroup of Aut(X) is that of Bir(X), see, e.g., [Po2011, Sect. 1]. The following reveals a relation between embeddability of finite subgroups of Bir(X) in connected affine algebraic subgroups of Bir(X) and Jordaness of Bir(X) (and the same holds for Aut(X)).

For every integer n > 0, consider the set of all isomorphism classes of connected reductive algebraic groups of rank ⩽ n, and fix a group in every class. The obtained set of groups R_{n} is finite. Therefore,

J_{⩽n} := sup_{G∈R_{n}}J_{G}  (18)

is a positive integer.

Theorem 13. Let X be an irreducible variety of dimension n. Then every finite subgroup G of every connected affine algebraic subgroup of Bir(X) has a normal abelian subgroup whose index in G is at most J_{⩽n}.

Proof. Let L be a connected affine algebraic subgroup of Bir(X) containing G. Being finite, G is reductive. Let R be a maximal reductive subgroup of L containing G. Then L is a semidirect product of R and the unipotent radical of L, see [Mo1956, Thm. 7.1]. Therefore, R is connected because L is. Faithfulness of the action R acts on X yields that rk R ⩽ dim X, see, e.g., [Po2011, Lemma 2.4]. The claim then follows from (18), Theorem 2, and Definition 1. □

Theorem 13 and Definition 1 imply

Corollary 11. Let X be an irreducible variety of dimension n such that Bir(X) (resp. Aut(X)) is non-Jordan. Then for every integer d >J_{⩽n}, there is a finite subgroup G of Bir(X) (resp. Aut(X)) with the properties:

(i) G does not lie in any connected affine algebraic subgroup of Bir(X) (resp. Aut(X));

(ii) G contains an abelian normal subgroup whose index in G is ⩾ d.

Corollary 12. If Cr_{n} (resp. Aut(A^{n})) is non-Jordan, then for every integer d > J_{⩽n}, there is a finite subgroup G of Cr_{n} (resp. Aut(A^{n})) with the properties:

(i) the action of G on A^{n} is nonlinearizable;

(ii) G contains an abelian normal subgroup whose index in G is ⩾ d.

Proof. This follows from Corollary 11 because GL_{n}(k) is a connected affine algebraic subgroup of Aut(A^{n}) and nonlinearizability of the action of G on A^{n} means that G is not contained in a subgroup of Cr_{n} (resp. Aut(A^{n})) conjugate to GL_{n}(k). □

2.3.5. Recent developments. The initiated in [Po2010] line of research of Jordaness of Aut(X) and Bir(X) for algebraic varieties X has generated interest of algebraic geometers in Moscow among whom I have promoted it, and led to a further progress in Problem B (hence A as well) in papers [Za2010], [PS2013_{1}], [PS2013_{2}]. In [Za2010], the earliest of them, the examples of non-Jordan groups Bir(X) only known to date (July 2013) have been constructed (see Theorem 11 above). Below are formulated the results obtained in [PS2013_{1}], [PS2013_{2}]. Some of them are conditional, valid under the assumption that the following general conjecture by A. Borisov, V. Alexeev, and L. Borisov holds true:

BAB Conjecture. All Fano varieties of a fixed dimension n and with terminal singularities are contained in a finite number of algebraic families.

Theorem 14 ([PS2013_{1}, Thm. 1.8]). If the BAB Conjecture holds true in dimension n, then, for every rationally connected n-dimensional variety X, the group Bir(X) is Jordan and, moreover, J_{Bir(X)} ⩽ u_{n} for a number u_{n} depending only on n.

Since for n = 3 the BAB Conjecture is proved [KMMT2000], this yields

Corollary 13 ([PS2013_{1}, Cor. 1.9]). The space Cremona group Cr_{3} is Jordan.

Proposition 1 ([PS2013_{1}, Prop. 1.11]). u_{3} ⩽ (25920 · 20736)^{20736}.

The pivotal idea of the proof of Theorem 14 is to use a technically refined form of the “fixed-point method” described in Subsection 2.2.3.

Theorem 15 ([PS2013_{2}, Thm. 1.8]). Let X be an irreducible n-dimensional variety.

(i) The group Bir(X) is Jordan in either of the cases: (a) X is non-uniruled;

(b) the BAB Conjecture holds true in dimension n and the irregula-

rity of X is 0. (ii) If X is non-uniruled and its irregularity is 0, then the group Bir(X)

is bounded (see Definition 2).

3. Appendix: Problems

Below I add a few additional problems to those which have already been formulated above (Problems A and B in Subsection 2.1, and Questions 1, 2).

3.1. Cr_{n}-conjugacy of finite subgroups of GL_{n}(k). Below GL_{n}(k) is identified in the standard way with the subgroup of Cr_{n}, which, in turn, is identified with the subgroup of Cr_{m} for any m = n + 1, n + 2, …, ∞ (cf. [Po2011, Sect. 1] or [Po2012_{1}, Sect. 1]).

Question 3. Consider the following properties of two finite subgroups A and B of GL_{n}(k):

(i) A and B are isomorphic,

(ii) A and B are conjugate in Cr_{n}. Does (i) imply (ii)?

Comments.

1. Direct verification based on the classification in [DI2009] shows that the answer is affirmative for n ⩽ 2.

2. By [Po2012_{1}, Cor. 5], if A and B are abelian, then the answer is affirmative for every n.

3. If A and B are isomorphic, then they are conjugate in Cr_{2n}. This is the corollary of the following stronger statement:

Proposition 2. For any finite group G and any injective homomorphisms

\xymatrix{G \ar[r]^{α_{1}} \ar[r]_{α_{2}} & GL_{n}(k),}  (19)

there exists an element φ ∈ Cr_{2n} such that α_{1} = Int(φ) ∘ α_{2}.

Proof. Every element g ∈ GL_{n}(k) is a linear transformation x ↦ g · x of A^{n} (with respect to the standard structure of k-linear space on A^{n}). The injections α_{1} and α_{2} determine two faithful linear actions of G on A^{n}: the ith action (i = 1, 2) maps (g, x) ∈ G × A^{n} to α_{i}(g) · x. Consider the product of these actions, i.e., the action of G on A^{n} × A^{n} defined by

G × A^{n} × A^{n} → A^{n} × A^{n}, (g, x, y) ↦ (α_{1}(g) · x, α_{2}(g) · y).  (20)

The natural projection of A^{n} × A^{n} → A^{n} to the ith factor is G-equivariant. By classical Speiser’s Lemma (see [LPR2006, Lemma 2.12] and references therein), this implies that A^{n} × A^{n} endowed with G-action (20) is Gequivariantly birationally isomorphic to A^{n}×A^{n} endowed with the G-action via the ith factor by means of α_{i}. Therefore, A^{n} × A^{n} endowed with the Gaction via the first factor by means of α_{1} is G-equivariantly birationally isomorphic to A^{n} × A^{n} endowed with the G-action via the second factor by means of α_{2}; whence the claim. □

Remark 8. In general, it is impossible to replace Cr_{2n} by Cr_{n} in Proposition 2. Indeed, in [RY2002] one finds the examples of finite abelian groups G and embeddings (19) such that α_{1} ∉ Int(Cr_{n}) ∘ α_{2}. However, since the images of these embeddings are isomorphic finite abelian subgroups of GL_{n}(k), by [Po2012_{1}, Cor. 5] they are conjugate in Cr_{n}.

3.2. Torsion primes. Let X be an irreducible variety. The following definition is based on the fact that the notion of algebraic torus in Bir(X) makes sense.

Definition 4 ([Po2012_{1}, Sect. 8]). Let G be a subgroup of Bir(X). A prime integer p is called a torsion number group of G if there exists a finite abelian p-subgroup of G that does not lie in any torus of G.

Let Tors(G) be the set of all torsion primes of G. If G is a connected reductive algebraic subgroup of Bir(X), this set coincides with that of the torsion primes of algebraic group G in the sense of classical definition, cf., e.g., [Se2000, 1.3].

Question 4 ([Po2012_{1}, Quest. 3]). What are, explicitly,

Tors(Cr_{n}), Tors(Aut A^{n}), Tors(Aut^{∗}A^{n}), n = 3, 4, …, ∞

where Aut^{∗}A^{n} := {g ∈ Aut A^{n} | Jac (g) = 1}?

Comments. By [Po2012_{1}, Sect. 8],

Tors(Cr_{1}) = {2},
Tors(Cr_{2}) = {2, 3, 5} (this coincides with Tors(E_{8})),
Tors(Cr_{n}) ⊇ {2, 3} for any n ⩾ 3,
Tors(Aut A^{n}) = Tors(Aut^{∗}A^{n}) = ∅ for n ⩽ 2.

Question 5 ([Po2012_{1}, Quest. 4]). What is the minimal n such that 7 lies in one of the sets Tors(Cr_{n}), Tors(AutA^{n}), Tors(Aut^{∗}A^{n})?

Question 6 ([Po2012_{1}, Quest. 5]). Are these sets finite?

Question 7. Are the sets

⋃_{n⩾1} Tors(Cr_{n}), ⋃_{n⩾1} Tors(AutA^{n}), ⋃_{n⩾1} Tors(Aut^{∗}A^{n})

finite?

3.3. Embeddability in Bir(X). Not every group G is embeddable in Bir(X) for some X. For instance, by [Co2013_{2}, Thm. 1.2], if G is finitely generated, its embeddability in Bir(X) implies that G has a solvable word problem. Another example: by [Ca2012], PGL_{∞}(k) is not embeddable in Bir(X) for k = C (I thank S. Cantat who informed me about these examples).

If Bir(X) is Jordan, then by Example 4 and Theorem 3(1)(i), N is not embeddable in Bir(X). Hence, by Theorem 15(i)(a), N is not embeddable in Bir(X) for any non-uniruled X.

Conjecture 1. The finitely generated group N defined in Example 4 is not embeddable in Bir(X) for every irreducible variety X.

3.4. Contractions. Developing the classical line of research, in recent years were growing activities aimed at description of finite subgroups of Bir(X) for various X; the case of rational X (i.e., that of the Cremona group Bir(X)) was, probably, most actively explored with culmination in the classification of finite subgroups of Cr_{2}, [DI2009]. In these studies, all the classified groups appear in the corresponding lists on equal footing. However, in fact, some of them are “more basic” than the others because the latter may be obtained from the former by a certain standard construction. Given this, it is natural to pose the problem of describing these “basic” groups.

Namely, let X_{1} and X_{2} be the irreducible varieties and let G_{i} ⊂ Bir(X_{i}), i = 1, 2, be the subgroups isomorphic to a finite group G. Assume that fixing the isomorphisms G → G_{i}, i = 1, 2, we obtain the rational actions of G on X_{1} and X_{2} such that there is a G-equivariant rational dominant map φ: X_{1} −−→ X_{2}. Let π_{X_{i}}: X_{i} −−→ X_{i}---G, i = 1, 2 be the rational quotients (see, e.g. [Po2011, Sect. 1]) and let φ_{G}: X_{1}---G −−→ X_{2}---G be the dominant rational map induced by φ. Then the following holds (see, e.g. [Re2000, Sect. 2.6]):

(1) The appearing commutative diagram

\xymatrix{X_{1} \ar@{-->}[r]^{φ} \ar@{-->}[d]_{π_{X_{1}}} & X_{2} \ar@{-->}[d]^{π_{X_{2}}} \\ X_{1}---G \ar@{-->}[r]^{φ_{G}} & X_{2}---G}  (21)

is, in fact, cartesian, i.e., π_{X_{1}} : X_{1} −−→ X_{1}---G is obtained from π_{X_{2}} : X_{2} −−→ X_{2}---G by the base change φ_{G}. In particular, X_{1} is birationally G-isomorphic

to X_{2} ×_{X_{2}---G} (X_{1}---G) := \overline{\{(x, y) ∈ X_{2} × (X_{1}---G) | π_{X_{2}}(x) = φ_{G}(y)\}}.

(2) For every irreducible variety Z and every dominant rational map β: Z → X_{2}---G such that X_{2} ×_{X_{2}---G} Z is irreducible, the variety X_{2} ×_{X_{2}---G} Z

inherits via X_{2} a faithful rational action of G such that one obtains commutative diagram (21) with X_{1} = X_{2} ×_{X_{2}---G} Z, φ_{G} = β, and φ = pr_{1}.

If such a φ exists, we say that G_{1} is induced from G_{2} by a base change. The latter is called trivial if φ is a birational isomorphism. If a finite subgroup G of Bir(X) is not induced by a nontrivial base change, we say that G is incompressible.

Example 12. The standard embedding Cr_{n} ↪ Cr_{n+1} permits to consider the finite subgroups of Cr_{n} as that of Cr_{n+1}. Every finite subgroup of Cr_{n+1} obtained this way is induced by the nontrivial base change determined by the projection A^{n+1} → A^{n}, (a_{1}, …, a_{n}, a_{n+1}) ↦ (a_{1}, …, a_{n}, a_{n+1}). □

Example 13. This is Example 6 in [Re2004]. Let G be a finite group that does not embed in Bir(Z) for any curve Z of genus ⩽ 1 (for instance, G = Sym_{5}) and let X be a smooth projective curve of minimal possible genus such that G is isomorphic to a subgroup of Aut(X). Then this subgroup of Bir(X) is incompressible.^{1}

Example 14. Consider two rational actions of G := Sym_{3} × ℤ/2ℤ on A^{3}. The subgroup Sym_{3} acts by natural permuting the coordinates in both cases. The nontrivial element of Z/2ℤ acts by (a_{1}, a_{2}, a_{3}) ↦ (−a_{1}, −a_{2}, −a_{3}) in the first case and by (a_{1}, a_{2}, a_{3}) ↦ (a_{1}^{−1}, a_{2}^{−1}, a_{3}^{−1}) in the second. The surfaces

P := {(a_{1}, a_{2}, a_{3}) ∈ A^{3} | a_{1} + a_{2} + a_{3} = 0},
T := {(a_{1}, a_{2}, a_{3}) ∈ A^{3} | a_{1}a_{2}a_{3} = 1}

are G-stable in, resp., the first and the second case. Since P and T are rational, these actions of G on P and T determine, up to conjugacy, resp., the subgroups G_{P} and G_{T} of Cr_{2}, both isomorphic to G. By [Is2003] (see also [LPR2006], [LPR2007]), these subgroups are not conjugate in Cr_{2}. However, by [LPR2007, Sect. 5], G_{T} is induced from G_{P} by a nontrivial base change (of degree 2). □

1The proof in [Re2004] should be corrected as follows. Assume that there is a faithful action of G of a smooth projective curve Y and a dominant G-equivariant morphism φ: X → Y of degree n > 1. By the construction, X and Y have the same genus g > 1, and the Hurwitz formula yields that the number of branch points of φ (counted with positive multiplicities) is the integer (n − 1)(2 − 2g). But the latter is negative, — a contradiction.

In fact, Example 14 is a special case (related to the simple algebraic group G_{2}) of the following

Example 15. Let G be a connected reductive algebraic group. Fix a maximal torus T of G and consider the natural actions of the Weyl group W = N_{G}(T)/T on T and on 𝔱 := Lie(T). Since these actions are faithful and T and 𝔱 are rational varieties, this determines, up to conjugacy, two embeddings of W in Cr_{r}, where r = dim T. Let W_{T} and W_{𝔱} be the images of these embeddings. By [LPR2006, Lemma 3.5(a) and Sect. 1.5], if G is not Cayley and W has no outer autormorphisms, then W_{T} and W_{𝔱} are not conjugate in Cr_{r}. On the other hand, by [LPR2006, Lemma 10.3], W_{T} is induced from W_{𝔱} by a (nontrivial) base change (see also Lemma 5 below).

This yields, for arbitrary n, the examples of isomorphic nonconjugate finite subgroups of Cr_{n} one of which is induced from the other by a nontrivial base change. For instance, if G = SL_{n+1}, then r = n and W = Sym_{n}. Since, by [LPR2006, Thm. 1.31], SL_{n+1} is not Cayley for n ⩾ 3 and Sym_{n} has no outer automorphisms for n ≠ 6, the above construction yields for these n two nonconjugate subgroups of Cr_{n} isomorphic to Sym_{n}, one of which is induced from the other by a nontrivial base change. □

The following gives a general way of constructing two finite subgroups of Cr_{n} one of which is induced from the other by a base change.

Consider an n-dimensional irreducible nonsingular variety X and a finite subgroup G of Aut(X). Suppose that x ∈ X is a fixed point of G. By Lemma 4, the induced action of G on the tangent space of X at x is faithful. Therefore this action determines, up to conjugacy, a subgroup G_{1} of Cr_{n} isomorphic to G. On the other hand, if X is rational, the action of G on X determines, up to conjugacy, another subgroup G_{2} of Cr_{n} isomorphic to G.

Lemma 5. G_{2} is induced from G_{1} by a base change.

Proof. By Lemma 3 we may assume that X is affine, in which case the claim follows from [LPR2006, Lemma 10.3]. □

Corollary 14. Let X be a nonrational irreducible variety and let G be an incompressible finite subgroup of Aut(X). Then X^{G} = ∅.

Question 8. Which finite subgroups of Cr_{2} are incompressible?

References

[Ad2011] S. I. Adian, The Burnside problem and related topics, Russian Math. Surveys 65 (2011), no. 5, 805–855.

[Ad2013] S. I. Adian, New bounds of odd periods for which we can prove the infinity of free Burnside groups, talk at the Internat. conf. “Contemporary Problems of Mathematics, Mechanics, and Mathematical Physics”, Steklov Math. Inst.

             RAS, May 13, 2013, Moscow, http://www.mathnet.ru/php/presentation.
             phtml?&presentid=6786&option lang=eng.

[AM1969] M. F. Atiyah, I. G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley, Ma, 1969.

[Ba1955] I. Barsotti, Structure theorems for group varieties, Ann. Mat. Pura Appl. (4) 38 (1955), 77–119.

[Bo1991] A. Borel, Linear Algebraic Groups, 2nd ed., Springer-Verlag, New York, 1991.

[CD2009] D. Cerveau, J. Deserti, Transformations birationnelles de petit degreé,

arXiv: 0811.2325 (April 2009).

[Ca2012] S. Cantat, Morphisms between Cremona groups and a characterization of rational varieties, preprint available at http://perso.univ-rennes1.

fr/serge.cantat/publications.html (2012).

[Co2013_{1}] Y. Cornulier, Nonlinearity of some subgroups of the planar Cremona group,

preprint, http://www.normalesup.org/∼cornulier/crelin.pdf (February

2013).

[Co2013_{2}] Y. Cornulier, Sofic profile and computability of Cremona groups, arXiv:

1305.0993 (May 2013).

[Co2007] M. J. Collins, On Jordans theorem for complex linear groups, J. Group Theory 10 (2007), 411–423.

[CR1962] C. W. Curtis, I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Wiley, New York, 1962.

[Do1986] I. Dolgachev, Infinite Coxeter groups and automorphisms of algebraic surfaces, Contemp. Math. 58 (1986), Part 1, 91–106.

[DI2009] I. Dolgachev, V. Iskovskikh, Finite subgroups of the plane Cremona group, in: Algebra, Arithmetic, and Geometry In Honor of Yu. I. Manin, Progress in Mathematics, Vol. 269, Birkhäuser Boston, Boston, MA, 2009, 443–548.

[Fi2011] D. Fisher, Groups acting on manifolds: around the Zimmer program, in: Geometry, Rigidity, and Group Actions, Chicago Lectures in Math., Univ. Chicago Press, Chicago, IL, 2011, pp. 72–157.

[Fu1993] W. Fulton, Introduction to Toric Varieties, Annals of Mathematics Studies, Vol. 131, Princeton University Press, Princeton, NJ, 1993.

[Hu1998] B. Huppert, Character Theory of Finite Groups, De Gruyter Expositions in Mathematics, Vol. 25, Walter de Gruyter, Berlin, 1998.

[Ig1977] T. Igarashi, Finite Subgroups of the Automorphism Group of the Affine Plane, M.A. thesis, Osaka University, 1977.

[Is2003] V. A. Iskovskikh, Two non-conjugate embeddings of Sym_{3} × ℤ_{2} into the Cremona group, Proc. Steklov Inst. of Math. 241 (2003), 93–97.

[IS1996] V. A. Iskovskikh, I. R. Shafarevich, Algebraic surfaces, in: Algebraic Geometry, II, Encyclopaedia Math. Sci., Vol. 35, Springer, Berlin, 1996, pp. 127– 262.

[Jo1878] C. Jordan, Mémoire sur les equations différentielle linéaire à intégrale algébrique, J. Reine Angew. Math. 84 (1878), 89–215.

[KMMT2000] J. Kollár, Y. Miyaoka, S. Mori, H. Takagi, Boundedness of canonical ℚ-Fano 3-folds, Proc. Japan Acad. Ser. A Math. Sci. 76 (2000), 73–77.

[La1965] S. Lang, Algebra, Addison-Wesley, Reading, Mass., 1965.

[LPR2006] N. Lemire, V. Popov, Z. Reichstein, Cayley groups, J. Amer. Math. Soc. 19 (2006), no. 4, 921–967.

[LPR2007] N. Lemire, V. Popov, Z. Reichstein, On the Cayley degree of an algebraic group, Proceedings of the XVIth Latin American Algebra Colloquium, Bibl. Rev. Mat. Iberoamericana, Rev. Mat. Iberoamericana, Madrid, 2007, 87– 97.

[Ma1971] M. Maruyama, On automorphisms of ruled surfaces, J . Math. Kyoto Univ. 11-1 (1971), 89–112.

[Ma1963] H. Matsumura, On algebraic groups of birational tansformations, Rend. Accad. Naz. Lincei, Ser. VIII 34 (1963), 151–155.

[Ma1958] T. Matsusaka, Polarized varieties, fields of moduli and generalized Kummer varieties of polarized varieties, Amer. J. Math. 80 (1958), 45–82.

[M-J2011] L. Moser-Jauslin, Automorphism groups of Koras–Russell threefolds of the first kind, in: Affine Algebraic Geometry, CRM Proc. Lecture Notes, Vol. 54, Amer. Math. Soc., Providence, RI, 2011, pp. 261–270.

[Mo1956] G. Mostow, Fully reducible subgroups of algebraic groups, Amer. J. Math. 78 (1956), 200–221.

[MiR2010] I. Mundet i Riera, Jordan’s theorem for the diffeomorphism group of some manifolds, Proc. Amer. Math. Soc. 138 (2010), no. 6, 2253–2262.

[Po1978] V. L. Popov, Algebraic curves with an infinite automorphism group, Math. Notes 23 (1978), 102–108.

[Po2010] V. L. Popov, On the Makar-Limanov, Derksen invariants, and finite automorphism groups of algebraic varieties, in: Affine Algebraic Geometry: The Russell Festschrift, CRM Proceedings and Lecture Notes, Vol. 54, Amer. Math. Soc., 2011, pp. 289–311, arXiv:1001.1311 (January 2010).

[Po2011] V. L. Popov, Some subgroups of the Cremona groups in: Affine Algebraic Geometry, Proceedings (Osaka, Japan, 36 March 2011), World Scientific, Singapore, 2013, pp. 213–242, arXiv:1110.2410 (October 2011).

[Po2012_{1}] V. L. Popov, Tori in the Cremona groups, to appear in Izvestiya: Mathe-

matics 77 (2013), no. 4, arXiv:1207.5205 (July 2012).

[Po2012_{2}] V. L. Popov, Problems for the problem session, CIRM Trento, http://www.

science.unitn.it/cirm/Trento postersession.html (November 2012).

[PS2013_{1}] Y. Prokhorov, C. Shramov, Jordan property for Cremona groups, arXiv:

1211.3563 (June 2013).

[PS2013_{2}] Y. Prokhorov, C. Shramov, Jordan property for groups of birational self-

maps, arXiv:1307.1784 (July 2013).

[Pu2007] V. Puppe, Do manifolds have little symmetry?, J. Fixed Point Theory Appl. 2 (2007), no. 1, 85–96.

[Re2000] Z. Reichstein, On the notion of essential dimension for algebraic groups, Transformation Groups 5 (2000), no. 3, 265–304.

[Re2004] Z. Reichstein, Compression of group actions, in: Invariant Theory in All Characteristics, CRM Proceedings and Lecture Notes, Vol. 35, Amererican Mathematical Society, Providence, RI, 2004, 199–202.

[RY2002] Z. Reichstein, B. Youssin, A birational invariant for algebraic group actions, Pacific J. Math. 204 (2002), 223–246.

[Ro1956] M. Rosenlicht, Some basic theorems on algebraic groups, Amer. J. Math. 78 (1956), 401–443.

[Ro1957] M. Rosenlicht, Some rationality questions on algebraic groups, Ann. Mat. Pura Appl. (4) 43 (1957), 25–50.

[Se2000] J-P. Serre, Sous-groupes finis des groupes de Lie, in: Séminaire N. Bourbaki 1998/–99, Exp. no. 864, Astérisque, Vol. 266, Société Mathématique de France, 2000, pp. 415–430.

[Se2008] J-P. Serre, Le groupe de Cremona et ses sous-groupes finis, Séminaire Bourbaki, no. 1000, Novembre 2008, 24 pp.

[Se2009_{1}] J-P. Serre, A Minkowski-style bound for the orders of the finite subgroups of the Cremona group of rank 2 over an arbitrary field, Moscow Math. J. 9 (2009), no. 1, 183–198.

[Se2009_{2}] J-P. Serre, How to use finite fields for problems concerning infinite fields,

Contemporary Math. 487 (2009), 183–193.

[Sp1998] T. A. Springer, Linear Algebraic Groups. 2nd ed., Progress in Mathematics, Vol. 9, Birkhäuser, Boston, 1998.

[Za2010] Y. G. Zarhin, Theta groups and products of abelian and rational varieties, to appear in Proc. Edinb. Math. Soc., arXiv:1006.1112 (June 2010).