Invariant Measures for Large Automorphism Groups of Projective Surfaces

Abstract

We classify invariant probability measures for non-elementary groups of automorphisms, on any compact Kähler surface X, under the assumption that the group contains a so-called “parabolic automorphism”. We also prove that except in certain rigid situations known as Kummer examples, there are only finitely many invariant, ergodic, probability measures with a Zariski dense support. If X is a K3 or Enriques surface, and the group does not preserve any algebraic subset, this leads to a complete description of orbit closures.

Appendix: . Abelian Surfaces

In this appendix, we consider the case when all parabolic automorphisms g of Γ induce an automorphism g_B of infinite order on the base of their invariant fibration π_g. In that case, we know from [14, Proposition 3.6] that X is a compact torus, and in fact an abelian surface since Γ is non-elementary. Thus, we assume that

• (i) X is an abelian surface, isomorphic to C²/Λ for some lattice Λ;

• (ii) Γ is a non-elementary group of automorphisms of X that contains a parabolic element g;

• (iii) every parabolic element g of Γ acts on the base of its invariant fibration π_g: X → B_g by an automorphism g_B: B_g → B_g of infinite order.

We provide an argument to complete the proof of Theorem A in that case; the strategy is the same as in Sections 4 and 5, but simpler since the dynamics is linear:

Proposition A.1

Under the above hypotheses (i), (ii), (iii), if μ is a Γ-invariant and ergodic measure, then either μ is the Haar measure on the abelian surface X, or there are finitely many subtori S_j ⊂ X of real dimension 2, and points a_j ∈ X, j = 1,…,k, such that

• (1) $\bigcup _{j} (a_{j}+S_{j})$ is Γ-invariant;

• (2) Γ permutes transitively the subsets a_j + S_j, j = 1,…,k;

• (3) μ is supported on $\bigcup _{j} (a_{j}+S_{j})$ and on each a_j + S_j, μ is given by $\frac {1}{k} m_{j}$ where m_j is the Haar measure on a_j + S_j.

Here, by Haar measure on a_j + S_j, we mean the image of the Haar measure on S_j by the translation s ∈ S_j↦a_j + s. With the results of Section 4, this proposition concludes the proof of Theorem A.

This proposition could be proved with techniques from homogeneous dynamics, but there are two subtleties: firstly the group Γ is not supposed to fix the origin of the torus so we have to deal with affine maps; secondly, the linear representation of Γ in GL₂(C) given by the linear part of its elements is not supposed to be irreducible as a subgroup of GL₄(R): totally real invariant subtori appear precisely in this case (see [15, Section 4] for this type of difficulty). Thus, we provide a proof in the spirit of this article.

1.1 A.1 Parabolic, Affine Transformations

The group Γ acts on X by affine transformations

$$ f(x,y)=A_{f}(x,y)+S_{f} \mod ({\Lambda} ) $$

(A.1)

where the linear part A_f ∈GL₂(C) preserves the lattice Λ ⊂C² and the translation part S_f is an element of C²/Λ. Now, pick a parabolic element g ∈Γ; its linear part is given by

$$ A_{g}=\left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \end{array}\right) $$

(A.2)

after a linear change of coordinates in C². In these coordinates, the fibration π_g is induced by the projection π₁: (x,y)↦x, and a conjugation by a translation reduces g to the form

$$ g(x,y)=(x+s,y+x) \mod ({\Lambda} ) $$

(A.3)

where s has infinite order in the elliptic curve B_g = C/π₁(Λ).

Lemma A.2

If the orbits of g_B: x↦x + s are dense in B_g, then g is uniquely ergodic: the unique g-invariant probability measure on X is the Haar measure.

This result is due to Furstenberg (see [30, Section 3.3]). Thus, in this case μ is the Haar measure on X and we are done. So in what follows, we assume that for every g ∈Hal(Γ) the orbits of the translation g_B are not dense: they equidistribute along circles x + T_g, where T_g is the closure of the group Zs ⊂ B_g; changing g into a positive iterate we may assume that this closure is isomorphic to R/Z (as a real Lie group). We let ℓ_g be the quotient map C²/Λ → B_g/T_g.

Lemma A.3

Every fiber of the linear projection ℓ_g is a 3-dimensional g-invariant torus, and g is uniquely ergodic on almost every fiber.

To prove Lemma A.3, we think of C² as a real vector space and fix a basis of Λ. Then C² is identified with R⁴ and Λ with Z⁴ ⊂R⁴. The eigenspace of A_g for the eigenvalue 1 is defined over Z with respect to Λ = Z⁴. Moreover, A_g acts trivially on the quotient space R⁴/Fix(A_g). Thus adapting the basis of Z⁴ to g, we may assume that

$$ g(x_{1},x_{2},x_{3},x_{4})=(x_{1}, x_{2}+s_{2}, x_{3}+ax_{1}+bx_{2}, x_{4}+cx_{1}+dx_{2}), $$

(A.4)

for some irrational number s₂ and some integers a, b, c, and d. The linear projection ℓ_g is now given by (x₁,x₂,x₃,x₄)↦x₁ and Lemma A.3 boils down to the following statement.

Lemma A.4

If 1, s₁ and s₂ are linearly independent over Q, then g is uniquely ergodic on the level set $\left \{x_{1} = s_{1}\right \}$.

Proof

Let us first observe that ad − bc≠ 0. Indeed, otherwise the linear part of g would have a fixed point set of dimension 3, which is impossible because g is holomorphic (see Eq. (A.2)). To prove unique ergodicity, we use the following criterion due to Furstenberg (see [30, Prop. 3.10]): let h be a homeomorphism on R/Z × (R²/Z²) of the form (x,y)↦(x + u,y + φ(x)), where u is irrational, then h is uniquely ergodic if and only if it is ergodic for the Haar measure. On the fiber x₁ = s₁, our map g is of the form

$$ (x_{2}, x_{3}, x_{4})\longmapsto (x_{2}+s_{2}, x_{3}+as_{1}+bx_{2}, x_{4}+cs_{1}+dx_{2}), $$

(A.5)

so we need to check that it is ergodic for the Haar measure. For this, we pick a measurable invariant subset A ⊂R³/Z³, we denote by $\mathbf 1_{A}\in L^{2}(\mathbf {R}^{3}/\mathbf {Z}^{3})$ its indicator function, and we expand it into a Fourier series $\mathbf 1_{A}(x_{2},x_{3},x_{4}) = {\sum }_{(k,\ell , m)\in \mathbf {Z}^{3}} c_{k,\ell , m} e^{2\textsf {i} \pi (kx_{2}+\ell x_{3}+ mx_{4})}$. Then

$$ \mathbf 1_{A}~\circ~ g (x_{2},x_{3},x_{4}) = \sum\limits_{(k,\ell, m)\in \mathbf{Z}^{3}} c_{k,\ell, m} e^{2{\mathsf{i}} \pi ks_{2}} e^{2{\mathsf{i}} \pi (\ell a+ mc)s_{1}} e^{2{\mathsf{i}} \pi (k+\ell b+ m d)x_{2} }e^{2{\mathsf{i}} \pi \ell x_{3}} e^{2\textsf{i} \pi m x_{4}} $$

(A.6)

and, from the g-invariance of 1_A and the uniqueness of the expansion, we get

$$ c_{k,\ell, m} = e^{2{\mathsf{i}} \pi ks_{2}} e^{2{\mathsf{i}} \pi (\ell a+ mc)s_{1}} c_{k- \ell b - md, \ell, m} $$

(A.7)

for all (k,ℓ,m) ∈Z³. For ℓ = m = 0, the irrationality of s₂ implies that c_k,0,0 = 0 unless k = 0. If ℓb + md≠ 0, iterating the relation $\left \vert c_{k,\ell , m}\right \vert = \left \vert c_{k- \ell b - md, \ell , m}\right \vert$ and using the fact that Fourier coefficients decay to zero at infinity, we infer that c_k,ℓ,m = 0. Finally, if ℓb + md = 0 and one of ℓ or m is nonzero, since ad − bc≠ 0 we get that ℓa + mc≠ 0 and Eq. (A.7) gives $c_{k,\ell , m} = e^{2{\mathsf {i}} \pi (ks_{2} + (\ell a+ mc)s_{1})} c_{k, \ell , m}$. Since 1, s₁, and s₂ are Q-linearly independent, we derive c_k,ℓ,m = 0. Thus, 1_A is a constant, which means that the Haar measure of A is 0 or 1. □

1.2 A.2 Using Distinct Parabolic Automorphisms

To complete the proof of Theorem A in the case of tori, one can now follow the same ideas as in Sections 4 and 5. We only need to replace the dimension $\dim _{\mathbf {R}}(\mu )$ by the minimal dimension of a real subtorus Z ⊂ X such that μ(q + Z) > 0 for some q ∈ X, the invariant fibration π_g by the R-linear projection ℓ_g, and the set $\mathrm {R}_{g}(B_{g}^{\circ })\subset B_{g}^{\circ }$ by

$$ \mathrm{R}(g)=\{ y \in B_{g}/T_{g} ; g \text{ is not uniquely ergodic in } \ell_{g}^{-1}(y)\} \subset B_{g}/T_{g}. $$

(A.8)

Lemma A.4 shows that R(g) is countable.

Lemma A.5

If there is a parabolic element g ∈Γ for which ((ℓ_g)_∗μ)(R(g)) < 1, then μ is the Haar measure on X.

Proof

The proof is the same as for Proposition 4.9. Pick another parabolic transformation h ∈Γ, such that ℓ_g and ℓ_h are linearly independent; such an h exist because Γ is non-elementary. The main tool is the disintegration of μ with respect to ℓ_g; for y in a subset ${\mathcal {Y}}_{g}\subset B_{g}/T_{g}$ of positive measure, the conditional measure λ_g,y is the Haar measure on the 3-dimensional torus $\ell _{g}^{-1}(y)$. Hence $\dim _{\mathbf {R}}(\mu )\geqslant 3$ and ((ℓ_g)_∗μ)(R(g)) = 0, as in Step 2 of the proof of Proposition 4.9. As in Steps 3 and 4, we infer that (ℓ_h)_∗μ does not charge R(h) and that (ℓ_h)_∗μ is absolutely continuous with respect to the Lebesgue measure on R/Z. This, implies that μ itself is invariant by all translations along the fibers of ℓ_h, because $\mu ={\int \limits }_{Y} \lambda _{h,y} d((\ell _{h})_{*}\mu )(y)$ and λ_h,y is the Haar measure for almost every y. Permuting the roles of g and h, μ is in fact invariant under all translations. Hence, μ is the Haar measure on X. □

Now, we are reduced to the case where (ℓ_g)_∗μ(R(g)) = 1 for every parabolic automorphism g in Γ. Since R(g) is countable, $d_{\mathbf {R}}(\mu )\leqslant 3$ and μ charges some fiber of ℓ_g. Using another parabolic automorphism h, we see that μ gives positive mass to a translate a₀ + S₀ of a 2-dimensional torus S₀ ⊂ X whose projections ℓ_g(a₀ + S₀) = ℓ_g(a₀) and ℓ_h(a₀) are in the countable sets R(g) ⊂ B_g/T_g and R(h) ⊂ B_h/T_h respectively. Thus, by ergodicity, we conclude that μ is supported on a finite union of translates of 2-dimensional tori a_j + S_j ⊂ X, $0\leqslant i\leqslant k-1$ for some $k\geqslant 1$.

A subgroup Γ₀ of index $\leqslant k!$ in Γ preserves a₀ + S₀, and g^k! and h^k! act on $a_{0}+ S_{0}\simeq \mathbf {R}^{2}/\mathbf {Z}^{2}$ as two linear parabolic transformations with respect to transverse linear fibrations. So, it follows that $\mu _{\vert a_{0}+S_{0}}$ is proportional to the Haar measure of a₀ + S₀, and the proof of Proposition A.1 is complete.

1.3 A.3 No or Infinitely Many Invariant Real Tori

Consider a compact complex torus X = C²/Λ of dimension 2. Let Γ be a subgroup of Aut(X). As in Appendix Appendix, write the elements f of Aut(X) in the form f(x,y) = A_f(x,y) + S_f, and denote by A_Γ ⊂GL₂(C) the image of Γ by the homomorphism f↦A_f. The group Γ is non-elementary if and only if A_Γ contains a free group, if and only if the Zariski closure of A_Γ in the real algebraic group GL₂(C) is semi-simple.

Now, assume that Γ is non-elementary and preserves at least one ergodic probability measure μ with $\dim _{\mathbf {R}}(\mu )=2$. Equivalently, after conjugation by a translation, there is a finite index subgroup Γ₀ ⊂Γ that preserves a real, two-dimensional subtorus Σ = π/Λ_π, where π ⊂C² is a real vector space of dimension 2 and Λ_R := π ∩Λ is a lattice in π (the restriction of μ to π/Λ_π is proportional to the Haar measure). The goal of this last section is to explain that, in fact, Γ preserves infinitely many ergodic measuresμ_j with $\dim _{\mathbf {R}}(\mu _{j})=2$. Two mechanisms can be used to establish this fact.

The first one relies on the fact that Γ₀ acts on the quotient $Q=X/{\Sigma }\simeq \mathbf {R}^{2}/\mathbf {Z}^{2}$, fixing the origin. Moreover, the action of Γ₀ on Q = R²/Z² is induced by an injective homomorphism Γ₀ →GL₂(Z) (to see this, note that C² = π ⊕_Riπ and iπ surjects onto Q). This implies that Γ₀ has arbitrarily large finite orbits in Q (coming from torsion points of Q). The preimages of these orbits in X provide surfaces Σ_j ⊂ X; they are “parallel” to Σ and have an arbitrarily large number of connected components; they are Γ₀-invariant; and each of them supports a unique invariant, ergodic, probability measure μ_j with $\dim _{\mathbf {R}}(\mu _{j})=2$.

For the second mechanism, we assume that Γ₀ fixes the origin and, changing Γ₀ in a finite index subgroup if necessary, we identify Γ₀ with a subgroup of SL₂(C). Identify (π,Λ_π) to (R²,Z²) and the restriction Γ₀|_π to a subgroup of GL₂(Z); since Γ₀ is non-elementary, Γ₀ is Zariski dense in SL₂(C) and Γ₀|_π is Zariski dense in SL₂(R) (resp. in SL₂(C)). In particular, the Q-algebra generated by Γ₀|_π is the algebra of 2 × 2 matrices with rational coefficients. The decomposition C² = π ⊕_Riπ is Γ₀-invariant, and the multiplication by i defines a Γ₀-equivariant map from π to iπ. Thus, Γ₀ preserves each of the real planes π_η = {(x,y) + η i(x,y); for (x,y) ∈π}, with η ∈R. Now, consider the (real) projection q of C² onto π parallel to iπ, and set ${\Lambda }^{\prime }=q({\Lambda })$. It is a Γ₀-invariant subgroup of π of rank at most 4, and it contains ${\Lambda }_{\Pi }\simeq \mathbf {Z}^{2}$. Then, one checks easily that

• (1) ${\Lambda }^{\prime }$ is commensurable to Λ_π ⊕ αΛ_π, for some α ∈R ∖Q, or to Λ_π, in which case we set α = 0;

• (2) Λ is commensurable to Λ_π ⊕ K_α,β(Λ_π) where K_α,β is the linear map from π to π ⊕iπ defined by K_α,β(u) = αu + βiu;

• (3) for m in Z, the real plane π_mα/β is Γ-invariant and intersects Λ on a cocompact lattice ${\Lambda }_{{\Pi }_{m\alpha /\beta }}$.

Then, the surfaces ${\Sigma }_{m}={\Pi }_{m\alpha /\beta }/{\Lambda }_{{\Pi }_{m\alpha /\beta }}$ form an infinite family of Γ-invariant tori in X.

Remark A.6

This second argument does not apply in the following case. Let E = C/Z[i], Λ = Z[i] ×Z[i] ⊂C², and X = C²/Λ = E × E. The group ${\Gamma }={\textsf {SL}}_{2}(\mathbf {Z})\ltimes \mathbf {R}^{2}/\mathbf {Z}^{2}$ is a subgroup of Aut(X) that preserves the torus π/Λ_π for π = R² ⊂C², but has no fixed point (because Γ contains π/Λ_π), and every Γ-invariant surface is a finite union of translates of this torus.

On the other hand, this second argument applies when X and Γ come from a genuine Kummer example, that is, a Kummer example defined on a surface that is not a compact torus. Indeed in that case Γ contains a finite index subgroup with a fixed point.