On the cohomological action of automorphisms of compact Kähler threefolds
aa r X i v : . [ m a t h . AG ] M a r ON THE COHOMOLOGICAL ACTION OF AUTOMORPHISMSOF COMPACT KÄHLER THREEFOLDS
FEDERICO LO BIANCO
Institut de Mathématiques de Marseille, University of Aix-Marseille,Centre de Mathématiques et Informatique (CMI)Technopôle Château-Gombert , Marseille (France)
AMS classification numbers: 14J30 (3-folds), 14J50 (Automorphisms ofsurfaces and higher-dimensional varieties), 32Q15 (Kähler manifolds).
Abstract.
Extending well-known results on surfaces, we give bounds on thecohomological action of automorphisms of compact Kähler threefolds. Moreprecisely, if the action is virtually unipotent we prove that the norm of ( f n ) ∗ grows at most as cn ; in the general case, we give a description of the spectrumof f ∗ , and bounds on the possible conjugates over Q of the dynamical degrees λ ( f ) , λ ( f ) . Examples on compact complex tori show the optimality of theresults. Résumé.
Nous Ãľtendons des rÃľsultats bien connus sur l’action en cohomo-logie des automorphismes des surfaces aux variÃľtÃľs compactes Kähler dedimension . Plus prÃľcisÃľment, si l’action est virtuellement unipotente nousmontrons que la norme de ( f n ) ∗ croît au plus comme cn ; dans le cas gÃľnÃľ-ral, nous donnons une description du spectre de f ∗ et des bornes sur les pos-sibles conjuguÃľs sur Q des degrÃľs dynamiques λ ( f ) , λ ( f ) . Des exemplessur les tores complexes compacts montrent l’optimalitÃľ de ces rÃľsultats. An automorphism f : X → X of a compact Kähler manifold induces by pull-backof forms a linear automorphism f ∗ : H ∗ ( X, Z ) → H ∗ ( X, Z ) which preserves the cohomology graduation, the Hodge decomposition, complexconjugation, wedge product and Poincaré duality. Question 1.
What else can one say on f ∗ ? More precisely, can one give constraintson f ∗ which depend only on the dimension of X (and not on the dimension of H ∗ ( X ) )? This is an interesting question in its own right since the cohomology of a man-ifold is a powerful tool to describe its geometry; furthermore, the cohomologicalaction of an automorphism is relevant when studying its dynamics: one can deduceits topological entropy from its spectrum (see Theorem 1.3.3), and in the surfacecase knowing f ∗ allows to establish the existence of f -equivariant fibrations (seeTheorem 2.2.1). It turns out that the restriction of f ∗ to the even cohomology E-mail address : [email protected] . encodes most of the interesting informations (see Section 1.3), therefore we focuson this part of the action; furthermore, in dimension the action on H ( X ) andon H ( X ) is trivial, and the action on H ( X ) can be deduced from the action on H ( X ) (see Proposition 1.1.(3)), so we only describe the latter.The situation of automorphisms (and, more generally, of birational transforma-tions) of curves and surfaces is well understood (see Section 2). We address herethe three-dimensional case.The first result describes the situation where f ∗ does not have any eigenvalue ofmodulus > , i.e. the dynamical degrees λ i ( f ) are equal to (see Definition 1.3.1). Theorem A.
Let X be a compact Kähler threefold and let f : X → X be an auto-morphism such that λ ( f ) = 1 and whose action on H ∗ ( X ) has infinite order. Thenthe induced linear automorphism f ∗ : H ( X, C ) → H ( X, C ) is virtually unipotentand has a unique Jordan block of maximal dimension m = 3 or . In particular,the norm of ( f n ) ∗ grows either as cn or as cn as n goes to infinity. For the proof of slightly more general results, see Theorem 3.0.1 and Proposition3.1.1.Next we give a description of the spectrum of f ∗ in terms of the dynamicaldegrees: Theorem B.
Let X be a compact Kähler threefold and let f : X → X be anautomorphism having dynamical degrees λ = λ ( f ) and λ = λ ( f ) (see Definition1.3.1). Let λ be an eigenvalue of f ∗ : H ( X, C ) → H ( X, C ) ; then there exists apositive integer N such that | λ | ( − N ∈ (cid:8) , λ , λ − , λ − λ (cid:9) . Furthermore, if N ≥ then | λ | ( − k is an eigenvalue of f ∗ for all k = 1 , . . . , N . For a proof see Proposition 4.3.1 (for the case where λ and λ are multiplica-tively independent) and 4.4.1 (for the case where λ and λ have a multiplicativedependency). Remark . While proving Theorem B, we will also obtain that if λ / ∈ { λ , √ λ } ,then λ and λ − are the only eigenvalues of f ∗ having modulus λ or λ − .This had already been proven in much wider generality by Truong in [Tru14].Finally, we describe the (moduli of) Galois conjugates of λ ( f ) over Q : Theorem C.
Let X be a compact Kähler threefold and let f : X → X be anautomorphism having dynamical degrees λ = λ ( f ) and λ = λ ( f ) . Then λ isan algebraic integer, all of whose conjugates over Q have modulus belonging to thefollowing set: (cid:26) λ , λ − , λ − λ , q λ − , p λ , q λ λ − (cid:27) . See Proposition 5.1.7 and 5.2.2 for a proof and for a more detailed descriptionof all possible subcases.In Section 1 we introduce the problem and the tools which will be used in theproofs, namely the generalized Hodge index theorem, an application of Poincaré’sduality and some elements of the theory of algebraic groups; in Section 2 we presentthe known results in dimension two. In the rest of the paper we treat the case ofdimension three: in Section 3 we give a proof of Theorem A and describe examples
OHOMOLOGY ACTION OF THREEFOLD AUTOMORPHISMS 3 on complex tori which show the optimality of the result; similarly, in Section 4and Section 5 we prove Theorem B and C respectively, and describe further ex-amples on tori which show the optimality of the claims; finally, in Section 6 weaddress the problem to determine whether f ∗ can be neither (virtually) unipotentnor semisimple (see Proposition 6.0.1). Acknowledgements.
I wish to express my deepest gratitude to my advisorSerge Cantat for proposing me this problem and a strategy of proof, and for all thehelp and suggestions he provided me at every stage of this work.1.
Introduction and main tools
Throughout this section, we denote by f : X → X an automorphism of a compactKähler manifold X of complex dimension d and by f ∗ : H ∗ ( X, R ) → H ∗ ( X, R ) the induced linear automorphism on cohomology. We still denote by f ∗ : H ∗ ( X, C ) → H ∗ ( X, C ) the complexification of f ∗ , and by f ∗ k (resp. f ∗ p,q ) the restriction of f ∗ to H k ( X, R ) (resp. to H p,q ( X ) ).1.1. First constraints.
The induced linear automorphism f ∗ preserves some ad-ditional structure on the cohomology space H ∗ ( X, R ) :(1) the graduation and Hodge decomposition: in other words f ∗ ( H i ( X, R )) = H i ( X, R ) for all i = 0 , . . . , d and f ∗ ( H p,q ( X )) = H p,q ( X ) for p, q = 0 , . . . d ;(2) the wedge (or cup) product: in other words f ∗ ( u ∧ v ) = f ∗ ( u ) ∧ f ∗ ( v ) forall u, v ∈ H ∗ ( X, R ) ;(3) Poincaré’s duality: in other words the isomorphism H i ( X, R ) ∼ = H d − i ( X, R ) ∨ induces an identification f ∗ i = (( f ∗ d − i ) − ) ∨ . Furthermore(4) f ∗ and ( f ∗ ) − are defined over Z ; in other words, the coefficients of f ∗ and ( f ∗ ) − with respect to an integral basis of H ∗ ( X, R ) are integers;(5) f ∗ preserves the convex salient cones K p ⊂ H p,p ( M, R ) generated by theclasses of positive currents (see [Dem97]).Remark that properties (1) − (4) are algebraic, while property (5) is not.The rough strategy for the proofs will be to consider the action of f ∗ on elementsof the form u ∧ v , u, v ∈ H ( X, C ) , and relate f ∗ with f ∗ by using property (3) .1.2. The unipotent and semisimple parts.
Let V be a finite dimensional realvector space and let g : V → V be a linear endomorphism. It is well-known thatthere exists a unique decomposition g = g u ◦ g s = g s ◦ g u , where g u is unipotent (i.e. ( g u − id V ) dim V = 0 ) and g s is semisimple (i.e. diago-nalizable over C ). This is a special case of the following more general statement: Theorem 1.2.1 (Jordan decomposition) . Let V be a finite dimensional real vectorspace and let G ⊂ GL( V ) be a commutative algebraic group. Then, denoting by G u ⊂ G (respectively by G s ⊂ G ) the subset of unipotent (respectively semisimple) COHOMOLOGY ACTION OF THREEFOLD AUTOMORPHISMS elements of G , G u and G s are closed subgroups of G and the product morphisminduces an isomorphism of real algebraic groups G ∼ = G u × G s . Let us go back to the context of automorphisms of compact Kähler manifold.Let f : X → X be an automorphism of a compact Kähler manifold, V = H ∗ ( X, R ) , f ∗ : V → V and G = [ n ∈ Z ( f n ) ∗ Zar ⊂ GL( V ); here A Zar denotes the Zariski-closure of a set A ⊂ GL( V ) , i.e. the smallest Zariski-closed subset of GL( V ) containing A . Then, since h g i is a commutative group, G isa commutative real algebraic group, and by Theorem 1.2.1 we have an isomorphismof real algebraic groups G ∼ = G u × G s ; this means in particular that, writing the Jordan decomposition f ∗ = f ∗ u ◦ f ∗ s , wehave f ∗ u , f ∗ s ∈ G , i.e. if f ∗ satisfies some algebraic constraint, then so do f ∗ u and f ∗ s .We have therefore: Lemma 1.2.2. f ∗ u and f ∗ s preserve the cohomology graduation, Hodge decomposi-tion, wedge product and Poincaré duality, and they are defined over Z (properties (1) − (4) in §1.1). Remark however that, since preserving a cone is not an algebraic property, f ∗ u and f ∗ s may not preserve the positive cones K p . Remark . Let g ∈ GL( H ∗ ( X, R )) be a linear automorphism preserving thecohomology graduation, Hodge decomposition, wedge product and Poincaré duality.Then we may put (the complexification of) g in Jordan form so that each Jordanblock corresponds to a subspace of H p,q ( X ) for some p, q .Furthermore, if all eigenvalues of g are real, we can put g itself in Jordan formso that each Jordan block corresponds to a subspace of ( H p,q ( X ) ⊕ H q,p ( X )) R forsome p, q .1.3. Dynamical degrees.
In this paragraph only, we allow f : M M to be adominant meromorphic self-map of a compact Kähler manifold M . Definition 1.3.1.
The p -th dynamical degree of f is defined as λ p ( f ) = lim sup n → + ∞ k ( f n ) ∗ p,p k n , where k · k is any matrix norm on the space L ( H p,p ( X, R )) of linear maps of H p,p ( X, R ) into itself. In the meromorphic case the pull-backs f ∗ p,p are defined in the sense of currents(see [Dem97]).One can prove that(1.1) λ p ( f ) = lim n → + ∞ (cid:18)Z M ( f n ) ∗ ω p ∧ ω d − p (cid:19) n for any Kähler form ω ; see [DS05], [DS04] for details.In the case of holomorphic maps, we have ( f n ) ∗ = ( f ∗ ) n , so that λ p ( f ) is thespectral radius (i.e. the maximal modulus of eigenvalues) of the linear map f ∗ p,p ; OHOMOLOGY ACTION OF THREEFOLD AUTOMORPHISMS 5 since f ∗ also preserves the positive cone K p ⊂ H p,p ( M, R ) , a theorem of Birkhoff[Bir67] implies that λ p ( f ) is a positive real eigenvalue of f ∗ p . In particular, λ p ( f ) isan algebraic integer. However it should be noted that in the meromorphic settingwe have in general ( f n ) ∗ = ( f ∗ ) n .At least in the projective case, the p -th dynamical degree measures the exponen-tial growth of the volume of f − n ( V ) for subvarieties V ⊂ M of codimension p , see[Gue04]. Remark . By definition λ ( f ) = 1 ; λ d ( f ) is called the topological degree of f :it is equal to the number of points in a generic fibre of f .The topological entropy of a continuous map of a topological space is a non-negative number, possibly infinite, which gives a measure of the chaos created bythe map and its iterates; for a precise definition see [HK02]. The computationof the topological entropy of a map is usually complicated, and requires ad hocarguments; however, in the case of dominant holomorphic self-maps of compactKähler manifolds one can apply the following result due to Yomdin [Yom87] andGromov [Gro90]: Theorem 1.3.3 (Yomdin-Gromov) . Let f : M → M be a dominant holomorphicself-map of a compact Kähler manifold of dimension d ; then the topological entropyof f is given by h top ( f ) = max p =0 ,...,d log λ p ( f ) . In the meromorphic setting one can only prove that the topological entropy isbounded from above by the maximum of the logarithms of the dynamical degrees(see [DS05], [DS04]).1.3.1.
Concavity properties.
Theorem 1.3.4 (Teissier-Khovanskii, see [Gro90]) . Let X be a compact Kählermanifold of dimension d , and Ω := ( ω , . . . , ω k ) be k -tuple of Kähler forms on X .For any multi-index I = ( i , . . . , i k ) let Ω I = ω i ∧ . . . ∧ ω i k k .Fix i , . . . , i k so that i := P h ≥ i h ≤ d , and let I p = ( p, d − i − p, i , . . . , i k ) ; thenthe function p log (cid:18)Z X Ω I p (cid:19) is concave on the set { , , . . . , d − i } . One can use Theorem 1.3.4 to prove the following log-concavity result:
Proposition 1.3.5.
Let f : X X be a dominant meromorphic self-map of acompact Kähler manifold X of dimension d . Then the function p log λ p ( f ) is concave on the set { , , . . . , d } . In particular, if λ ( f ) = 1 then λ p ( f ) = 1 forall p = 0 , . . . , d . The following result was proven in [Din05] for automorphisms, but the sameproof adapts to the case of dominant meromorphic self-maps.
COHOMOLOGY ACTION OF THREEFOLD AUTOMORPHISMS
Proposition 1.3.6.
Let f : X X be a dominant meromorphic self-map of acompact Kähler manifold X , and let r p,q ( f ) = lim n → + ∞ k ( f n ) ∗ p,q k n ; here we use the convention that the norm of the identity on the null vector space isequal to .Then r p,q ≤ q λ p ( f ) λ q ( f ) . In particular lim n + ∞ k ( f n ) ∗ k n = max p λ p ( f ) . In other words, the exponential growth of the matrix norm of ( f n ) ∗ can bedetected on the restriction to the intermediate Hodge spaces H p,p ( X ) . Corollary 1.3.7.
Let f : X → X be a dominant holomorphic endomorphism of acompact Kähler manifold X . Then the following are equivalent: (1) λ ( f ) = 1 ; (2) f ∗ is virtually unipotent; (3) r p,q ( f ) = 1 for all p, q .Proof. The implications (2) ⇒ (3) and (3) ⇒ (1) are evident; let us show that (1) ⇒ (2) .Since by Proposition 1.3.5 λ ( f ) = λ ( f ) = . . . = λ d ( f ) = 1 , Proposition 1.3.6implies that r p,q ( f ) ≤ for all p, q . Therefore the spectral radii of the linearautomorphisms f ∗ k : H k ( X, Z ) → H k ( X, Z ) are equal to . It follows from a lemma of Kronecker that the roots of the charac-teristic polynomial of f ∗ k are roots of unity; therefore, some iterate of f ∗ has asits only eigenvalue, i.e. it is unipotent. This concludes the proof. (cid:3) Remark that in the situation of Corollary 1.3.7 we have λ d ( f ) = 1 , so that f isan automorphism.1.3.2. Polynomial growth.
Suppose now that f : X → X is a dominant holomor-phic endomorphism, and assume that λ ( f ) = 1 ; then by Corollary 1.3.7 all theeigenvalues of f ∗ have modulus (and in particular f is an automorphism), andan easy linear algebra argument implies that k ( f n ) ∗ p,p k ∼ cn µ p ( f ) , where µ p ( f ) + 1 is the maximal size of Jordan blocks of f ∗ p,p . Then one can definea number measuring the polynomial growth of ( f n ) ∗ . Definition 1.3.8.
Suppose that λ p ( f ) = 1 ; the p -th polynomial dynamical degreeis defined as µ p ( f ) = lim n → + ∞ log k ( f n ) ∗ p,p k log n . The following question is still open even for birational maps of P k ( C ) , k ≥ . Question 2.
Let f : X X be a meromorphic self-map of a compact Kählermanifold such that λ ( f ) = 1 ; is it true that k ( f n ) ∗ k grows polynomially? OHOMOLOGY ACTION OF THREEFOLD AUTOMORPHISMS 7
The inequalities of Tessier-Khovanskii allow to prove an equivalent of Proposition1.3.5 and Proposition 1.3.6:
Proposition 1.3.9.
Let f : X → X be an automorphism of a compact Kählermanifold such that λ p ( f ) = 1 ; then (1) the function p µ p ( f ) is concave on the set { , . . . , d } ; (2) lim n → + ∞ log k ( f n ) ∗ k log n = max p µ p ( f ) . Generalized Hodge index theorem.
The classical Hodge index theoremasserts that if S is a compact Kähler surface, then the intersection product on H , ( X, R ) is hyperbolic, i.e. it has signature (1 , h , ( S ) − ; this is a consequence ofthe Hodge-Riemann bilinear relations, which can be generalized in higher dimensionin order to obtain an analogue of the classical result. We will focus on the secondcohomology group, but analogue results exist for cohomology of any order (see[DN06]).Let ( X, ω ) be a compact Kähler manifold of dimension d ≥ ; we define aquadratic form q on H ( X, R ) by q ( α, β ) := Z X ( α , ∧ β , − α , ∧ β , − α , ∧ β , ) ∧ ω d − α, β ∈ H ( X, R ) , where α i,j (resp. β i,j ) denotes the ( i, j ) -part of α (resp. of β ).Remark that the decomposition H ( X, R ) = H , ( X, R ) ⊕ ( H , ( X ) ⊕ H , ( X )) R is q -orthogonal. Theorem 1.4.1 (Generalized Hodge index theorem) . Let ( X, ω ) be a compactKähler manifold of dimension d ≥ and let q be defined as above. Then therestriction of q to H , ( X, R ) has signature (1 , h , ( X ) − ; its restriction to ( H , ( X ) ⊕ H , ( X )) R is negative definite. An immediate consequence, which we will use constantly in the rest of the paper,is the following:
Corollary 1.4.2. If V ⊂ H ( X, R ) is a q -isotropic space, then dim V < . Inparticular, if u, v ∈ H , ( X, R ) ∪ ( H , ( X ) ⊕ H , ( X )) R are linearly independentclasses, then the classes u ∧ u, u ∧ v, v ∧ v ∈ H ( X, R ) cannot all be null. If furthermore u ∈ ( H , ( X ) ⊕ H , ( X )) R , then u ∧ u = 0 .Analogously, if u, v ∈ H , ( X ) ∪ ( H , ( X ) ⊕ H , ( X )) are linearly independentclasses, then the classes u ∧ ¯ u, u ∧ ¯ v, v ∧ ¯ v ∈ H ( X, C ) cannot all be null. If furthermore u ∈ H , ( X ) ⊕ H , ( X ) , then u ∧ ¯ u = 0 . COHOMOLOGY ACTION OF THREEFOLD AUTOMORPHISMS The case of surfaces
Remark first that the case of automorphisms of curves is dynamically not veryinteresting: indeed, if the genus of the curve is g ≥ , then the group of automor-phism is finite; the only non-trivial dynamics arise from automorphisms of P andfrom automorphisms of elliptic curves (which, up to iteration, are translations),and both are well-understood.Let us focus then on the surface case: let S be a compact Kähler surface and let f : S → S be an automorphism. By the Hodge index theorem (Theorem 1.4.1), thegeneralized intersection form q makes H ( X, R ) into a hyperbolic space; further-more, q is preserved by f , so that we may consider g = f ∗ : H ( X, R ) → H ( X, R ) an element of O ( H ( X, R ) , q ) .2.1. Automorphisms of hyperbolic spaces.
Let ( V, q ) be a hyperbolic vectorspace of dimension n and let k · k be a norm on the space L ( V ) of linear endomor-phisms of V . Definition 2.1.1.
Let g ∈ O ( V, q ) . We say that g is • loxodromic (or hyperbolic ) if it admits an eigenvalue of modulus strictlygreater than ; • parabolic if all its eigenvalues have modulus and k g n k is not bounded as n → + ∞ ; • elliptic if all its eigenvalues have modulus and k g n k is bounded as n → + ∞ . In each of the cases above, simple linear algebra arguments allow to furtherdescribe the situation. For the following result see for example [Gri16].
Theorem 2.1.2.
Let g ∈ O + ( V, q ) , and suppose that g preserves a lattice Γ ⊂ V . • If g is loxodromic, then it is semisimple and it has exactly one eigenvalue λ with modulus > and exactly one eigenvalue λ − with modulus < ; theseeigenvalues are real and simple, so that in particular k g n k ∼ cλ n . Theeigenvalue λ is an algebraic integer whose conjugates over Q are λ − andcomplex numbers of modulus , i.e. λ is a quadratic or Salem number. • If g is parabolic, then all the eigenvalues of g are roots of unity, and someiterate of g has Jordan form
00 0 0 I d − . In particular k g n k ∼ cn . • If g is elliptic, then it has finite order. An automorphism f : S → S of a compact Kähler surface S is called loxodromic,parabolic or elliptic if g = f ∗ is loxodromic, parabolic or elliptic respectively. Re-mark that g preserves the integral lattice H ( X, Z ) / ( torsion ) , so that Theorem 2.1.2can be applied to g .Remark that, if f is homotopic to the identity, then its action on cohomologyis trivial. Conversely, if f acts trivially on cohomology, then some of its iterates ishomotopic to the identity. More precisely: OHOMOLOGY ACTION OF THREEFOLD AUTOMORPHISMS 9
Theorem 2.1.3 (Fujiki, Liebermann [Fuj78, Lie78]) . Let M be a compact Kählermanifold. If [ κ ] is a Kähler class on M , the connected component of the identity Aut( X ) has finite index in the group of automorphisms of M fixing [ κ ] . This implies that a surface automorphism is elliptic if and only if one of itsiterates is homotopic to the identity.2.2.
Equivariant fibrations.
In the case of surfaces, it turns out that the coho-mological action of an automorphism has consequences on the following propertyof decomposability of its dynamics. Let f : M → M be an automorphism of a com-pact Kähler manifold M ; we say that a fibration π : M → B (i.e. a surjective mapwith connected fibres) is f -equivariant if there exists an automorphism g : B → B such that π ◦ f = g ◦ π , i.e. the following diagram commutes: M MB B fπ πg . The following theorem was stated and proved in the present form by Cantat[Can01], and follows from a result of Gizatullin (see [Giz80], or [Gri16] for a survey);see also [DF01] for the birational case.
Theorem 2.2.1.
Let S be a compact Kähler surface and let f be an automorphismof S . (1) If f is parabolic, there exists an f -equivariant elliptic fibration π : S → C ; f doesn’t admit other equivariant fibrations. (2) Conversely, if a non-elliptic automorphism of a surface f : S → S admitsan equivariant fibration π : S → C onto a curve, then f is parabolic. Inparticular, the fibration π is elliptic, and it is the only equivariant fibration. In other words, a non-elliptic automorphism of a surface admits an equivariantfibration if and only if its topological entropy is zero.In higher dimension, one can ask the following question:
Question 3.
Let f : X → X be an automorphism of a compact Kähler manifold M .Suppose that f ∗ is virtually unipotent of infinite order. Does f admit an equivariantfibration? Apart from the case of surfaces, the only situation where the answer is known(and affirmative) is that of irreducible holomorphic symplectic (or hyperkähler)manifolds of deformation type K [ n ] or generalized Kummer (see [HKZ15]); theproof uses the hyperkähler version of the abundance conjecture, which was proven inthis context by Bayer and Macrí [BM14]. See also [LB17] for a converse statement,which holds for any irreducible holomorphic symplectic manifold.3. Automorphisms of threefolds: the unipotent case
Throughout this section let X be a compact Kähler threefold and let g : H ∗ ( X, R ) → H ∗ ( X, R ) be a unipotent linear automorphism preserving the cohomology graduation, theHodge decomposition, the wedge-product and Poincaré’s duality (properties (1) − (3) in §1.1). If f : X → X is an automorphism such that λ ( f ) = 1 , then the linear auto-morphism f ∗ : H ∗ ( X, R ) is virtually unipotent, and therefore an iterate g = ( f N ) ∗ satisfies the assumptions above.More generally, if f : X → X is any automorphism and f ∗ = g u g s = g s g u is the Jordan decomposition of f ∗ , then by Lemma 1.2.2 the unipotent part g = g u satisfies the assumptions above.Theorem A is thus a special case of the following theorem and of Proposition3.1.1. Theorem 3.0.1.
Let X be a compact Kähler threefold and let g : H ∗ ( X, R ) → H ∗ ( X, R ) be a non-trivial unipotent linear automorphism preserving the cohomologygraduation, the Hodge decomposition, the wedge-product and the Poincaré duality.Then (1) the maximal Jordan block of g (for the eigenvalue ) has dimension ≤ ; (2) if furthermore g preserves the cone C = { v ∈ H ( X, R ) ; q ( v ) ≥ } , thenits maximal Jordan block has odd dimension.In particular the norm of g n grows as cn k with k ≤ ; and if furthermore g preserves the positive cone, then k is even.Remark . Let f ∈ Aut( X ) be an automorphism such that λ ( f ) = 1 , so that,up to iterating f , g = f ∗ satisfies the assumptions of Theorem 3.0.1. In this case,by Proposition 1.3.9, the growth of k g n k is the same as the maximal growths of the k g np,p k , i.e. the growth of k g n , k by Poincaré duality.Furthermore, g preserves the cone C , therefore the maximal Jordan block has odddimension by [Bir67]. Proof.
Let u , . . . , u k ∈ H ( X, R ) be a basis of a maximal Jordan block satisfying g ( u ) = u , g ( u h ) = u h + u h − for h = 2 , . . . , k. By Remark 1.2.3 we can suppose that u i ∈ H , ( X, R ) ∪ ( H , ( X ) ⊕ H , ( X )) R for i = 1 , . . . , k .We show that the norm of g n grows at least as cn k − . Let us consider theelements u k ∧ u k , u k − ∧ u k − ∈ H ( X, R ) .Using the notation of Lemma 3.0.3 below, let P h = P h ( n ) ; then g n ( u k ∧ u k ) = g n ( u k ) ∧ g n ( u k ) == ( P k − u + P k − u + P k − u + . . . ) ∧ ( P k − u + P k − u + P k − u + . . . ) == P k − ( u ∧ u ) + 2 P k − P k − ( u ∧ u ) + (2 P k − P k − u ∧ u + P k − u ∧ u ) + . . . If u ∧ u = 0 or u ∧ u = 0 , then the norm of g n would grow at least as n k − ;we can thus assume that u ∧ u = u ∧ u = 0 . Thus, by Corollary 1.4.2 we have u ∧ u = 0 .Let us apply Lemma 3.0.3 and look at the dominant terms of P k − , P k − and P k − ;if we had(3.1) u ∧ u ( k − k − u ∧ u (( k − = 0 , then the norm of g n would grow at least as cn k − . We may then assume that (3.1)is not satisfied. OHOMOLOGY ACTION OF THREEFOLD AUTOMORPHISMS 11
Now, g n ( u k − ∧ u k − ) = g n ( u k − ) ∧ g n ( u k − ) == ( P k − u + P k − u + P k − u + . . . ) ∧ ( P k − u + P k − u + P k − u + . . . ) == 2 P k − P k − ( u ∧ u ) + P k − ( u ∧ u ) + . . . By the same argument, if(3.2) u ∧ u ( k − k − u ∧ u (( k − = 0 , then the norm of g n would grow at least as cn k − . Since u ∧ u = 0 and the twolinear relations 3.1 and 3.2 are independent, at least one between (3.1) and (3.2) issatisfied.This shows that the norm of g n grows at least as cn k − .Now, by Poincaré duality (property (3) in §1.1), the norm of g n = ( g − n ) ∨ growsexactly as the norm of g n . In particular k − ≥ k − ⇒ k ≤ , which concludes the proof. (cid:3) Lemma 3.0.3.
Let A = . . .
00 1 1 . . . ... . . . . . . . . . ... . . . . . . be a Jordan block of dimension k. Then A n = P ( n ) P ( n ) P ( n ) . . . P k − ( n )0 P ( n ) P ( n ) . . . P k − ( n ) ... . . . . . . . . . ... . . . P ( n ) P ( n )0 . . . P ( n ) , where P i ( n ) is a polynomial of degree i in n whose leading term is n i /i ! .Proof. The functions P i satisfy the equation P i +1 ( n ) = P i +1 ( n −
1) + P i ( n − , therefore one obtains recursively(3.3) P i +1 ( n ) = n X j =0 P i ( j ) i = 0 , . . . , k − . The claim is clear fo i = 0 ; reasoning inductively, we may assume that P i is apolynomial of degree i whose leading coefficient is /i ! .In order to conclude, one needs only recall the well-known fact that for any fixed m the function n n X k =0 k m is a polynomial q m of degree m + 1 with leading coefficient / ( m + 1) . This can beproven by writing n m +1 = n − X k =0 (( k + 1) m +1 − k m +1 ) = m X h =0 (cid:18) m + 1 h (cid:19) q h ( n ) and by applying induction to m .Therefore, writing P i − ( n ) = 1( i − n i − + a i − n i − + . . . + a , we have P i ( n ) = 1( i − q i − ( n ) + a i − q i − ( n ) + . . . + a q ( n ) , and the claim follows easily. (cid:3) Bound on the dimension of non-maximal Jordan blocks.Proposition 3.1.1.
Let X be a compact Kähler threefold and let g : H ∗ ( X, R ) → H ∗ ( X, R ) be a non-trivial unipotent linear automorphism preserving the cohomologygraduation, the Hodge decomposition, the wedge-product and the Poincaré duality(properties (1) − (3) in §1.1). Then there exists a unique Jordan block of g ofmaximal dimension k ≤ (for the eigenvalue ); more precisely, all other Jordanblocks have dimension ≤ k +12 .Proof. Let v , . . . , v k ∈ H ( X, R ) form a basis for a maximal Jordan block of g , andlet w , . . . , w l ∈ H ( X, R ) form a Jordan basis for another Jordan block satisfying g ( v ) = v , g ( v i ) = v i + v i − for i = 2 , . . . , k,g ( w ) = w , g ( w j ) = w j + w j − for j = 2 , . . . , l. By Remark 1.2.3 we can suppose that v i , w j ∈ H , ( X, R ) ∪ ( H , ( X ) ⊕ H , ( X )) R for i = 1 , . . . , k and j = 1 , . . . , l .We will suppose that l > (otherwise the claim is evident), and consider the actionof g on the classes v k ∧ v k , v k ∧ w l , w l ∧ w l ∈ H ( X, R ) .By Corollary 1.4.2, the classes v ∧ v , v ∧ w , w ∧ w ∈ H ( X, R ) cannot be allnull; since g n v k ∼ cn k − v and g n w l ∼ c ′ n h − w , this implies that k g n k grows atleast as c ′′ n l − . Since by Poincaré duality k g n k and k g n k have the same growth,we get l − ≤ k − ⇒ l ≤ k + 12 , which concludes the proof. (cid:3) Unipotent examples on complex tori.
Examples on complex tori of di-mension show the optimality of Theorem 3.0.1 and Proposition 3.1.1. Let E = C / Λ be an elliptic curve, where Λ is a lattice of C , and let X := E × E × E = C (cid:30) Λ × Λ × Λ . Every matrix M ∈ SL ( Z ) acts linearly on C preserving the lattice Λ × Λ × Λ ,and therefore induces an automorphism f : X → X . One can easily show that, if dx, dy, dz are holomorphic linear coordinates on the three factors respectively, then OHOMOLOGY ACTION OF THREEFOLD AUTOMORPHISMS 13 the matrix of f ∗ , with respect to the basis dx, dy, dz of H , ( X ) is exactly thetransposed M T . Since the wedge product of forms induces an isomorphism H , ( X ) ∼ = H , ( X ) ⊗ H , ( X ) , the matrix of f ∗ , with respect to the basis dx ∧ d ¯ x, dx ∧ d ¯ y, . . . , dz ∧ d ¯ z is M , = M t ⊗ M t := ( m j,i m l,k ) i,j,k,l =1 , . Example . Let M = . Then M , = M , is unipotent and its Jordan blocks have dimension , and . Example . Let M = . Then M , = .M , is unipotent and its Jordan blocks have dimension , , and .4. Automorphisms of threefolds: the semisimple case, proof ofTheorem B
Throughout this section let X be a compact Kähler threefold and let g : H ∗ ( X, R ) → H ∗ ( X, R ) be a semisimple linear automorphism preserving the cohomology graduation, theHodge decomposition, the wedge-product and Poincaré’s duality (properties (1) − (3) in §1.1).If f : X → X is an automorphism and f ∗ = g u g s = g s g u is the Jordan decomposition of f ∗ , then by Lemma 1.2.2 the semisimple part g = g s satisfies the assumptions above.Let λ = λ ( g ) and λ = λ ( g ) be the dynamical degrees of g , i.e. the spectralradii of g and g respectively, and let Λ be the spectrum of g , i.e. the set of complexeigenvalues of g with multiplicities; we will say that two elements λ, λ ′ ∈ Λ aredistinct if either λ = λ ′ or λ = λ ′ is an eigenvalue with multiplicity ≥ .The main ingredient of the proofs in the rest of this section is the followinglemma. Lemma 4.0.1.
Let g ∈ GL( H ∗ ( X, R )) be a semisimple linear automorphism pre-serving the cohomology graduation, the Hodge decomposition, the wedge-product andPoincaré’s duality and let Λ be its spectrum (with multiplicities taken into account).If λ, λ ′ ∈ Λ are distinct elements, then (cid:26) | λ | , λ ¯ λ ′ , | λ ′ | (cid:27) ∩ Λ = ∅ . Proof.
By Remark 1.2.3, we may pick eigenvectors v, v ′ ∈ H , ( X ) ∪ H , ( X ) ∪ H , ( X ) for the eigenvalues λ, λ ′ respectively. By Corollary 1.4.2, the wedge pro-ducts v ∧ ¯ v, v ∧ ¯ v ′ , v ′ ∧ ¯ v ′ ∈ H ( X, C ) cannot all be null. A non-null wedge product gives rise to an eigenvector for g ; inparticular, denoting by Λ the spectrum of g , we have (cid:8) | λ | , λ ¯ λ ′ , | λ ′ | (cid:9) ∩ Λ = ∅ . Now, by assumption (3) g can be identified with ( g − ) ∨ , and in particular Λ = Λ − = { λ − , λ ∈ Λ } . This concludes the proof. (cid:3)
Before passing to the actual proof of Theorem B let us outline the strategy wewill adopt.(1) In §4.1 we will study the number r ( g ) of multiplicative parameters whichdescribe the moduli of eigenvalues of g ; this can be formally defined as thesplit-rank of the real algebraic group G generated by g .(2) In §4.2 we define the weights of g (or rather of g ) as the real characters w λ : G → R ∗ such that g
7→ | λ | for any eigenvalue λ ; for the sake of clarity we will adoptan additive notation on weights.Using an immediate consequence of Lemma 4.0.1 (Lemma 4.2.5), in Lemma4.2.7 we prove that r ( g ) ≤ . (3) We conclude by considering the cases r ( g ) = 2 and r ( g ) = 1 separately(§4.3 and §4.4 respectively). In both cases the proof relies essentially onLemma 4.2.5. OHOMOLOGY ACTION OF THREEFOLD AUTOMORPHISMS 15
Structure of the algebraic group generated by g . For the content ofthis Section we refer to [Bor91, §8]. Let g be as above and let G = h g i Zar ≤ GL( H ( X, R )) be the Zariski-closure of the group generated by g ; it is a real algebraic group by[Bor91, Proposition I.1.3]; furthermore, since h g i is diagonalizable over C and com-mutative, so is G .The Zariski-connected component of the identity G of G is thus a real algebraictorus; we define G d ≤ G as the subgroup generated by real one-parameter sub-groups of G , and G a ≤ G as the intersection of the kernels of real characters of G . Then we have the following classical result: Proposition 4.1.1.
Let G be as above; then (1) G d ∼ = ( R ∗ ) r is the maximal split subtorus; (2) G a ∼ = ( S ) s is the maximal anisotropic subtorus; (3) the product morphism G d × G a → G is an isogeny (i.e. it is surjective andwith finite kernel). The number r ≥ is the (real) split-rank of G ; we will denote it by r ( g ) and callit the rank of g ; informally, r ( g ) (respectively s ( g ) ) is the number of multiplicativeparameters which are necessary to describe the moduli (respectively, the arguments)of the complex eigenvalues of g (see Lemma 4.2.3).4.2. Weights of g . Let λ ∈ Λ be a complex eigenvalue of g ; then the grouphomomorphism h g i → R ∗ g n
7→ | λ | n is algebraic, and therefore can be extended to a non-trivial real character of G .Upon restriction to G and pull-back to G d × G a ∼ = ( R ∗ ) r × ( S ) s , this yields anon-trivial morphism of real algebraic groups ρ λ : ( R ∗ ) r × ( S ) s → R ∗ . Since all morphisms of real algebraic groups S → R ∗ are trivial, we have ρ λ ( x . . . , x r , θ , . . . , θ s ) = x m ( λ )1 · · · x m r ( λ ) r , m i ∈ Z . For the sake of simplicity, we will adopt an additive notation, so that the character ρ λ is identified with the vector w λ = ( m ( λ ) , . . . , m r ( λ )) ∈ R r . Definition 4.2.1.
The weight of the eigenvalue λ ∈ Λ of g is the vector w λ =( m ( λ ) , . . . , m r ( λ )) ∈ R r . We denote by W the set of all weights of eigenvalues of g with multiplicities; as for the elements of Λ , we say that two elements w, w ′ of W are distinct if either w = w ′ or w = w ′ has multiplicity > .Remark . Remark that w λ = w ¯ λ . Therefore, if λ is a non-real eigenvalue of g , the weight w λ will be counted twice, once for λ and once for ¯ λ .We say that a weight w ∈ W is maximal for a linear functional α ∈ ( R r ) ∨ if | α ( w ) | = max w ∈ W | α ( w ) | ; we simply say that w is maximal if it is maximal forsome linear functional. The maximal weights are exactly those belonging to theboundary of the convex hull of W in R r . Lemma 4.2.3.
The weights of g span a real vector space of dimension r ( g ) . Proof.
Remark first that r ( g ) is equal to the rank over Z of the group of realcharacters G → R ∗ . Therefore, in order to prove the claim one only needs to show that any real characterof G can be written as a product of the ρ λ . Of course, one can check such propertyon g and its iterates.Since g is semisimple, one can find a real basis of V = H ∗ ( X, R ) in which g iswritten as g = λ R θ λ R θ . . . λ k R θ k λ k +1 . . . λ n , where R θ denotes the rotation matrix for the angle θ and λ i ∈ R ∗ ; actually, afterpossibly replacing g by g (which doesn’t change G ) we may assume that λ i ∈ R + .Now let G C be the complexification of G ; then, upon diagonalizing g in GL( H ∗ ( X, C )) ,we can naturally see G C as a subgroup of a complex torus ( C ∗ ) n + k , where the first n coordinates correspond to the λ i ’s and the last k correspond tothe rotations R θ j . More explicitly, considering the above base of H ∗ ( X, R ) and thecorresponding matrix coordinates a i,j of GL( H ∗ ( X, R )) , define x i = a i − , i − a i, i − a i, i − a i − , i i = 1 , . . . , kx i = a i + k,i + k i = k + 1 , . . . , ny j = ( a j − , j − + ia j, j − ) /x j j = 1 , . . . , k. Then it is not hard to see that the x i ’s and the y j ’s are multiplicative coordinatesof a complex torus D n + k containing G C .The ( x, y ) coordinates of the element g are x i = λ i i = 1 , . . . , n, y j = e iθ j j = 1 , . . . , k. Now, every complex character G C → C ∗ can be written as the restriction to G C of a product χ : ( x , . . . , x n , y , . . . , y k ) x A · · · x A n n y B · · · y B k k . In order for χ to be real in restriction to G , we need to have χ ( λ , . . . , λ n , e iθ , . . . , e iθ k ) ∈ R . This implies that k X j =1 B j θ j ∈ π Z , OHOMOLOGY ACTION OF THREEFOLD AUTOMORPHISMS 17 meaning that, in restriction to g and its iterates, χ coincides, maybe up to a sign,with χ ′ : ( x , . . . , x n , y , . . . , y k ) x A · · · x A n n . However, two real characters on a connected group which can at most differ by asign are equal, thus χ | G = χ ′| G . By the initial remark, this proves the claim. (cid:3) Lemma 4.2.4.
There exist a basis w , . . . , w r of R r and a basis α , . . . , α r of ( R r ) ∨ such that (1) the w i belong to W ; (2) w i is α i -maximal for all i ; (3) if i > j , then α i ( w j ) = 0 .Proof. First, by Lemma 4.2.3 the elements of W span the vector space R r .We construct the adapted basis inductively. Since W is finite, there exists amaximal weight, say w , for a functional α .Now, suppose that w , . . . , w k ∈ W ⊂ R r and α , . . . , α k ∈ ( R r ) ∨ are linearlyindependent and satisfy properties (1) − (3) . Pick any α k +1 ∈ { α ∈ ( R r ) ∨ | α ( w ) = . . . = α ( w k ) = 0 } \ { } ⊂ ( R r ) ∨ , and let w k +1 ∈ W be α k +1 -maximal. By the condition on α k +1 , w k +1 does notbelong to the span of w , . . . , w k . This completes the proof by induction. (cid:3) In the language of weights, Lemma 4.0.1 becomes the following:
Lemma 4.2.5.
Let w, w ′ ∈ W be distinct elements; then {− w, − w − w ′ , − w ′ } ∩ W = ∅ . If furthermore w = w λ is the weight of an eigenvalue λ of f ∗ , or f ∗ , , then − w ∈ W .Remark . If λ ∈ Λ has maximal weight, then | λ | − / ∈ Λ (and in particular theweight w λ ∈ W is simple by Lemma 4.0.1); therefore, if w, w ′ are maximal weights,by Lemma 4.2.5, − w − w ′ ∈ W .As a preliminary result, we bound the rank of g : Lemma 4.2.7.
The rank of g satisfies r ( g ) ≤ .Proof. Let us fix bases w , . . . , w r and α , . . . , α r of R r and ( R r ) ∨ respectively asin Lemma 4.2.4, and suppose by contradiction that r ≥ .Since w , w and w are maximal, we have − w , − w , − w / ∈ W , and thereforeby Lemma 4.2.5 − w − w , − w − w ∈ W. Since − w − w and − w − w are both maximal for α , by Remark 4.2.6 we have w + w + 2 w ∈ W. However this contradicts the α -maximality of w . (cid:3) We will show later that the rank of g is < if and only if its dynamical degrees λ ( g ) and λ ( g ) satisfy a multiplicative relation: λ ( g ) m = λ ( g ) n , ( m, n ) ∈ N \ { (0 , } ; see Corollary 4.3.2. ✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁ ❆❆❆❆❆❆❆❆❆❆❆❆❆❆❆❆✑✑✑✑✑✑✑✑✑✑✑✑ ◗◗◗◗◗◗◗◗◗◗◗◗ s ss✉ ss s ✁✁✁✁✁✁✁✁❆❆❆❆❆❆❆❆ s s ✁✁✁✁ s w w w − w − w − w w w − w Figure 1.
An example of the structure of W ⊂ R (without takingmultiplicities into account) in the case r ( g ) = 2 ; here, with thenotation of Proposition 4.3.1, n = 2 , n = 1 , n = 3 .4.3. The case r ( g ) = 2 . Throughout this section, we assume that the rank of g (i.e. the split-rank of G = h g i Zar , see Section 4.1) is equal to ; in other words, theelements of W span a real vector space of dimension . Proposition 4.3.1.
Let g ∈ GL( H ∗ ( X, R )) be a semisimple linear automorphismpreserving the cohomology graduation, Hodge decomposition, wedge product andPoincaré duality. Denote by λ = λ ( g ) and λ = λ ( g ) the spectral radii of g and g respectively, by Λ the spectrum of g (with multiplicities) and by W the set ofweights of eigenvalues λ ∈ Λ (with multiplicities).Let w (respectively w ) be the weight of W associated to the eigenvalue λ ∈ Λ (respectively λ − ∈ Λ ).Assume that the rank of g is equal to . Then (1) λ − λ is an element of Λ , whose weight is w := − w − w ; (2) w , w and w are maximal weights of W , and in particular they have mul-tiplicity in W ; in other words, λ , λ − and λ − λ are simple eigenvalues g , and no other eigenvalue of g has modulus λ , λ − or λ − λ ; (3) for any eigenvalue λ ∈ Λ such that | λ | / ∈ { λ , λ − , λ − λ , } , we have | λ | − ∈ Λ ; (4) there exist n , n , n ≥ such that, up to multiplicities, W \ { } = [ i =1 , , (cid:26) w i ( − n ; n = 0 , . . . , n i (cid:27) . Proof.
Let us fix an adapted basis w , w of R as in Lemma 4.2.4, and let w := − w − w . We show first properties (2) − (4) for these w , w , w , and then that, afterpossibly permuting indices, w , w and w are the weights of λ , λ − , λ − λ ∈ Λ respectively.The maximality of w , w is part of Lemma 4.2.4; since α ( w ) = 0 , w is also α -maximal. Property (2) then follows from Remark 4.2.6. OHOMOLOGY ACTION OF THREEFOLD AUTOMORPHISMS 19
Now let λ ∈ Λ be an eigenvalue of g whose weight w is different from w , w , w ;we want to show that | λ | − ∈ Λ . • Suppose first that w = 0 (i.e. | λ | = 1 ), and let λ ′ ∈ Λ be an eigenvaluewhose weight is w ; recall that by maximality − w / ∈ W and w is a simpleweight. If | λ | − = 1 / ∈ Λ , then by Lemma 4.0.1 we would have λλ ′ ∈ Λ ,which contradicts the simplicity of w . • Now suppose that α ( w ) = 0 ; since α ( w ) and α ( w ) have different sign,we have either | α ( − w − w ) | > | α ( w ) | or | α ( − w − w ) | > | α ( w ) | = | α ( w ) | , so that by maximality − w − w and − w − w cannot be bothweights of W . Since, again by maximality, − w , − w / ∈ W , by Lemma4.0.1 | λ | − ∈ Λ . • Finally suppose that w = 0 and α ( w ) = 0 , so that w ∈ R w . We repeatthe inductive construction of an adapted basis as in the proof of Lemma4.2.4 starting with w ′ := w , which is maximal for α ′ := α ; pick anynon-trivial α ′ ∈ w ⊥ ⊂ ( R ) ∨ and w ′ ∈ W maximal for α ′ . If we hadagain α ′ ( w ) = 0 , then w ∈ R w ∩ R w = { } , a contradiction; thus we canconclude as above.This shows property (3) .Property (4) follows from property (3) : indeed, if a w didn’t satisfy property (4) ,by (3) we would have ( − n w ∈ W \ { } for all n ∈ N , contradicting the finitenessof W .Now let us show that, after permuting indices, w and w are the weights of λ and λ − respectively. Since λ ( g ) = λ ( g − ) , it is enough to show that the weightof λ is one of the w i .Suppose by contradiction that the weight w of λ is not one of the w i ; then byproperty (4) there exist k > and i ∈ { , , } such that w = w i ( − k . Since λ is the spectral radius of g , we have λ / ∈ Λ ; since by property (3) we have ( − h w ∈ W for all h = 0 , . . . k , we must have k = 1 . Up to permuting the indices,we may suppose that i = 1 , so that w = − w = w + w . Denoting by λ, λ ′ ∈ Λ the eigenvalues associated to w and w , this means that | λλ ′ | = λ ; since λ is the spectral radius of g , this implies that | λ | = | λ ′ | = λ , contradictingthe assumption that r ( g ) = 2 .This shows that we may assume that w and w are the weights of the eigenvalues λ , λ − ∈ Λ . Since w = − w − w has multiplicity in W , it is associated to a realsimple eigenvalue, which is λ − λ by Lemma 4.0.1. This concludes the proof. (cid:3) Proposition 4.3.1 shows in particular that, if r ( g ) = 2 , then λ and λ aremultiplicatively independent: λ ( g ) m = λ ( g ) n , m, n ∈ Z ⇔ m = n = 0 . Indeed, the weights w and w form a base of R .Conversely, if r = 1 , since all the weights of g can be interpreted as integers, λ and λ satisfy a non-trivial equation λ ( g ) m = λ ( g ) n . Thus we have the following: ✉ s w s − w s w s w s − w s w s − w Figure 2.
Example of weights of W ⊂ R (without taking multi-plicities into account) in the case r ( g ) = 1 ; here, with the notationof Proposition 4.4.1, n = 2 , n = 1 , n = 1 . Corollary 4.3.2.
The rank of g is equal to if and only if the dynamical degreesof g are multiplicatively independent. The case r ( g ) = 1 . Recall that we denote by λ = λ ( g ) and λ = λ ( g ) thedynamical degrees of g , by Λ the spectrum of g (with multiplicities) and by W theset of weights of eigenvalues λ ∈ Λ (with multiplicities).Throughout all this section, we assume that the rank of g (i.e. the split-rank of G = h g i Zar , see Section 4.1) is equal to ; in other words, the elements of W span areal vector space of dimension . In this case the weights are equipped with a naturalorder: w λ > w λ ′ if and only if | λ | > | λ ′ | ; for w ∈ W we set | w | := max { w, − w } . Proposition 4.4.1.
Let g ∈ GL( H ∗ ( X, R )) be a semisimple linear automorphismpreserving the cohomology graduation, Hodge decomposition, wedge product andPoincaré duality. Denote by λ = λ ( g ) and λ = λ ( g ) the spectral radii of g and g respectively, by Λ the spectrum of g (with multiplicities) and by W the set ofweights of eigenvalues λ ∈ Λ (with multiplicities).Let w (respectively w ) be the weight of W associated to the eigenvalue λ ∈ Λ (respectively λ − ∈ Λ ).Assume that the rank of g is equal to (i.e. λ > and λ and λ are notmultiplicatively independent). Then (1) λ − λ is an element of Λ , whose weight is w := − w − w ; (2) for any eigenvalue λ ∈ Λ such that | λ | / ∈ { λ , λ − , λ − λ , } , we have | λ | − ∈ Λ ; (3) there exist n , n , n ≥ such that, up to multiplicities, W \ { } = [ i =1 , , (cid:26) w i ( − n ; n = 0 , . . . , n i (cid:27) . Remark . Let g = f ∗ s , where f : X → X is an automorphism and f ∗ s denotesthe semisimple part of the induced linear automorphism f ∗ ∈ GL( H ∗ ( X, R )) . Then w / ∈ {− w , − w / } if and only if the log-concavity inequalities p λ ( f ) ≤ λ ( f ) ≤ λ ( f ) are strict (see Proposition 1.3.5). If this is the case, then by Proposition 4.3.1 and4.4.1 λ and λ − are the only eigenvalues of f ∗ having such modulus and they aresimple. This was already proven in greater generality by Truong in [Tru14]. Proof.
After possibly replacing g by g − , we may suppose that w = | w | ≥ | w | = − w , so that w is maximal; let v , v ∈ H ( X, R ) denote eigenvectors for theeigenvalues λ , λ − ∈ Λ .Remark that, if v ∈ H ( X, C ) is an eigenvector whose eigenvalue λ ∈ Λ has weight w ∈ ]0 , w [ , then by Lemma 4.0.1 applied to v and v and by maximality of w wehave v ∧ ¯ v = 0 . OHOMOLOGY ACTION OF THREEFOLD AUTOMORPHISMS 21
Let us prove first that λ − λ ∈ Λ . If w = − w , then λ − λ = λ − and theclaim is evident; therefore we may suppose that w = − w . We observe first that,since λ − is the minimal modulus of eigenvalues of g , w is the minimal weight forthe natural order introduced above. Now, if we had λ − λ / ∈ Λ , then by Lemma4.0.1 we would have λ ∈ Λ and in particular − w ∈ W ; since we supposed that − w = w , by the above remark this implies that w ∈ W , contradicting theminimality of w . This shows (1) .Now let us show that for any λ ∈ Λ whose weight is w ∈ W \ { , w , w , w } andfor any eigenvector v ∈ H ( X, C ) with eigenvalue λ we have v ∧ ¯ v = 0 . The case w > (i.e. | λ | ≥ ) has been treated above; let w < , and suppose by contradictionthat v ∧ ¯ v = 0 . Then by Lemma 4.0.1 we get − w − w ∈ W , and since − w − w > and − w − w = w by assumption, we also have w + 2 w ∈ W (see the remark atthe beginning of the proof); this contradicts the minimality of w for the naturalorder, and concludes the proof of (2) .Property (3) follows from (2) by induction: indeed, if a w didn’t satisfy property (3) , by (2) we would have ( − n w ∈ W \ { } for all n ∈ N , contradicting thefiniteness of W .Now assume that w = − w and suppose by contradiction that w has multi-plicity > in W . Then by Lemma 4.2.5 − w ∈ W , and since − w > we alsohave w ∈ W . This contradicts the minimality of λ for the natural order andproves (4) . (cid:3) Automorphisms of threefolds: the semisimple case, proof ofTheorem C
As in §4, let X be a compact Kähler threefold and let g ∈ GL( H ∗ ( X, R )) be a semisimple linear automorphism preserving the Hodge decomposition, the wedgeproduct and Poincaré duality; furthermore, suppose now that g and g − are definedover Z . These are properties (1) − (4) in §1.1.We denote as usual by λ and λ the dynamical degrees of g (i.e. the spectral radiiof g and g respectively), by Λ the spectrum of g (with multiplicities) and by W the set of weights of g (with multiplicities).Recall that we pick all eigenvectors of g inside the union of subspaces H , ( X ) ∪ ( H , ( X ) ⊕ H , ( X )) (see Remark 1.2.3).Let P ( T ) be the minimal polynomial of g ; since g is defined over Z , we have P ( T ) ∈ Z [ T ] . Since g is semisimple, we can write P ( T ) = P ( T ) · · · P n ( T ) , where the P i ∈ Z [ T ] are distinct and irreducible over Q . Let P be the factor having λ ( g ) as a root, and denote by Λ i ⊂ Λ (respectively W i ⊂ W ) the set of roots of P i (respectively the set of weights of roots of P i ).For i = 1 , . . . , n let V i = ker P i ( g ) ⊂ V := H ( X, C ) . Since g is semisimple, we have V = n M i =1 V i . For a polynomial Q ∈ C [ T ] define Q ∨ ( T ) = T deg Q · Q ( T − ) . Poincaré’s duality allows to identify H ( X, C ) with H ( X, C ) ∨ = V ∨ ; under thisidentification, we have g = ( g − ) ∨ , so that the minimal polynomial of g is P ∨ ( T ) = P ∨ ( T ) · · · P ∨ n ( T ) . Since g is semisimple, we have V ∨ = H ( X, C ) = n M i =1 V ∨ i . Finally, let us define the bilinear map θ : H ( X, C ) × H ( X, C ) → H ( X, C )( u, v ) u ∧ v. Remark . The V i and the V ∨ i are g -invariant subspaces defined over Q ; fur-thermore, if the roots of some P i are simple eigenvalues of g (or, equivalently, if P i is a simple factor of the characteristic polynomial of g ), then V i is minimal for suchproperty: { } is the only proper subspace of V i which is g -invariant and definedover Q . The same holds for the action of g on V ∨ i .The goal of this section is to describe which are the possible (moduli of) rootsof a given P i , most importantly for the factor having λ as a root.In what follows, we say for short that λ, λ ′ ∈ Λ are conjugate if they are conjugateover Q . Remark . Since P i (0) = ± , we have Y λ ∈ Λ i λ = ± , X w ∈ W i w = 0 . Definition 5.0.3.
Let λ ∈ Λ ; we say that a weight w ∈ W is conjugate to λ if oneof the conjugates of λ has weight w . The main technical tool for the proofs in this section is the following basic resultin Galois theory (see for example [Lan02]).
Lemma 5.0.4.
Let α, β ∈ Q be two algebraic numbers. If α and β are conjugate,then there exists ρ ∈ Gal( Q / Q ) = { ρ ∈ Aut( Q ) | ρ | Q = id Q } such that ρ ( α ) = β . Remark that, since elements of
Gal( Q / Q ) act as the identity on Q , the polyno-mials P i are fixed; in particular Gal( Q / Q ) acts by permutations on each Λ i and oneach W i .Recall that the rank r ( g ) of g (i.e. the split-rank of G = h g i Zar , see Section 4.1),which is the number of multiplicative parameters necessary to describe the moduliof the eigenvalues of g , is no greater than by Lemma 4.2.7.Furthermore, we saw in Corollary 4.3.2 that r ( g ) = 2 if and only if λ and λ are multiplicatively independent. We will distinguish the two cases r ( g ) = 2 and r ( g ) = 1 .5.1. The case r ( g ) = 2 . Let us treat first the case where the rank of g (see Section4.1) is equal to .We denote as usual by w , w , w ∈ W the weights of the eigenvalues of g α := λ , α := λ − , α := λ − λ , and fix non-null eigenvectors v , v , v ∈ H ( X, R ) for these eigenvalues. OHOMOLOGY ACTION OF THREEFOLD AUTOMORPHISMS 23
Algebraic properties of the eigenvalues.
Lemma 5.1.1.
Let r = 2 and λ ∈ Λ . If one of the conjugates of λ has modulus ,then λ is a root of unity.Proof. By a lemma of Kronecker, if all the conjugates of an algebraic integer λ havemodulus , then λ is a root of unity. Therefore, we only need to show that, if aconjugate of λ has modulus , then λ has also modulus . Suppose by contradictionthat this is not the case, and let µ be a conjugate of λ such that | µ | = 1 .Let ρ ∈ Gal( Q , Q ) be such that ρ ( µ ) = λ ; since µ · ¯ µ = 1 , we have λ · ρ (¯ µ ) = 1 , so that ρ (¯ µ ) = λ − . In terms of weights, this means that w λ and w λ − = − w λ areboth non-trivial weights of W . This contradicts Proposition 4.3.1 and concludesthe proof. (cid:3) Proposition 5.1.2.
Let r = 2 . Then for all ≤ k ≤ n there exist n i = n i ( k ) , i = 1 , , , such that, without taking multiplicities into account, W k ⊂ (cid:26) w ( − n , w ( − n +1 , w ( − n , w ( − n +1 , w ( − n , w ( − n +1 (cid:27) . Proof.
Let λ ∈ Λ i , and let w = w λ be its weight. We will prove that if a weight w ′ collinear to w is conjugate to λ , then w ′ ∈ n − w , w, − w o . The claim then follows easily.Suppose by contradiction that w ′ = w λ ′ / ∈ {− w/ , w, − w } ; remark first thatby Lemma 5.1.1 w and w ′ are both non-trivial. By Proposition 4.3.1, after maybeswapping λ and λ ′ , we have w ′ = ( − k w, k ≥ , which means that λ ¯ λ = ( λ ′ ¯ λ ′ ) ( − k . Now, let ρ ∈ Gal( Q / Q ) be an automorphism such that ρ ( λ ) is a conjugate of λ whose weight can be written as w a / ( − n a , a ∈ { , , } , with n a maximal. Let α, β, γ, δ denote the images of λ, ¯ λ, λ ′ , ¯ λ under ρ , and let w α = w a ( − n a , w β = w b ( − n b , w γ = w c ( − n c , w δ = w d ( − n d denote their weights; here a, b, c, d ∈ { , , } , n a , n b , n c , n d ≥ and n a is maximal.Since αβ = ( γδ ) ( − k , in terms of weights we get w a ( − n a + w b ( − n b = ( − k w c ( − n c + ( − k w d ( − n d , so that w a + ( − n a − n b w b = ( − k + n a − n c w c + ( − k − n a − n d w d . Let
Γ = Z w ⊕ Z w ⊂ R be the lattice generated by w , w . Since k ≥ wehave w a + ( − n a − n b w b ≡ , which is impossible. This leads to a contradiction and concludes the proof. (cid:3) Corollary 5.1.3.
Let r = 2 and λ ∈ Λ . If λ is not a root of unity, then its degreeover Q is a multiple of .Proof. Fix λ ∈ Λ which is not a root of unity, and let P j be the unique factor of P having λ as a root; according to Proposition 5.1.2, there exist n , n , n ≥ suchthat the weights of conjugates of λ are elements of the set (cid:26) w ( − n , w ( − n +1 , w ( − n , w ( − n +1 , w ( − n , w ( − n +1 (cid:27) . Since by Remark 5.0.2 we have X w ∈ W i w = 0 , we get (cid:18) k ( − n + h ( − n +1 (cid:19) w + (cid:18) k ( − n + h ( − n +1 (cid:19) w + (cid:18) k ( − n + h ( − n +1 (cid:19) w = 0 , where the k i and the h i are the multiplicities of the weights in W j . Since the onlylinear dependency among the w i is w + w + w = 0 , this implies that there existsa constant c ∈ Z [1 / such that k i ( − n i + h i ( − n i +1 = c, i = 1 , , . This implies that k i + h i ≡ c mod 3 , so that in particular P i ( k i + h i ) ≡ modulo . (cid:3) Algebraic properties of λ . Now let us focus on the factor P having λ as aroot. Recall that we denoted by θ the map on H ( X, C ) × H ( X, C ) defined by θ : ( u, v ) u ∧ v ∈ H ( X, C ) . Lemma 5.1.4.
Let r = 2 , and let P be the factor of P having λ as a root. Ifeither v , v ∈ V or v , v / ∈ V , then θ ( V × V ) = V ∨ . If either v ∈ V , v ∈ V i or v ∈ V i , v ∈ V for some i = 1 , then θ ( V × V ) = V ∨ ⊕ V ∨ i . Proof.
Without loss of generality, in the second case we may assume that i = 2 .Let us first prove the ⊆ inclusions. Denote by θ the restriction of θ to V × V ;let π : V ∨ = n M i =1 V ∨ i → n M i =2 V ∨ i be the projection onto the last n − factors and π , : V ∨ = n M i =1 V ∨ i → n M i =3 V ∨ i be the projection onto the last n − factors.For π ∈ { π , π , } , the subspace ker( π ◦ θ ) := { u ∈ V ; π ◦ θ ( u, v ) = 0 for all v ∈ V } ⊂ V OHOMOLOGY ACTION OF THREEFOLD AUTOMORPHISMS 25 is g -invariant and defined over Q . By minimality of V (see Remark 5.0.1), wethen have either ker( π ◦ θ ) = 0 or ker( π ◦ θ ) = V . Therefore, in order to showthe inclusions, we only need to prove that v ∈ ker( π ◦ θ ) ( π = π in the first caseand π = π , in the second case); since g is semisimple, it is enough to check that π ◦ θ ( v , v ) = 0 for all eigenvectors v ∈ V .Let β be the eigenvalue associated to an eigenvector v ∈ V , and let w = w β beits weight. We distinguish the following subcases: • w = 0 is excluded by Lemma 5.1.1; • if w / ∈ { w , w , − w / } , then − w − w / ∈ W , so that v ∧ v = 0 and inparticular π ◦ θ ( v , v ) = 0 ; • if w = − w / and v ∧ v = 0 , then v ∧ v is an eigenvector with eigenvalue βλ = ¯ β − . Since β is conjugated to λ , so is ¯ β , and thus v ∧ v ∈ V ∨ and π ◦ θ ( v , v ) = 0 ; • if w = w , by simplicity of the weight w we have β = α ; in particular v ∈ V . Then v ∧ v = v ∧ v is an eigenvector for g with eigenvalue α α = α − . If also v ∈ V , then v ∧ v ∈ V ∨ ; if v / ∈ V , say v ∈ V ,then v ∧ v ∈ V ∨ . In both cases, choosing the right π ∈ { π , π , } we get π ◦ θ ( v , v ) = 0 ; • the case w = w is analogous to the case w = w .This concludes the proof of the ⊆ inclusions.Let us now prove the other inclusions ⊇ .Suppose first that either v , v ∈ V or v , v / ∈ V , so that θ ( V × V ) ⊆ V ∨ . Since θ ( V × V ) is g -invariant and defined over Q , by minimality of V ∨ we only need toshow that θ ( V × V ) = { } . Since dim V ≥ , this follows from Lemma 4.0.1.Now suppose that v ∈ V , v ∈ V i for some i = 1 (the proof of the other case beinganalogous), so that θ ( V × V ) ⊆ V ∨ ⊕ V ∨ i . Then v ∧ v ∈ V ∨ is an eigenvectorwith eigenvalue α − , so that v ∧ v ∈ V ∨ i . Since θ ( V × V ) is g -invariant anddefined over Q , by minimality of V ∨ and V ∨ i we have either θ ( V × V ) = V ∨ i or θ ( V × V ) = V ∨ ⊕ V ∨ i . The first case contradicts Lemma 5.1.5 below, so equality must hold and the proofis complete. (cid:3)
Lemma 5.1.5.
Let r = 2 , and let P be the factor of P having λ as a factor.Suppose that θ ( V × V ) ⊆ V ∨ i for some ≤ i ≤ n . Then i = 1 .Proof. Assume by contradiction that i = 1 , say i = 2 . By the ⊆ inclusions inLemma 5.1.4 (whose proof is independent on the result we want to prove here), wemay then assume that v ∈ V , v ∈ V .Let us prove first that − w / , − w / / ∈ W . Indeed, suppose for example that − w / ∈ W , and let v ∈ V be an eigenvector whose eigenvalue λ has weight − w / . Then by Lemma 4.0.1 we have v ∧ ¯ v = 0 , so that | λ | = λ − is aneigenvalue of the restriction of g to θ ( V × V ) = V ∨ . This contradicts the factthat λ − is an eigenvalue of g restricted to V ∨ , and proves that − w / / ∈ W ; theproof for − w / is analogous. Now let us prove that w / ( − n / ∈ W for n ≥ . Suppose by contradiction that v ∈ V is an eigenvector whose eigenvalue λ has weight − w / ( − n , n ≥ . Thenby Lemma 4.0.1 v ∧ ¯ v = 0 is a non-trivial eigenvector with eigenvalue µ = | λ | ; since θ ( V × V ) = V ∨ , µ is conjugated to µ ′ = α − , and these two algebraic integerssatisfy an algebraic equation µ ( − N = µ ′ N ≥ . Using Lemma 5.0.4 it is not hard to see that for this to happen we need to have µ = µ ′ = 1 , a contradiction. This proves that w / ( − n / ∈ W for n ≥ .Now, by Proposition 5.1.2 this implies that, up to multiplicities, W ⊂ n w , w , − w o . This however contradicts the equation X w ∈ W w = 0 . The claim is then proven. (cid:3)
Remark . Lemma 5.1.4 and 5.1.5 still hold if one permutes α , α and α ; theproofs are completely analogous.We are ready to state and prove Theorem C in the case where λ and λ aremultiplicatively independent (i.e. the rank of g is equal to ). Proposition 5.1.7.
Let g ∈ GL( H ( X, R )) be a semisimple linear automorphismpreserving the cohomology graduation, Hodge decomposition, wedge product andPoincaré duality, and such that g and g − are defined over Z . Let λ and λ be the spectral radii of g and g respectively, and suppose that λ and λ aremultiplicatively independent.Then the conjugates of λ ( g ) have all modulus belonging to the following set: (cid:26) λ , λ − , λ − λ , q λ − , p λ , q λ λ − (cid:27) . More accurately, there exists a permutation α , α , α of the eigenvalues λ , λ − , λ − λ of g , such that one of the following is true: (1) α , α and α are all cubic algebraic integers without real conjugates; (2) α is a cubic algebraic integer without real conjugates; α and α are con-jugate to one another, and their other conjugates are pairs of conjugatecomplex numbers with modulus α − / , α − / , α − / ( k, k + 1 and k + 1 pairsrespectively, k ≥ ); (3) α , α and α are conjugate, and their other conjugates are pairs of conju-gate complex numbers with modulus α − / , α − / , α − / ( k pairs for eachmodule, k ≥ ).Proof. Thanks to Lemma 5.1.4 and Remark 5.1.6, up to permutation of indicesonly three situations are possible:(1) α , α and α are not mutually conjugate. In this case, denoting by P i thefactor of P having α i as a factor, Lemma 5.1.4 implies that θ ( V i × V i ) = V ∨ i . OHOMOLOGY ACTION OF THREEFOLD AUTOMORPHISMS 27
Then W i = { w i , − w i / } ; indeed, suppose by contradiction that w = w j / ( − n ∈ W i for some j = i , and let v ∈ V i be an eigenvector whose eigenvalue hasweight w . Then by Lemma 4.0.1, either n = 0 or v ∧ ¯ v = 0 , so that − w ∈ W i . By a recursive argument, this proves that w j ∈ W i , whichcontradicts the assumption that α j and α i are not conjugate. Therefore,by Proposition 5.1.2, W i ⊆ n w i , − w i o . Since P w ∈ W i w = 0 , the multiplicity of − w i / must be , which implies thatthe conjugates of α i are two conjugate complex numbers. This concludesthe proof of case (1) .(2) α and α are conjugate, while α is not. The above proof shows that α iscubic without real conjugates. Let P (respectively P ) be the factor of P having α (respectively α and α ) as a root; by Lemma 5.1.4 and Remark5.1.6 we have θ ( V × V ) = V ∨ ⊕ V ∨ . Suppose by contradiction that w = w / ( − n ∈ W for some n ≥ , and let v ∈ V be an eigenvector whose eigenvalue λ has weight w . Then by Lemma4.0.1 v ∧ v = 0 , so that | λ | − ∈ Λ ∪ Λ ; since α is a non-trivial powerof | λ | − , these two numbers cannot be conjugate, therefore | λ | − ∈ Λ .Inductively, this shows that β = α − / ∈ Λ is conjugate to α and α . Wecan write β = α α . Since { α , α } ∩ { λ , λ − } 6 = ∅ , we can find ρ ∈ Gal( Q / Q ) such that ρ ( β ) = λ or ρ ( β ) = λ − ; both caseslead to a contradiction by maximality of λ (respectively, by minimality of λ − ) among the moduli of eigenvalues of g . Therefore w / ( − n / ∈ W for n ≥ and n = 0 .By Proposition 5.1.2 this implies that, up to multiplicities, W ⊂ n w , w , − w , − w , − w o . The equation X w ∈ W w = 0 implies that the multiplicities of − w / , − w / , − w / are h, h + 2 , h + 2 respectively for some h ≥ . Since α cannot be conjugate to α − / , h = 2 k is even, which concludes the proof of case (2) .(3) α , α and α are conjugate. Then, by Proposition 5.1.2, up to multiplici-ties, W ⊂ n w , w , w , − w , − w , − w o . The equation X w ∈ W w = 0 implies that the multiplicities of − w / , − w / , − w / are alle equal to h for some h ≥ . Since α cannot be conjugate to α − / , h = 2 k is even,which concludes the proof of case (3) . (cid:3) The case r ( g ) = 1 . Let us now suppose that the rank of g is equal to (seeSection 4.1). Recall that in this case the weights are equipped with a natural order: w λ > w λ ′ if and only if | λ | > | λ ′ | ; for w ∈ W we set | w | := max { w, − w } .Denote as usual by w , w , w ∈ W the weights of the eigenvalues λ , λ − , λ − λ ∈ Λ respectively. Lemma 5.2.1.
Suppose that r = 1 and let λ = λ be a real conjugate of λ ; then λ = λ − . In this case λ = λ .Proof. Since r = 1 , there exist integers m, n , not both equal to , such that λ m = λ n . Suppose that | m | ≥ | n | (the case | n | ≥ | m | is proven in the same way) and let ρ ∈ Gal( Q / Q ) be such that ρ ( λ ) = µ , where µ is a conjugate of λ whose weight hasmaximal modulus. Denoting by w and w ′ the weights of µ and ρ ( λ ) respectively,the above equation implies that mw = nw ′ ; by maximality of | w | we get | m | = | n | , so that λ = λ − as claimed.In order to prove that in this case λ = λ , it suffices to apply Proposition 4.4.1:if this were not the case, since λ − ∈ Λ , then either λ − = λ − λ , a contradiction,or λ ∈ Λ , contradicting the maximality of λ . (cid:3) The following proposition implies Theorem C in the case where λ and λ arenot multiplicatively independent. Proposition 5.2.2.
Let g ∈ GL( H ( X, R )) be a semisimple linear automorphismpreserving the cohomology graduation, Hodge decomposition, wedge product andPoincaré duality, and such that g and g − are defined over Z . Let λ and λ be thespectral radii of g and g respectively, and suppose that the rank of g is equal to (i.e. λ > and λ and λ are not multiplicatively independent). Then • either λ and λ are both cubic without real conjugates; • or λ = λ = λ , λ and λ − are conjugate, and all of their other conjugatesare pairs of conjugate complex numbers of modulus √ λ, or √ λ − ( k, k ′ and k pairs respectively, k, k ′ ≥ ).Proof. Denote by P the factor of P having λ as a root. Case 1: λ / ∈ {√ λ , λ , λ } . We show first that θ ( V × V ) = V ∨ . Since λ is a simple eigenvalue of g by Proposition 4.4.1, V is minimal among the g -invariant subspaces defined over Q (see Remark 5.0.1), and so is V ∨ ; therefore,as in the proof of Lemma 5.1.4, in order to prove that θ ( V × V ) ⊆ V ∨ we onlyneed to show that v ∧ v ∈ V ∨ for all eigenvectors v ∈ V . Let v ∈ V be an eigenvector with eigenvalue λ and let w = w λ , and let as usual w := − w − w . OHOMOLOGY ACTION OF THREEFOLD AUTOMORPHISMS 29 • If w / ∈ {− w / , w , w } , then v ∧ v = 0 . Indeed, if this were not the case,then − w − w ∈ W ; by the assumption and Proposition 4.4.1,the smallestweights of W (with respect to the natural order) are w < − w . Therefore w > − w / ⇒ − w − w < − w , which implies that − w − w = w , i.e. w = w , contradicting the assump-tion. • If w = − w / , i.e. λ ¯ λ = λ − , then v ∧ ¯ v is an eigenvector with eigenvalue ¯ λ − ; since λ and ¯ λ are conjugate, this implies that v ∧ v ∈ V ∨ . • If w = w or w = w , then λ = λ − or λ = λ − λ is a real conjugateof λ ; but then by Lemma 5.2.1 we have λ = λ − , which contradicts theassumptions on λ . Thus this case cannot occur.We have showed that θ ( V × V ) ⊆ V ∨ .The vector space θ ( V × V ) is non-empty by Lemma 4.0.1, g -invariant and definedover Q . Therefore, equality follows from minimality.Now let us show that λ is cubic without real conjugates. Since X w ∈ W w = 0 and since w has multiplicity in W , we only need to show that the conjugates of λ have weight − w / . Let λ be a conjugate of λ with weight w and let v be aneigenvector for λ . • If we had w = w , then by simplicity of such weight we have λ = λ − ,contradicting Lemma 5.2.1 since λ = λ . • If we had w = 0 , a conjugate λ of λ would satisfy λ ¯ λ = 1 . Applying ρ ∈ Gal( Q / Q ) such that ρ ( λ ) = λ , we would have that λ − is aconjugate of λ , so that λ = λ , a contradiction. • If w = w , since λ − λ = λ , λ − , Lemma 5.2.1 implies that λ / ∈ R . Lemma 4.0.1 applied to v and v impliesthat v ∧ ¯ v = 0 : indeed otherwise we would have v ∧ ¯ v = 0 (contradictingthe minimality of λ − > λ − ) or v ∧ ¯ v = 0 (contradicting the simplicity ofthe weight w ).Therefore, since θ ( V , V ) = V ∨ , we have | λ | − = λ λ − ∈ Λ and by Lemma 5.2.1 either λ λ − = λ , i.e. λ = λ , a contradiction or | λ | − = λ − ∈ Λ and λ = λ , again a contradiction. • Finally, if w / ∈ { , w , − w / , w , w } , then by Proposition 4.4.1 v ∧ ¯ v = 0 would be an eigenvector with (real) eigenvalue | λ | . Since θ ( V × V ) = V ∨ ,this implies that | λ | − ∈ Λ ; by Lemma 5.2.1 we would have | λ | = λ ± , acontradiction. This shows that λ is cubic without real conjugates; the proof for λ is completelyanalogous. Case 2: λ ∈ {√ λ , λ } . Up to replacing g by g − , we may assume that λ = √ λ ; let us show that λ and λ are both cubic without real conjugates.Remark that by Proposition 4.4.1, up to multiplicities W \ { } = (cid:26) w ( − n , n = 0 , . . . , N (cid:27) . Let us show first that λ is cubic without real conjugates. Since X w ∈ W w = 0 and since w has multiplicity in W , we only need to show that W ⊂ { w , − w / } .Let w ∈ W be the weight of an eigenvalue λ ∈ Λ . • If w = 0 , we show as in Case 1 that λ = λ , a contradiction. • If w = w / ( − n and n ≥ , then we argue as in the proof of Proposition5.1.2 to obtain a contradiction: indeed in this case ( λ ¯ λ ) k = λ , | k. Applying ρ ∈ Gal( Q / Q ) such that ρ ( λ ) has weight w / ( − n with n max-imal and letting w ′ and w ′′ be the weights of ρ ( λ ) and ρ (¯ λ ) respectively,we would have k ( w ′ + w ′′ ) = w ( − n , a contradiction modulo Z w / ( − n − .Therefore W ⊂ { w , − w / } and thus λ is cubic without real conjugates.Now let us prove that λ is also cubic without real conjugates; this is equivalentto λ − being cubic without real conjugates. Let P be the factor of P having λ − as a root. Since λ = √ λ , λ has degree or over Q ; the same proof as aboveand the simplicity of the weight w show that W ⊂ n w , − w o . Since X w ∈ W w = 0 , if λ had degree then the multiplicity of the weight w in W would be equal to , contradicting the fact that λ − is a real eigenvalue with weight w . Therefore λ is cubic, and by Lemma 5.2.1 it doesn’t have any real conjugate. Case 3: λ = λ . Suppose that λ is not a cubic algebraic integer without realconjugates. Denote by P the factor of P having λ as a root; since X w ∈ W w = 0 , and since the weight w ∈ W has multiplicity , λ is not cubic without realconjugates if and only if some conjugate λ of λ has weight w / ∈ { w , − w / } .Let us prove first that in this case λ and λ − are conjugate. We distinguish thefollowing sub-cases: OHOMOLOGY ACTION OF THREEFOLD AUTOMORPHISMS 31 • w = − w . Then, since λ − is the only eigenvalue with weight − w , λ and λ − are conjugate. • ≤ | w | < w / . Since the rank r is equal to , λ and λ satisfy an equation ( λ ¯ λ ) m = λ n , and since | w | < w / we have | n | < | m | . By Lemma 5.0.4 there exists ρ ∈ Gal( Q / Q ) such that ρ ( λ ) = λ ; let λ ′ = ρ (¯ λ ) , λ ′′ = ρ ( λ ) , and let w ′ , w ′′ be their weights respectively. Then the above equation implies that(5.1) mw + mw ′ = nw ′′ ⇔ w = − w ′ + nm w ′′ . This implies that either w ′ = w or w ′′ = w ; indeed, if this were not thecase, by Proposition 4.4.1 we would have | w ′ | , | w ′′ | ≤ max (cid:26) | w | , | w | , | w + w | (cid:27) = | w | this would contradict equation 5.1 because | n/m | < .We have shown that w is a conjugate weight of λ ; since λ − is the onlyeigenvalue with weight w , this means that λ and λ − are conjugate asclaimed. • w = w / . We may assume that we don’t fall in one of the cases above, i.e.that W ⊂ n w , w , − w o . We show that θ ( V × V ) = V ∨ ; in order to do that it suffices to show that v ∧ v ∈ V ∨ for every eigenvector v ∈ V (see Case 1 above and the proofof Lemma 5.1.4). If the eigenvalue µ of the eigenvector v has weight w or w / , then v ∧ v = 0 ; if the weight is − w / , then v ∧ v is an eigenvectorwith eigenvalue ¯ µ − , hence v ∧ v ∈ V ∨ . Therefore θ ( V × V ) = V ∨ asclaimed.Now let v ∈ V be an eigenvector with eigenvalue of weight w / ; by Propo-sition 4.4.1, v ∧ ¯ v = 0 , so that − w ∈ W . This shows that we fall in one ofthe above cases, and in particular λ and λ − are conjugate.We have shown that λ and λ − are conjugate. By Lemma 5.2.1 there are noother real conjugates, therefore in order to complete the proof we only need to showthat ± w n / ∈ W for n ≥ . This can be proven exactly as in Case 2. (cid:3)
Examples on tori.
In this section we provide examples of automorphismsof compact complex tori of dimension which show that (almost) all of the sub-cases of Proposition 5.1.7 and 5.2.2 can actually occur. For more examples see[OT15, Ogu17]. Lemma 5.3.1.
Let P ∈ Z [ T ] be a monic polynomial of degree n all of whose rootsare distinct and non-real and such that P (0) = 1 . Then there exists a compactcomplex torus X of dimension n and an automorphism f : X → X such that the characteristic polynomial of the linear automorphism f ∗ : H ( X, C ) → H ( X, C ) is equal to P . Proof.
Let P ( T ) = T n + a n − T n − + . . . + a T + 1 ∈ Z [ T ] be any polynomial.We will prove first that there exists a linear diffeomorphism f of the real torus M = R n / Z n such that the induced linear automorphism f ∗ ∈ GL( H ( M, R )) hascharacteristic polynomial P . Indeed, the companion matrix A = A ( P ) = . . . −
11 0 0 . . . − a . . . − a ... . . . . . . . . . ... ...... − a n − · · · − a n − has characteristic polynomial P ; since A ∈ SL n ( Z ) , the induced linear automor-phism f of R n preserves the lattice Z n and so does its inverse. Hence, A inducesa linear automorphism, which we denote again by f : f : R n (cid:30)Z n → R n (cid:30)Z n . Let dx i be a coordinate on the i -th factor of M = R n / Z n = ( R / Z ) n . In thebasis dx , . . . , dx n of H ( X, R ) , the matrix of f ∗ is exactly the transposed A T ; inparticular, the characteristic polynomial of f ∗ is equal to P .In order to conclude the proof, we will show that, if the roots of P are all distinctand non-real, then M can be endowed with a complex structure J such that f isholomorphic with respect with the structure J . Let β , ¯ β , . . . , β n , ¯ β n ∈ C \ R be the roots of P , and let V i = ker( f − β i I )( f − ¯ β i I ) ⊂ R n , where we have identified f with the linear automorphism of R n induced by thematrix A .The V i are planes such that R n = n M i =1 V i . The restriction of f to V i is diagonalizable with eigenvalues β i and ¯ β i ; therefore thereexists a unique complex structure J i on V i such that, with respect to a holomorphiccoordinate z i on V i ∼ = C , the action of f is the multiplication by β i : f | V i ( z i ) = β i z i . The complex structures J i induce a complex structure on R n ; by canonically iden-tifying R n with the tangent space at any point of M , we get an almost-complexstructure on M . It is not hard to see that J is integrable, and that f is holomorphicwith respect to J . This concludes the proof. (cid:3) Let us apply Lemma 5.3.1 to the three-dimensional case: fix a monic polynomial P ∈ Z [ T ] of degree such that P (0) = 1 , and suppose that its roots β , β , β , β = ¯ β , β = ¯ β , β = ¯ β OHOMOLOGY ACTION OF THREEFOLD AUTOMORPHISMS 33 are all distinct and non-real.By Lemma 5.3.1, there exists a -dimensional complex torus X = C / Λ and an au-tomorphism f : X → X such that the induced linear automorphism f ∗ : H ( X, C ) → H ( X, C ) has characteristic polynomial P . Remark that the proof of the Lemmashows something more precise: the restriction of f ∗ to H , ( X ) (respectively to H , ( X ) ) is diagonalizable with eigenvalues β , β , β (respectively ¯ β , ¯ β , ¯ β ).Since for a complex torus the wedge-product of forms induces isomorphisms H , ( X ) ∼ = V H , ( X ) , H , ( X ) ∼ = H , ( X ) ⊗ H , ( X ) , H , ( X ) ∼ = V H , ( X ) , the eigenvalues of f ∗ ∈ GL( H ( X, R )) are exactly the numbers β i β j , ≤ i In this last section, we deal with the case of an automorphism f : X → X of acompact Kähler threefold X such that the action in cohomology f ∗ : H ( X, R ) → H ( X, R ) is neither (virtually) unipotent nor semisimple. As we saw in Section 2,this situation is not possible in the surface case; in the threefold case, we manage togive some constraints but not to completely exclude this situation. However, dueto restriction on the dimension, no examples can be produced on complex tori, andto the best of my knowledge no examples are known at all. Conjecture 4. Let f : X → X be an automorphism of a compact Kähler threefold.Then f ∗ : H ( X, R ) → H ( X, R ) is either semisimple or virtually unipotent. Proposition 6.0.1. Let X be a compact Kähler threefold and let f : X → X be anautomorphism such that λ ( f ) > and f ∗ is not semisimple. Then (1) λ ( f ) ∈ { p λ ( f ) , λ ( f ) } ; in particular, r ( f ) = 1 and λ = λ ( f ) and λ = λ ( f ) are both cubic without real conjugates; OHOMOLOGY ACTION OF THREEFOLD AUTOMORPHISMS 35 (2) if λ = λ , then the eigenvalue λ has a unique non-trivial Jordan blockwhose dimension is m ≤ ; the other eigenvalues having non-trivial Jor-dan blocks have modulus / √ λ , and their non-trivial Jordan blocks havedimension at most m − ; (3) analogously, if λ = √ λ , then the eigenvalue λ − has a unique non-trivialJordan block whose dimension is m ≤ ; the other eigenvalues having non-trivial Jordan blocks have modulus √ λ , and their non-trivial Jordan blockshave dimension at most m − . In what follows denote by g = f ∗ : H ∗ ( X, R ) → H ∗ ( X, R ) the linear automor-phism induced by f , and by g i the restriction of g to H i ( X, R ) . We will denote by λ i = λ i ( f ) the dynamical degrees and we will assume that λ ≥ λ > the case λ ≥ λ follows from the previous one by replacing f by f − . Lemma 6.0.2. If λ = λ , then λ has no non-trivial Jordan block for g . If λ = λ , then g has at most one non-trivial Jordan block for the eigenvalue λ ,whose dimension is at most . In either case, g does not have non-trivial Jordanblocks for other eigenvalues of modulus λ .Proof. Suppose first that λ = λ ; then, by Theorem B, w is a simple weight of(the semisimple part of) g . Therefore in particular λ has no non-trivial Jordanblock.Now suppose that λ = λ . We prove first that the Jordan blocks for theeigenvalue λ have dimension at most . Suppose by contradiction that there existsa Jordan block of dimension at least ; then, as in the proof of Theorem 3.0.1, wemay pick u , u , u , u ∈ H , ( X, R ) ∪ ( H , ( X ) ⊕ H , ( X )) R such that g ( u ) = λ u , g ( u i ) = u i +1 + λ u i i = 1 , , . Considering g n ( u ∧ u ) , g n ( u ∧ u ) ∈ H ( X, R ) , and applying Lemma 4.0.1 as in the proof of Theorem 3.0.1, we obtain a class v ∈ H ( X, R ) such that k g n v k ∼ cn k λ n = cn k λ n for some k ≥ . This means that the eigenvalue λ has a non-trivial Jordan block for g , and since g = ( g − ) ∨ , the eigenvalue λ − has a non-trivial Jordan block for g . Applying thefirst part of the claim to g − , we obtain that λ = λ , contradicting the assumptionthat λ = λ .This proves that Jordan blocks of g for the eigenvalue λ have dimension at most . Now let us prove that there exists a unique non-trivial Jordan block of g forthe eigenvalue λ . Suppose by contradiction that we can find linearly independentelements u , u , v , v ∈ H , ( X, R ) ∪ ( H , ( X ) ⊕ H , ( X )) R such that g ( u ) = λ u , g ( u ) = u + λ u , g ( v ) = λ v , g ( v ) = v + λ v . Then, considering g n ( u ∧ u ) , g n ( u ∧ v ) , g n ( v ∧ v ) and applying Lemma 4.0.1 to the classes u and v , we get as before a class v ∈ H ( X, R ) such that k g n v k ∼ cn k λ n = cn k λ n for some k ≥ , which yields a contradiction as above. This concludes the proof. (cid:3) Proof of Proposition 6.0.1. Let λ ∈ Λ be an eigenvalue of g with weight w suchthat g has a non-trivial Jordan block for λ of dimension k > . As in the proofof Theorem 3.0.1, we can take a Jordan basis u , . . . , u k ∈ H , ( X ) ∪ ( H , ( X ) ⊕ H , ( X )) such that g ( u ) = λu , g ( u i +1 ) = u i + λu i +1 i = 1 , . . . , k − . Suppose that λ ≥ λ , so that by Lemma 6.0.2 applied to f − the eigenvalue λ hasno non-trivial Jordan block. Let as usual w , w , w the weights of the eigenvalues α = λ , α = λ − , α = λ − λ ∈ Λ .We distinguish the following cases: • w / ∈ { , w , w , w } : by Propositions 4.3.1.(3) and 4.4.1.(2) we have u ∧ ¯ u = 0 . In particular, g n ( u k ∧ ¯ u k ) ∼ cn k − | λ | n ( u ∧ ¯ u ) , which means that g has a Jordan block of dimension ≥ k − for theeigenvalue | λ | . Since g = ( g − ) ∨ , g has a Jordan block of dimension ≥ k − > k for the eigenvalue | λ | − ∈ Λ ; • w = 0 ; take λ ∈ Λ with weight such that the dimension k of its maximalJordan block is maximal, and let v ∈ H , ( X, R ) ∪ ( H , ( X ) ⊕ H , ( X )) R be an eigenvector for the eigenvalue λ .Since v ∧ v = 0 , by Lemma 4.0.1 we have either u ∧ v = 0 or u ∧ ¯ u = 0 . Inthe first case, considering g n ( u k ∧ v ) we obtain a non-trivial Jordan blockfor an eigenvalue λ ′ of weight − w ; this implies that w = − w , and byProposition 4.4.1 the weight w is simple, contradicting the existence ofa non-trivial Jordan block. In the second case, considering g n ( u k ∧ ¯ u k ) we obtain a Jordan block of dimension ≥ k − > k for the eigenvalue | λ | − = 1 ; since has weight , this contradicts maximality. Therefore,this case cannot occur; • w = w : in this case, by Lemma 6.0.2 we have λ = λ , λ = λ and k ≤ ; • w = w : by Lemma 6.0.2 applied to f − this case cannot occur; • w = w : let v ∈ H , ( X, R ) ∪ ( H , ( X ) ⊕ H , ( X )) R be an eigenvectorfor the eigenvalue λ − . Since v ∧ v = 0 , by Lemma 4.0.1 we have either v ∧ ¯ u = 0 or u ∧ ¯ u = 0 .In the first case we obtain a non-trivial Jordan block for an eigenvalue ofweight w ; by Lemma 6.0.2 we have then λ = λ , thus w = w and, againby Lemma 6.0.2, λ = λ .In the second case, we get a non-trivial Jordan block for the eigenvalue | λ | − with dimension > k .The above computation show that, if g has a non-trivial Jordan block of dimension k for the eigenvalue λ ∈ Λ , then • either λ = λ , in which case λ = λ ; • or | λ | 6 = 1 and there is a Jordan block of dimension > k for the eigenvalue | λ | − . OHOMOLOGY ACTION OF THREEFOLD AUTOMORPHISMS 37 One proves inductively that g admits a non-trivial Jordan block for the eigenvalue λ . By Lemma 6.0.2, such block has dimension at most ; the claim follows fromthe fact that, as we proved above, the dimension of a non-trivial Jordan block of λ = λ is strictly smaller than that of a non-trivial Jordan block of | λ | − . 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