Dynamics of automorphisms on projective complex manifolds
aa r X i v : . [ m a t h . AG ] O c t DYNAMICS OF AUTOMORPHISMS ON PROJECTIVECOMPLEX MANIFOLDS
DE-QI ZHANG
Abstract.
We show that the dynamics of automorphisms on all projectivecomplex manifolds X (of dimension 3, or of any dimension but assuming theGood Minimal Model Program or Mori’s Program) are canonically built upfrom the dynamics on just three types of projective complex manifolds: com-plex tori, weak Calabi-Yau manifolds and rationally connected manifolds. Asa by-product, we confirm the conjecture of Guedj [20] for automorphisms on3-dimensional projective manifolds, and also determine π ( X ). Introduction
We work over the field C of complex numbers.We show that the dynamics of automorphisms on all projective complexmanifolds (of dimension 3, or of any dimension but assuming the GoodMinimal Model Program or Mori’s Program) are canonically built up fromthe dynamics on just three types of projective complex manifolds: complextori, weak Calabi-Yau manifolds, and rationally connected manifolds.For a similar phenomenon on the dynamics in dimension 2, we refer to[6]. Here a projective manifold X is weak Calabi-Yau or simply wCY ifthe Kodaira dimension κ ( X ) = 0 and the first Betti number B ( X ) = 0.A projective manifold X is rationally connected (the higher dimensionalanalogue of a rational surface) if any two points on X are connected by arational curve; see [5] and [31]. For a smooth projective surface X , it is wCYif and only if X itself or its ´etale double cover is birational to a K3 surface,while X is rationally connected if and only if it is a rational surface.For the recent development on complex dynamics, we refer to the surveyarticles [13] and [59] and the references therein. See also [7], [10], [37], [42]and [56]. For algebro-geometric approach to dynamics of automorphismsdue to Oguiso, see [45], [46] and [47].We shall consider dynamics of automorophisms on projective complexmanifolds of dimension ≥
3. To focus on the dynamics of genuinely high di-mension, we introduce the notions of rigidly parabolic pairs (
X, g ) and pairs(
X, g ) of primitively positive entropy , where X is a projective manifold ofdim X ≥
3, and g ∈ Aut( X ). In other words, these are the pairs where thedynamics are not coming from the dynamics of lower dimension; see Conven-tion 2.1, and also Lemma 2.20 for the classification of the rigidly parabolic Mathematics Subject Classification.
Key words and phrases. automorphism of variety, Kodaira dimension, topological entropy. pairs in the case of surfaces. These notions might be the geometrical incar-nations of McMullen’s lattice-theoretical notion ”essential lattice isometry”in [35] §
4. By the way, all surface automorphisms of positive entropy areautomatically primitive.In Theorem 1.1 below and Theorem 3.1 of §
3, it is shown that a pair (
X, g )of rigidly parabolic or of primitively positive entropy exists only when theKodaira dimension κ ( X ) ≤ q ( X ) ≤ dim X . If ( Y, g ) isjust of positive entropy, then one can only say that κ ( Y ) ≤ dim Y − §
3, we determine the g -structure of manifolds X of dimension ≥
3, with κ ( X ) = 0 and −∞ , respectively. The difficult partin Theorem 3.2 is to show that the regular action of g on the initial manifold X is equivariant to a nearly regular action on another better birationalmodel X ′ ; see Convention 2.1. Such difficulty occurs only in dimension ≥ Z with mild terminal singularities is minimal if the canonical divisor K Z isnef (= numerically effective).We now state the main results. The result below says that dynamicsoccur essentially only on those X with Kodaria dimension κ ( X ) ≤ X, g ) is of primitively positive entropy if g is of positiveentropy but g is not induced from an automorphism on a manifold of lowerdimension; ( X, g ) is rigidly parabolic if g is parabolic and every descentof g to a lower and positive dimensional manifold is parabolic, while anautomorphism h is parabolic if it is of null entropy and ord( h ) = ∞ ; see 2.1for the precise definitions. Theorem 1.1.
Let X be a projective complex manifold of dim X ≥ , andwith g ∈ Aut( X ) . Then we have: (1) Suppose that ( X, g ) is either rigidly parabolic or of primitively posi-tive entropy ( see ( )) . Then the Kodaira dimension κ ( X ) ≤ . (2) Suppose that dim X = 3 and g is of positive entropy. Then κ ( X ) ≤ ,unless d ( g − ) = d ( g ) = d ( g ) = e h ( g ) and it is a Salem number.Here d i ( g ) are dynamical degrees and h ( g ) is the entropy ( see ( )) . If (
X, g ) does not satisfy the conditions in Theorem 1.1(1), then a positivepower of g is induced from some automorphism on a manifold of lowerdimension. So by Theorem 1.1 and by the induction on the dimenion, wehave only to treat the dynamics on those X with κ ( X ) = 0 or −∞ . This isdone in Theorems 3.2 and 3.3. See the statements in § § YNAMICS OF AUTOMORPHISMS 3
Theorem 1.2.
Let X ′ be a smooth projective complex threefold. Supposethat g ∈ Aut( X ′ ) is of positive entropy. Then there is a pair ( X, g ) bira-tionally equivariant to ( X ′ , g ) , such that one of the cases below occurs. (1) There are a -torus ˜ X and a g -equivariant ´etale Galois cover ˜ X → X . (2) X is a weak Calabi-Yau threefold. (3) X is a rationally connected threefold in the sense of [5] and [30] . (4) d ( g − | X ) = d ( g | X ) = d ( g | X ) = e h ( g | X ) and it is a Salem number. The higher dimensional analogue of the result above is summarized inTheorems 3.2 and 3.3 in §
3, and it confirms the claim about the buildingblocks of dynamics made in the abstract of the paper. In the results there,we need to assume the existence of Good Minimal Models for non-uniruledvarieties. The recent breakthrough in [3] on the existence of flips and thefinite generation of canonical rings suggests that such existence problem ofa usual minimal model is quite within the reach in a near future, but thequestion about the Goodness of a minimal model in dimension ≥ Theorem 1.3.
Let X be a smooth projective complex threefold admittinga cohomologically hyperbolic automorphism g in the sense of [20] page .Then the Kodaira dimension κ ( X ) ≤ . More precisely, either X is a weakCalabi-Yau threefold, or X is rationally connected, or there is a g -equivariantbirational morphism X → T onto a Q -torus. We can also determine the fundamental group below. For the case of κ ( X ) = 0 we refer to Namikawa-Steenbrink [44, Corollary (1.4)]. Theorem 1.4.
Let X be a smooth projective complex threefold with g ∈ Aut( X ) of primitively positive entropy. Suppose that the Kodaira dimension κ ( X ) = 0 . Then either π ( X ) = (1) , or π ( X ) = Z ⊕ . For examples (
X, g ) of positive entropy with X a torus (well known case),or a rational manifold (take product of rational surfaces), or a Calabi-Yaumanifold, we refer to [7], [36] and [37], and Mazur’s example of multi-degreetwo hypersurfaces in P × · · · × P as in [10] Introduction. See also Remark1.7 below.We define the following sets of dynamical degrees for automorphisms ofpositive entropy, where X is wCY = weak Calabi-Yau if κ ( X ) = 0 = B ( X ),where rat.conn. = rationally connected is in the sense of [5] and [31], wheretype (*) = type (t) (torus), or type (cy) (weak Calabi-Yau), or type (rc)(rational connected). Note that D ∗ ( n ) ⊂ D ∗ ( n ) ′ . D t ( n ) := { λ ∈ R > | λ = d ( g ) for g ∈ Aut( X ) with X an n -torus } , DE-QI ZHANG D cy ( n ) := { λ ∈ R > | λ = d ( g ) for g ∈ Aut( X ) with X a wCY n -fold } ,D rc ( n ) := { λ ∈ R > | λ = d ( g ) for g ∈ Aut( X ) with X a rat.conn. n -fold } ,Sa = { λ ∈ R > | λ is a Salem number } . Denote by D ∗ ( n ) ′ the set of those λ ∈ R > satisfying the following: there area type (*) n -fold X , an ample Cartier integral divisor H on X , a sublattice L ⊂ NS( X ) / (torsion) containing H and a σ ∈ Hom Z ( L, L ) which is bijectiveand preserves the induced multi product form on L such that σ ∗ P ≡ λP fora nonzero nef R -divisor P in L ⊗ Z R .We conclude the introduction with the result and question below whichsuggest a connection between the existence of meaningful dynamics and thetheory of algebraic integers like the dynamical degrees d i ( g ).See McMullen [35] for the realization of some Salem numbers as dynamicaldegrees of K Theorem 1.5.
Let X be a smooth projective complex threefold. Suppose g ∈ Aut( X ) is of positive entropy. Then the first dynamical degree satisfies d ( g ) ∈ Sa ∪ D t (3) ∪ D cy (3) ∪ D rc (3) . Further, for some s > , d ( g s ) ∈ D rc (2) ′ ∪ k =2 { D t ( k ) ∪ D cy ( k ) ∪ D rc ( k ) } . The question below has a positive answer in dimension 2; see [6].
Question 1.6.
Let X be a projective complex manifold of dimension n ≥ and g ∈ Aut( X ) of primitively positive entropy. Does the first dynamicaldegree d ( g ) satisfy the following d ( g ) ∈ ∪ nk =2 { D t ( k ) ∪ D cy ( k ) ∪ D rc ( k ) } ? Remark 1.7.
Take the elliptic curve E i := C / ( Z + Z √− Z / (2)) ⊕ on the abelian variety A := E i × E i × E i such that X := A/ Γ is a smooth Calabi-Yau variety with K X ∼
0. The group SL ( Z ) acts on A , and as observed in [9, (4.5)], itcontains a free abelian subgroup G of rank 2 such that the action on A byeach id = g ∈ G is of positive entropy. If we can find such a g normalizingΓ, then g | A descends to a ¯ g ∈ Aut( X ) of positive entropy. Remark 1.8.
Like [45] - [47], [29], [43], [57] and [58], our approach isalgebro-geometric in nature.
Acknowledgement.
I would like to thankTien-Cuong Dinh and Nessim Sibony for the explanation of the paper [9]and the reference [8], Noboru Nakayama for answering questions on nef andbig divisors, Keiji Oguiso for sending me the papers [45]-[47] on dynamics of
YNAMICS OF AUTOMORPHISMS 5 automorphism groups on projective varieties, Eckart Viehweg for patientlyanswering my persistent questions on moduli spaces and isotrivial families,Kang Zuo for the explanation on families of Calabi-Yau manifolds, and thereferee for the suggestions for the improvement of the paper.I also like to thank the Max Planck Institute for Mathematics at Bonnfor the warm hospitality in the first quarter of the year 2007. This projectis supported by an Academic Research Fund of NUS.2.
Preliminary results
In this section we recall definitions and prove some lemmas.Results like Lemmas 2.20 and 2.23 are not so essentially used by thepaper, but are hopefully of independent interest. (1) For a linear transformation T : V → V of a vector space V , let ρ ( T )be the spectral radius of T , i.e., the largest modulus of eigenvalues of T .(2) We shall use the terminology and notation in Kawamata-Matsuda-Matsuki [28] and Koll´ar - Mori [32]. Most of the divisors are R -Cartierdivisors: P si =1 r i D i with r i ∈ R and D i Cartier prime divisor.Let X be a projective manifold. Set H ∗ ( X, C ) = ⊕ ni =0 H i ( X, C ). There isthe Hodge decomposition: H k ( X, C ) = ⊕ i + j = k H i,j ( X, C ) . Denote by H i,ia ( X, C ) the subspace of H i,i ( X, C ) spanned by the algebraicsubvarieties of complex codimension i .(3) Let Pic( X ) be the Picard group , and NS( X ) = Pic( X ) / (algebraicequivalence) = H ( X, O ∗ X ) / Pic ( X ) ⊆ H ( X, Z ) the Neron-Severi group .NS( X ) is a finitely generated abelian group whose rank is the Picard number .Set NS B ( X ) = NS( X ) ⊗ Z B ⊂ H ( X, B ) for B = Q and R . Let Nef( X ) bethe closed cone of nef divisors in NS R ( X ). So Nef( X ) is the closure of theample cone. Also Nef( X ) ⊆ K ( X ), the closure of the K¨ahler cone K ( X ).Let N ( X ) be the R -space generated by algebraic 1-cycles modulo numer-ical equivalence; see [32] (1.16). When X is a surface, N ( X ) = NS R ( X ).(4) Let g ∈ Aut( X ). Denote by ρ ( g ) the spectral radius of g ∗ | H ∗ ( X, C ). Itis known that either ρ ( g ) >
1, or ρ ( g ) = 1 and all eigenvalues of g ∗ | H ∗ ( X, C )are of modulus 1. When log ρ ( g ) > ρ ( g ) = 0) we say that g is of positive entropy (resp. null entropy ).We refer to Gromov [18], Yomdin [55], Friedland [12], and Dinh - Sibony[10] page 302, for the definition of the i -th dynamical degree d i ( g ) for 1 ≤ i ≤ n = dim X (note that d n ( g ) = 1 now and set d ( g ) = 1) and the actualdefinition of the topological entropy h ( g ) which turns out to be log ρ ( g ) inthe setting of our paper.(5) Let Y be a projective variety and g ∈ Aut( Y ). We say that g is of positive entropy , or null entropy , or parabolic , or periodic , or rigidly parabolic ,or of primitively positive entropy (see the definitions below), if so is g ∈ DE-QI ZHANG
Aut( ˜ Y ) for one (and hence all) g -equivariant resolutions as guaranteed byHironaka [24]. The definitions do not depend on the choice of ˜ Y becauseevery two g -equivariant resolutions are birationally dominated by a thirdone, by the work of Abramovich - Karu - Matsuki - Wlodarczyk; see Matsuki[34] (5-2-1); see also Lemma 2.6.(6) We use g | Y to signify that g ∈ Aut( Y ).(7) In this paper, by a pair ( Y, g ) we mean a projective variety Y and anautomorphism g ∈ Aut( Y ). Two pairs ( Y ′ , g ) and ( Y ′′ , g ) are g - equivariantlybirational , if there is a birational map σ : Y ′ · · · → Y ′′ such that the bira-tional action σ ( g | Y ′ ) σ − : Y ′′ · · · → Y ′′ extends to a biregular action g | Y ′′ .(8) g ∈ Aut( Y ) is periodic if the order ord( g ) is finite. g is parabolic iford( g ) = ∞ and g is of null entropy.(9) ( Y ′ , g ) is rigidly parabolic if ( g | Y ′ is parabolic and) for every pair ( Y, g )which is g -equivariantly birational to ( Y ′ , g ) and for every g -equivariantsurjective morphism Y → Z with dim Z >
0, we have g | Z parabolic.(10) Let Y ′ be a projective variety and g ∈ Aut( Y ′ ) of positive entropy(so dim Y ′ ≥ Y ′ , g ) is of primitively positive entropy if it is not ofimprimitive positive entropy, while a pair ( Y ′ , g ) is of imprimitively positiveentropy if it is g -equivariantly birational to a pair ( Y, g ) and if there is a g -equivariant surjective morphism f : Y → Z such that either one of thetwo cases below occurs.(10a) 0 < dim Z < dim Y , and g | Z is still of positive entropy.(10b) 0 < dim Z < dim Y , and g | Z is periodic.(11) Remark.
We observe that in Case(10b), for some s > g s | Z = id and that g s acts faithfully on the general fibre Y z of Y → Z , suchthat g s | Y z is of positive entropy. To see it, we replace g s by g for simplicity.In view of Lemma 2.6, we may assume that Y z is connected by making useof the Stein factorization, and also that both Y and Z are smooth after g -equivariant resolutions as in Hironaka [24]. Let 0 = v g ∈ Nef( Y ) be a nefdivisor as in Lemma 2.4 such that g ∗ v g = d ( g ) v g with d ( g ) := d ( g | Y ) > v g | Y z = 0. Indeed, take very ample divisors H , . . . , H k with k = dim Z . Then f ∗ H . . . f ∗ H k = cY z homologously with c =( H . . . H k ) >
0. Noting that f ∗ H . . . f ∗ H i = 0 and g ∗ ( f ∗ H j ) = f ∗ ( g ∗ H j ) = f ∗ H j and applying Lemma 2.3 repeatedly, we get f ∗ H . . . f ∗ H i .v g = 0 forall i ≤ k . In particular, 0 = f ∗ H . . . f ∗ H k .v g = cY z .v g = cv g | Y z homolo-gously; see Fulton [15] (8.3) for the last equality. This claim is proved.Next we claim that d k +1 ( g ) := d k +1 ( g | Y ) ≥ d ( g | Y z ) ≥ d ( g ) with k =dim Z , so that g | Y z is of positive entropy. Indeed, g ∗ ( v g | Y z ) = d ( g )( v g | Y z )implies that d ( g | Y z ) ≥ d ( g ). By Lemma 2.4, g ∗ ( v ( g | Y z ) ) = ( g | Y z ) ∗ v ( g | Y z ) = d ( g | Y z ) v ( g | Y z ) , so d k +1 ( g ) ≥ d ( g | Y z ).In [42, Appendix, Theorem D], we will show that d ( g s | Y ) = d ( g s | Y z )for the s > YNAMICS OF AUTOMORPHISMS 7 (12)
Remark.
By the observation above and Lemma 2.2, if dim Y ≤ Y, g ) is of positive entropy, then dim Y = 2 and the pair( Y, g ) is always of primitively positive entropy.(13) We refer to Iitaka [25] for the definition of D - dimension κ ( X, D ); the
Kodaira dimension κ ( X ) = κ ( ˜ X ) = κ ( ˜ X, K ˜ X ) with ˜ X → X a projectiveresolution; and the Iitake fibring (of X ): X ′ → Y ′ with X ′ birational to X ,both X ′ and Y ′ smooth projective, dim Y ′ = κ ( X ) (= κ ( X ′ )) and κ ( X ′ y ) = 0for a general fibre X ′ y over Y ′ . Note that κ ( X ) attains one of the values: −∞ , 0 , , . . . , dim X . We say that X is of general type if κ ( X ) = dim X .(14) Remark.
The Iitaka fibring is defined by the pluri-canonical system | rK X | for r >> g -equivariant blowup to resolve base points in the sys-tem; see Hironaka [24]. So we can always replace ( X, g ) by its g -equivariantblowup ( X ′ , g ) such that there is a g -equivariant Iitaka fibring X ′ → Y ′ withprojective manifolds X ′ and Y ′ and with dim Y ′ = κ ( X ′ ) = κ ( X ). Note that κ ( X ) is a birational invariant.(15) A projective manifold X of dimension n is uniruled if there is adominant rational map P × Y · · · → X , where Y is a projective manifoldof dimension n − X is a Q - torus in the sense of [40] if there isa finite ´etale cover T → X from a torus T .(17) A projective manifold is a weak Calabi-Yau manifold (or wCY forshort) if the Kodaira dimension κ ( X ) = 0 and if the irregularity q ( X ) = h ( X, O X ) = 0. A normal projective variety ¯ X with only terminal singu-larity is a Calabi-Yau variety if the canonical divisor K ¯ X satisfies sK ¯ X ∼ s > q ( ¯ X ) = 0. So a projective resolution X of aCalabi-Yau variety ¯ X is a weak Calabi-Yau manifold. Conversely, assumingthe minimal model program, every weak Calabi-Yau manifold is birational toa Calabi-Yau variety. We refer to [28] or [32] for the definition of singularityof type: terminal, canonical, log terminal , or rational .(18) An algebraic integer λ > r + 1) over Q with r ≥
0, is a
Salem number (see [50] or [36] §
3) if all conjugates of λ over Q (including λ itself) are given as follows, where | α i | = 1: λ, λ − , α , ¯ α , . . . , α r , ¯ α r . The following result is fundamentally important in the study of complexdynamics. For the proof, we refer the readers to [18], [55], [12], [10] (2.5)and page 302, [8] Proposition 5.7, [11] before (1.4), [13] the Introduction,[19] (1.2), (1.5), (1.6).
Lemma 2.2.
Let X be a projective manifold of dimension n , and g ∈ Aut( X ) . Then the following are true. (1) d i ( g ) = ρ ( g ∗ | H i,i ( X, R )) = ρ ( g ∗ | H i,ia ( X, R )) , ≤ i ≤ n . DE-QI ZHANG (2) d ( g ) = ρ ( g ∗ | NS R ( X )) ( see also [10] (3 . . (3) h ( g ) = log ρ ( g ) = max ≤ i ≤ n log d i ( g ) . (4) The entropy h ( g ) > holds if and only if the dynamical degree d ℓ ( g ) > for one ( and hence for all ) 1 ≤ ℓ ≤ n − by (5)) . (5) The map ℓ d ℓ ( g ) /d ℓ +1 ( g ) with ≤ ℓ ≤ n − , is non-decreasing.So d ℓ ( g ) ≤ d ( g ) ℓ and d n − ℓ ( g ) ≤ d ℓn − ( g ) for all ≤ ℓ ≤ n . Alsothere are integers m, m ′ such that: d ( g ) < d ( g ) < · · · < d m ( g ) = · · · = d m ′ ( g ) > · · · > d n ( g ) = 1 . The following very useful result is proved in Dinh-Sibony [9] (3.2), (4.4).
Lemma 2.3.
Let X be a projective manifold of dimension n . Let Nef( X ) ∋ P , P ′ , P i (1 ≤ i ≤ m ; m ≤ n − be nef divisors. Then we have: (1) Suppose that P .P = 0 in H , ( X, R ) . Then one of P , P is amultiple of the other. (2) We have P · · · P m .P.P ′ = 0 ∈ H m +2 ,m +2 ( X, R ) if the two conditionsbelow are satisfied. (2a) P . . . P m .P = 0 and P . . . P m .P ′ = 0 . (2b) One has g ∗ ( P . . . P m .P ) = λ ( P . . . P m .P ) and g ∗ ( P . . . P m .P ′ ) = λ ′ ( P . . . P m .P ) , for some g ∈ Aut( X ) and distinct (positive) realnumbers λ and λ ′ . We refer to Dinh-Sibony [10] (3.5) for a result including the one belowand with more analytical information.
Lemma 2.4. (Generalized Perron-Frobenius Theorem) Let X be a projectivemanifold and g ∈ Aut( X ) . Then there are non-zero nef divisors v g and v g − in Nef( X ) such that: g ∗ v g = d ( g ) v g , ( g − ) ∗ v g − = d ( g − ) v g − . Proof.
To get the first equality, we apply to the nef cone Nef( X ) of thePerron - Fobenius Theorem for cones as in Schneider - Tam [51] page 4,item 5. The second is the application of the first to g − . This proves thelemma. (cid:3) Here is the relation between dynamical degrees of automorphisms.
Lemma 2.5.
Let X be a projective manifold of dimension n , and g ∈ Aut( X ) . Then we have: (1) Denote by Σ( g ) i = Σ( g | X ) i the set of all eigenvalues of g ∗ | H i,i ( X, C ) (including multiplicities). Then Σ( g ) = Σ( g − ) n − . (2) The dynamical degrees satisfy d ( g ) = d n − ( g − ) . (3) g is of positive entropy (resp. null entropy; periodic; parabolic) if andonly if so is g m for some (and hence for all) m = 0 .Proof. There is a natural perfect pairing H , ( X, C ) × H n − ,n − ( X, C ) → C YNAMICS OF AUTOMORPHISMS 9 induced by the cup product, via the Hodge decomposition. This pairing ispreserved by the action of g ∗ ; see Griffiths - Harris [17] page 59. So a simplelinear algebraic calculation shows that if g ∗ | H , ( X, C ) is represented by amatrix A then g ∗ | H n − ,n − ( X, C ) is represented by the matrix ( A t ) − . Thusthe lemma follows; see Lemma 2.2. (cid:3) The result below shows that the first dynamical degree of an automor-phism is preserved even after lifting up or down by a generically finite andsurjective morphism.
Lemma 2.6.
Let f : X → Y be a g -equivariant generically finite surjectivemorphism between projective manifolds of dimension n ≥ . Then we have: (1) d ( g | X ) = d ( g | Y ) . (2) g | X is of positive entropy (resp. null entropy; periodic) if and onlyif so is g | Y . (3) g | X is of parabolic if and only if so is g | Y . (4) If g | X is rigidly parabolic then so is g | Y .Proof. (1) Set Σ X = Σ( g ∗ | NS R ( X )) = { λ ∈ R | g ∗ D = λD for a divisor0 = D ∈ NS R ( X ) } . We show first that Σ Y ⊆ Σ X , whence d ( g | Y ) ≤ d ( g | X )by Lemma 2.2. Indeed, assume that g ∗ ¯ L = λ ¯ L for some 0 = ¯ L ∈ NS R ( Y )and λ ∈ Σ Y . Set L := f ∗ ¯ L . Then g ∗ L = f ∗ g ∗ ¯ L = λL . Note that 0 = L ∈ NS R ( X ) ⊂ H ( X, R ) because f ∗ : H ∗ ( Y, R ) → H ∗ ( X, R ) is an injective ringhomomorphism; see [1] I (1.3). Therefore, λ ∈ Σ X .Conversely, let 0 = L := v g ∈ Nef( X ) such that g ∗ L = d L with d = d ( g | X ), as in Lemma 2.4. Set ¯ L := f ∗ L . For any H ∈ H n − ( Y, R ),by the projection formula, we have g ∗ ¯ L.H = ¯
L.g ( − ∗ H = L.f ∗ g ( − ∗ H = L.g ( − ∗ f ∗ H = g ∗ L.f ∗ H = d L.f ∗ H = d ¯ L.H . So ( g ∗ ¯ L − d ¯ L ) .H = 0 for all H ∈ H n − ( Y, R ). Hence g ∗ ¯ L = d ¯ L in H ( Y, R ).We claim that ¯ L = 0 in NS R ( Y ), whence d = d ( g | X ) ∈ Σ Y , d ( g | X ) ≤ d ( g | Y ) by Lemma 2.2, and we conclude the assertion (1). Assume the con-trary that ¯ L = 0. Take an ample divisor H Y on Y . Then f ∗ H Y is nef and bigon X . So f ∗ H Y = A + D for an ample Q -divisor A and an effective Q -divisoron X , by Kodaira’s lemma. By the projection formula and the nefness of L ,one has 0 = ¯ L.H n − Y = L.f ∗ H n − Y = L.f ∗ H n − Y . ( A + D ) ≥ L.f ∗ H n − Y .A ≥· · · ≥ L.A n − ≥
0. Applying the Lefschetz hyperplane section inductivelyto reduce to the Hodge index theorem for surfaces and using the nefness of L , we see that L = 0 ∈ NS R ( X ) ⊆ H ( X, R ), a contradiction. So the claimand hence the assertion (1) are proved.With (1), the assertion (2) follows. Now (3) follows from (1) and (2).(4) Assume that g | X is rigidly parabolic. Modulo g -equivariant birationalmodification, we have only to show that g | Y is parabolic whenever Y → Y isa g -equivariant surjective morphism with dim Y >
0. This follows from theassumption on g | X and the g -equivariance of the composition X → Y → Y .This proves the lemma. (cid:3) We now describe the behavior of automorphisms dynamics in a fibration.
Lemma 2.7.
Let X → Y be a g -equivariant surjective morphism betweenprojective manifolds with dim X > dim
Y > . Then we have: (1) If g | X is of null entropy (resp. periodic), then so is g | Y . (2) Suppose that the pair ( X, g ) is either rigidly parabolic or of primi-tively positive entropy. Then g | Y is rigidly parabolic.Proof. The proof of (1) is similar to that of Lemma 2.6. Suppose the contrarythat (2) is false for some Y in (2). Then, after g -equivariant birationalmodification, there is a g -equivariant surjective morphism Y → Z withdim Z > g | Z is not parabolic. Thus, g | Z is periodic or of positiveentropy. This contradicts the rigidity or primitivity of ( X, g ) because 0 < dim Z ≤ dim Y < dim X . (cid:3) Here is the relation between Salem numbers and dynamical degrees in 2.1:
Lemma 2.8.
Let X be a projective manifold and g ∈ Aut( X ) of positiveentropy. Then we have: (1) If dim X = 2 , then d ( g ) = d ( g − ) = e h ( g ) and it is a Salem number. (2) Suppose dim X = 3 and there is a g -equivariant morphism f : X → Y onto a smooth projective curve Y with connected general fibre F .Then all e h ( g ± ) , d ( g ± ) , d ( g ± ) are equal and it is a Salem number.Proof. The result in Case(1) is well known and follows from Lemmas 2.2and 2.5 and the proof of McMullen [36] Theorem 3.2.We consider Case(2). Set L := (NS( X ) | F ) / (torsion) ⊂ NS( F ) / (torsion).We define the following intersection form h , i L on the lattice L : h D | F, D | F i L := D .D .F ∈ H ( X, Z ) ∼ = Z . This h , i L is compatible with the intersection form on NS( F ) via the restric-tion H ( X, Z ) → H ( F, Z ). This compatibility, the Hodge index theoremfor the smooth projective surface F , and the fact that H | F = 0 in L with H an ample divisor of X , imply that the lattice L is non-degenerate andhas signature (1 , r ) with 1 + r ≤ rank NS( F ). There is a natural action g ∗ | L on L given by g ∗ ( D | F ) = ( g ∗ D ) | F . Since g ∗ F = F in NS( X ), this action iswell defined and preserves the intersection form h , i L .Since g ∗ F = F and g ∗ v g = d v g with d = d ( g ) > F and v g are not proportional. So the Lefschetz hyperplanesection theorem on cohomology and the Cauchy-Schwarz inequality or theHodge index theorem for surfaces imply that v g .F.A = ( v g | A ) . ( F | A ) = 0 fora very ample divisor A on X . So v g .F = v g | F gives a non-zero v := v g | F ∈ L ⊗ Z R . Further, g ∗ v = ( g ∗ v g ) | F = d v . Since L is an integral lattice ofsignature (1 , r ) and g ∗ | L is an isometry of L , by the proof of McMullen [36]Theorem 3.2, d is a Salem number and all eigenvalues of g ∗ | L are given as: d , d − , α , ¯ α , . . . , α t , ¯ α t with | α i | = 1 and 2( t + 1) = r + 1. Arguing with g − | L , we get d ( g | X ) = d ( g − | X ) (= d ( g | X ) by (2.5)). The result follows; see Lemma 2.2. (cid:3) YNAMICS OF AUTOMORPHISMS 11
The following result (though it will not be used in the sequel) is a gener-alization of a well-known result in the case of surfaces.
Lemma 2.9.
Let X be a projective manifold of dimension n ≥ and g ∈ Aut( X ) of positive entropy. Let = v i ∈ Nef( X ) (1 ≤ i ≤ s ) be nef divisorssuch that g ∗ v i = λ i v i for scalars λ i with λ i > and that λ i are pairwisedistinct. Then we have: (1) s ≤ n − . (2) If s = n − , then d ( g ) = max ≤ i ≤ s { λ i } .Proof. (1) Applying Lemma 2.3 repeatedly, we see that u ( s ) := Q s i =1 v i = 0if s ≤ n . Note that g ∗ = id on H n,n ( X, R ) ∼ = R . If s ≥ n , then u ( n )is a non-zero scalar in H n,n ( X, R ), whence u ( n ) = g ∗ u ( n ) = λu ( n ) with λ := Q ni =1 λ i >
1. This is a contradiction.(2) Assume that s = n −
1. If d := d ( g ) is one of λ i , then (2) is trueby the maximality of d ( g ) as in Lemma 2.2. Suppose that d = λ i for all i .one gets a contradiction to (1) if one sets v n = v g in the notation of Lemma2.4. The lemma is proved. (cid:3) The result below shows that one may tell about the im/primitivity ofthreefold automorphisms by looking at the algebraic property of its dynam-ical degrees or entropy.
Lemma 2.10.
Let X be a smooth projective threefold and g ∈ Aut( X ) suchthat the pair ( X, g ) is of imprimitively positive entropy. Then we have: (1) All scalars e h ( g ± ) , d ( g ± ) , d ( g ± ) are equal and it is a Salem number. (2) For some s > , we have d ( g s ) ∈ D t (2) ∪ D cy (2) ∪ D rc (2) . We now prove Lemma 2.10. After g -equivariant birational modification,there is a g -equivariant surjective morphism f : X → Y such that dim X > dim Y > g | Y is of positive entropy or g | Y is periodic.Let X → Y → Y be the Stein factorization. After g -equivariant blowups asin Hironaka [24], we may assume that X , Y and Y are all smooth, X → Y has connected fibres F and Y → Y is generically finite and surjective.By Lemma 2.6, either g | Y is of positive entropy or g | Y is periodic. SinceQuestion 1.6 has a positive answer in dimension 2 as in Cantat [6], ourlemma follows from Lemma 2.8 and the claim below. Claim 2.11.
We have:(1) Suppose that g | Y is of positive entropy. Then the four scalars d ( g ± | X ), d ( g ± | Y ) coincide and we denote it by d .(2) Suppose that g | Y is periodic say g s | Y = id for some s >
0. Then thefour scalars d ( g ± s | X ), d ( g ± s | F ) coincide and we denote it by d s or d ( g s ). So d ( g | X ) = d ( g − | X ) = d .(3) For both cases in (1) and (2), if 0 = P ∈ Nef( X ) is a nef divisor suchthat g ∗ P = λP then λ ∈ { , d ± } .(4) For both cases in (1) and (2), d = d ( g ± | X ) = d ( g ± | X ) = e h ( g ± | X ) . Let us prove the claim. (4) follows from (1) - (2) and Lemmas 2.5 and 2.2.Consider the case where g | Y is of positive entropy. Thus dim Y ≥
2, whencedim Y = 2 and the fibre F is of dimension 1. The two scalars d ( g ± | Y )coincide and we denote it by d ; see Lemma 2.5. Set L ± := f ∗ v ( g ± | Y ) ∈ Nef( X ); see Lemma 2.4 for the notation. Note that L ± = 0 in NS R ( X ); see[1] I (1.3). Further, g ∗ L ± = f ∗ g ∗ v ( g ± | Y ) = d ± L ± . Thus, to prove the claimin the present case, we only have to show (3); see Lemma 2.4. If λ = d ± ,then u := L + .L − .P ∈ H , ( X, R ) ∼ = R is a non-zero scalar by Lemma 2.3and hence u = g ∗ u = d d − λu = λu . So λ = 1. This proves the claim forthe present case.Consider the case where g s | Y = id. As remarked in (2.1), g s | F is ofpositive entropy, so dim F = 2 and dim Y = 1. Further, d ( g − s | X ) = d ( g s | X ) ≥ d ( g s | F ) ≥ d ( g s | X ); see also Lemma 2.5. Arguing with g − ,we get d ( g s | X ) ≥ d ( g − s | F ) ≥ d ( g − s | X ). Since the two scalars d ( g ± s | F )coincide for surface F by Lemma 2.5, the above two sequences of inequalitiesimply (2). Consider u = v ( g | X ) .v ( g − | X ) .P as in the early case, one proves(3) for the present case. This proves the claim and also the lemma.The following lemma is crucial, which is derived from a result of Nakamura- Ueno, and Deligne as in Ueno [52] Theorem 14.10. See [42, Theorem A] forthe generalization of [52] Theorem 14.10 to meromorphic dominant maps onK¨ahler manifolds. Lemma 2.12.
Let X → Y a g -equivariant surjective morphism betweenprojective manifolds with dim Y > . Suppose that the pair ( X, g ) is ei-ther rigidly parabolic or of primitively positive entropy. Then the Kodairadimension κ ( Y ) ≤ . In particular, κ ( X ) ≤ .Proof. If κ ( Y ) = dim Y , then Aut( X ) is a finite group; see Iitaka [25] Theo-rem 11.12. Thus g | Y is periodic, which is impossibe by our assumption on g .Suppose that 0 < κ ( Y ) < dim Y . After replacing with g -equivariant blowupsof X and Y as in Hironaka [24], we may assume that Y → Z is a well-definedIitaka fibring with Y and Z smooth projective and dim Z = κ ( Y ) >
0. Thenatural homomorphism Bir( Y ) → Bir( Z ) between birational automorphismgroups, has a finite group as its image; see [52] Theorem 14.10. In particu-lar, g | Z is periodic. This contradicts the assumption on g , noting that thecomposition X → Y → Z is g -equivariant. This proves the lemma. (cid:3) For g of positive entropy (not necessarily being primitive), we have: Lemma 2.13.
The following are true. (1)
Let X be a projective manifold of dimension n . Suppose that g ∈ Aut( X ) is of positive entropy. Then the Kodaira dimension κ ( X ) ≤ n − . (2) Conversely, for every n ≥ and every k ∈ {−∞ , , , . . . , n − } ,there are a projective manifold X and g ∈ Aut( X ) of positive entropysuch that dim X = n and κ ( X ) = k . YNAMICS OF AUTOMORPHISMS 13
Proof. (1) Assume the contrary that κ ( X ) ≥ n −
1. After g -equivariantblowup as in Hironaka [24], we may assume that for some s >
0, one has | sK X | = | M | + F ix with
F ix the fixed part and with the movable part | M | base point free, so that f := Φ | M | : X → Y ⊂ P N with N = h ( X, M ),is the ( g -equivariant) Iitaka fibring. Note that M κ ( X ) is homologous to apositive multiple of a fibre of f and hence M r = 0 for every r ≤ κ ( X ). Also g ∗ M ∼ M (linearly equivalent). With the notation of Lemma 2.4 and byLemma 2.3, we have v g .M r = 0 in H r +1 ,r +1 ( X, R ) for all r ≤ n −
1. Since g ∗ = id on H n,n ( X, R ) ∼ = R , for the scalar u := v g .M n − ∈ H n,n ( X, R ), wehave u = g ∗ u = d u with d = d ( g ) >
1, so u = 0, a contradiction.(2) Let S be a surface with g ∈ Aut( S ) of positive entropy. Let Z be any( n − X := S × Z and g | X := ( g | S ) × (id Z ). Then g | X is of positiveentropy by looking at the K¨unneth formula for H ( X, C ) as in Griffiths-Harris [17] page 58; see also [8] Proposition 5.7. Also κ ( X ) = κ ( S ) + κ ( Z ).All values in {−∞ , } (resp. {−∞ , , , . . . , n − } ) are attainable as theKodaira dimension of a suitable S (resp. Z ); see, for instance, Cantat [7]and McMullen [37]. Thus (2) follows. This proves the lemma. (cid:3) We need the following result on the eigenvalues of g ∗ | H ∗ ( X, C ). Lemma 2.14.
Let X be a projective manifold of dimension n , and g ∈ Aut( X ) of null entropy. Then there is an integer s > such that ( g s ) ∗ | H ∗ ( X, C ) is unipotent, i.e., all eigenvalues are equal to .Proof. By Lemmas 2.2 and 2.5, every eigenvalue λ of g ∗ | H ∗ ( X, C ) has modu-lus 1. Since g ∗ is defined over ⊕ ni =0 H i ( X, Z ) / (torsion) the monic characteris-tic polynomial of g ∗ | H ∗ ( X, C ) has integer coefficients, whence all eigenvalues λ of g ∗ are algebraic integers. So every eigenvalue λ of g ∗ is an algebraicinteger and all its conjugates (including itself) have modulus 1. Thus these λ are all units of 1 by Kronecker’s theorem. The lemma follows. (cid:3) The result below says that a rigidly parabolic action on an abelian varietyis essentially the lifting of a translation.
Lemma 2.15.
Let A = 0 be an abelian variety and g ∈ Aut variety ( A ) .Suppose that the pair ( A, g ) is rigidly parabolic. Then there are integers s > , m ≥ and a sequence of abelian subvarieties B ⊂ B · · · ⊂ B m ⊂ A such that the following are true (setting B = 0 ). (1) The homomorphisms below are all g s -equivariant: A → A/B → · · · → A/B m = 0 . (2) g s | ( A/B m ) is a translation of infinite order (so the pointwise fixedpoint set ( A/B i ) g r = ∅ for all r ∈ s N and all ≤ i ≤ m ).Proof. We may assume that g ∗ | H ∗ ( A, C ) is already unipotent; see Lemma2.14. Assume that the pointwise fixed locus A g = ∅ . Then we may assumethat g | A is a homomorphism after changing the origin. By [4] (13.1.2), A g is a subgroup of positive dimension equal to that of the eigenspace of g ∗ | H , ( A, C ). Let B be the identity component of A g . Then 0 < dim B < dim A by the parabolic rigidity of g | A . The homomorphism A → A/B is g -equivariant. But now ( A/B ) g contains the origin (= the image of B ),and g | ( A/B ) is again rigidly parabolic by the definition. The parabolicrigidity of g | A helps us to continue this process forever. This contradictsthe finiteness of dim A .Therefore, A g = ∅ . Write g = t h with a translation t and a homomor-phism h . Then 0 = | A g | = | A h | in the notation of [4] (13.1.1), whence A h is ofpositive dimension equal to that of Ker( h ∗ − id) = Ker( g ∗ − id) ⊂ H , ( A, C ).Let B be the identity component of A h . Then g ( x + B ) = g ( x ) + B andhence the homomorphism A → A/B is g -equivariant. If B = A , then h = id and we are done. Otherwise, g | ( A/B ) is again rigidly parabolic bythe definition. Also ( A/B ) g r = ∅ for all r > B m +1 = A for some m . So some positive power g s | ( A/B m ) is a translationof infinte order. The lemma is proved. (cid:3) The density result (3) below shows that a rigidly parabolic action on anabelian variety is very ergodic.
Lemma 2.16.
Let A → Y be a g -equivariant generically finite surjectivemorphism from an abelian variety A onto a projective manifold Y . Assumethat g | A is rigidly parabolic. Then we have: (1) No proper subvariety of Y is stabilized by a positive power of g . (2) g has no periodic points. (3) For every y ∈ Y , the Zariski closure D := { g s ( y ) | s > } equals Y . (4) Suppose that f : X → Y is a g -equivariant surjective morphism froma projective manifold onto Y . Then f is a smooth morphism. Inparticular, if f is generically finite then it is ´etale.Proof. It suffices to show (1). Indeed, (2) is a special case of (1). If (3)is false, then some positive power g s fixes an irreducible component of D ,contradicting (1). If (4) is false, then the discriminant D = D ( X/Y ) isstabilized by g and we get a contradiction as in (3).For (1), since A → Y is g -equivariant, it is enough to show (1) for A ;see Lemma 2.6. Suppose the contrary that a positive power g v stabilizes aproper subvariety Z of A . To save the notation, rewrite g v as g . If Z isa point, then g s | ( A/B m ) fixes the image on A/B m of Z in the notation ofLemma 2.15, absurd. Assume that dim Z >
0. If the Kodaira dimension κ ( Z ) = 0, then Z is a translation of a subtorus by Ueno [52] Theorem 10.3,and we may assume that Z is already a torus after changing the origin; welet Z be the B in Lemma 2.15 and then g s | ( A/B m ) fixes the origin (= theimage of Z ), absurd.Suppose that κ ( Z ) ≥
1. By Ueno [52] or Mori [38] (3.7), the identitycomponent B of { a ∈ A | a + Z = Z } has positive dimension such that Z → Z/B ⊂ A/B is birational to the Iitaka fibring with dim( Z/B ) = κ ( Z ) YNAMICS OF AUTOMORPHISMS 15 and
Z/B of general type. We can check that g ( a + B ) = g ( a ) + B (write g as the composition of a translation and a homomorphism and then argue),so the homomorphism A → A/B is g -equivariant and g | ( A/B ) stabilizesa subvariety Z/B of general type (having finite Aut( Z/B ) by Iitaka [25]Theorem 11.12). Thus a positive power g v | ( A/B ) fixes every point in Z/B .So in Lemma 2.15, another positive power g s | ( A/B m ) fixes every point inthe image of Z/B , absurd. (cid:3) Here are two applications of Lemma 2.12 and Viehweg-Zuo [54] (0.2).
Lemma 2.17.
Let X be a projective manifold of Kodaira dimension κ ( X ) ≥ and g ∈ Aut( X ) . Suppose that f : X → P is a g -equivariant surjectivemorphism. Then g | P is periodic. In particular, ( X, g ) is neither rigidlyparabolic nor of primitively positive entropy.Proof. By [54] Theorem 0.2, f has at least three singular fibres lying overa set of points of P on which g | P permutes. Thus a positive power g s | P fixes every point in this set and hence is equal to the identity. (cid:3) Lemma 2.18.
Let f : X → Y be a g -equivariant surjective morphismfrom a projective manifold onto a smooth projective curve. Suppose that theKodaira dimension κ ( X ) ≥ . Suppose further that the pair ( X, g ) is eitherrigidly parabolic, or of primitively positive entropy ( so dim X ≥ .Then Y is an elliptic curve, g s | Y (for some s | ) is a translation of infiniteorder, and f is a smooth morphism.Proof. By Lemma 2.12, the Kodaira dimension κ ( Y ) ≤
0. So the arithmeticgenus p a ( Y ) ≤
1. By Lemma 2.17, Y is an elliptic curve. So g s | Y isa translation for some s |
6, which is of infinite order since g | Y is rigidlyparabolic by Lemma 2.7. In view of Lemma 2.16 the lemma follows. (cid:3) The following are sufficient conditions to have rational pencils on surfaces.
Lemma 2.19.
Let X be a smooth projective surface of Kodaira dimension κ ( X ) ≥ , and let g ∈ Aut( X ) . Let X → X m be the smooth blowdown to theminimal model. Then there is a unique g -equivariant surjective morphism τ : X → P such that g | P is periodic, if either Case (1) or (2) below occurs. (1) X m is a hyperelliptic surface. (2) g is parabolic. X m is K or Enriques.Proof. Since κ ( X ) ≥ X m of X is unique and hence X → X m is g -equivariant. So we may assume that X = X m .There are exactly two elliptic fibrations on a hyperelliptic surface X (seeFriedman-Morgan [14] § X )) andthe other is onto P . Thus Lemma 2.17 implies the result in Case (1).If X m is K
3, then the existence of τ follows from Cantat [7] (1.4).An Enrques X can be reduced to the K g | X lifts to aparabolic g | Y (see Lemma 2.6) on the universal K Y of X so that a positive power g s | Y stabilizes every fibre of an elliptic fibration on Y . This fibration descends to one on X fibre-wise stabilized by g s | X .For the uniqueness of τ in Case (2), if F i are fibres of two distinct g -equivariant fibrations, then g ∗ stabilizes the class of the nef and big divisor F + F and hence some positive power g r ∈ Aut ( X ) = (1) (see Lemma2.23). This is absurd. This proves the lemma. (cid:3) Now we classify rigidly parabolic actions on surfaces.
Lemma 2.20.
Let X be a smooth projective surface and g ∈ Aut( X ) suchthat the pair ( X, g ) is rigidly parabolic. Then there is a g s -equivariant (forsome s > ) smooth blowdown X → X m such that one of the following casesoccurs (the description in (3) or (4) will not be used in the sequel). (1) X m is an abelian surface ( so q ( X ) = 2) ; see also ( ) . (2) X m → E = Alb( X m ) is an elliptic ruled surface (so q ( X ) = 1 and E an elliptic curve). g s | E is a translation of infinite order. (3) X m is a rational surface such that K X m = 0 , the anti-canonical divi-sor − K X m is nef and the anti-Kodaira dimension κ ( X m , − K X m ) = 0 .For a very general point x ∈ X m , the Zariski closure D ( x ) := { g r ( x ) | r > } equals X m . (4) X m is a rational surface with K X m = 0 and equipped with a ( uniqueand relatively minimal ) elliptic fibration f : X m → P such that f is g s -equivariant. (5) One has X m = F e the Hirzebruch surface of degree e ≥ such thata/the ruling F e → P is g s -equivariant. (6) One has X = X m = P . So there are a g -equivariant blowup F → P of a g | P -fixed point and the g -equivariant ruling F → P . We now prove Lemma 2.20. By Lemma 2.12, the Kodaira dimension κ ( X ) ≤
0. Consider first the case κ ( X ) = 0. Then X contains only finitelymany ( − X m . So g | X descends to a biregular action g | X m . Rewrite X = X m . Then X is Abelian,Hyperelliptic, K X is an abelian surface.Consider next the case where X is an irrational ruled surface. Thenthere is a P -fibration f : X → E = alb X ( X ) with genral fibre X e ∼ = P ,so that p a ( E ) = q ( X ) ≥
1. All rational curves (especially ( − g | X permutes finitely many such ( − g s | X stabilizes every ( − X → X m be the g s -equivariant blowdown to a relatively minimal P -fibration f : X m → E where all fibres are P . By the proof of Lemma 2.18and replacing s , we may assume that E is an elliptic curve and g s | E is atranslation of infinite order. So Case(2) occurs.Consider the case where X is a rational surface. So Pic( X ) = NS( X ). As-sume that g ∗ | Pic( X ) is finite, then Ker(Aut( X ) → Aut(Pic( X ))) is infinite.Hence X has only finitely many ( − YNAMICS OF AUTOMORPHISMS 17 X → X m be a g s -equivariant smooth blow-down to a relatively minimal rational surface so that X m = P , or F e theHirzebruch surface of degree e ≥
0. Note that a/the ruling F e → P is g s -equivariant (the ′′ ′′ is to take care of the case e = 0 where there are tworulings on F e ). If X m = P but X = P , then we are reduced to the case F . If X = X m = P , the last case in the lemma occurs (one trianglizes tosee the fixed point).Assume that X is rational and g ∗ | Pic( X ) is infinite. By [57] Theorem4.1 (or by Oguiso [46] Lemma 2.8 and the Riemann-Roch theorem appliedto the v and the adjoint divisor K X + v there as well as Fujita’s unique-ness of the Zariski-decomposition for pseudo-effective divisors like v and K X + v as formulated in Kawamata [28] Theorem 7-3-1), there is a g -equivariant smooth blowdown X → X m such that K X m = 0, − K X m isnef and κ ( X m , − K X m ) ≥ X = X m .If κ ( X, − K X ) ≥
1, then Case(4) occurs by the claim (and the uniquenessof f ) below. Claim 2.21.
Let X be a smooth projective rational surface such that − K X is nef, K X = 0 and κ ( X, − K X ) ≥
1. Then X is equipped with a uniquerelatively minimal elliptic fibration f : X → P such that − K X is a positivemultiple of a fibre.We now prove the claim. Write | − tK X | = | M | + F ix for some t >>
F ix is the fixed part. Note that 0 ≤ M + M.F ix ≤ ( − tK X ) = 0.Thus M ∼ rF (linearly equivalent) with | F | a rational free pencil, notingthat q ( X ) = 0. Also M.F ix = 0 and 0 = ( − tK X ) = ( F ix ) . Hence F ix is a rational multiple of F ; see Reid [49] page 36. Thus − K X is Q -linearlyequivalent to a positive multiple of F , and − K X .F = 0 = F . So F iselliptic. Since K X = 0 and by going to a relative minimal model of theelliptic fibration and applying Kodaira’s canonical divisor formula there, wesee that f := Φ | F | : X → C ( ∼ = P ) is already relatively minimal. Theuniqueness of such f again follows from Kodaira’s this formula. This provesthe claim.We return to the proof of Lemma 2.20. We still have to consider thecase where X is rational, g ∗ | Pic( X ) is infinite, K X = 0, − K X is nef and κ ( X, − K X ) = 0. We shall show that Case(3) occurs. Take x ∈ X whichdoes not lie on any negative curve or the anit-pluricanonical curve in some | − tK X | or the set ∪ r> X g r of g -periodic points. Suppose the contrarythat the Zariski-closure D ( x ) in Case(3) is not the entire X . Then D ( x )is 1-dimensional and we may assume that a positive power g s stabilizes acurve D ∋ x in D ( x ). By the choice of x , our D ≥
0. If − K X isa rational multiple of D , then we have κ ( X, − K X ) ≥
1, a contradiction.Otherwise, the class of H := D − K X is ( g s ) ∗ -stable and H > theorem, whence g s acts on H ⊥ := { L ∈ Pic( X ) | L.H = 0 } which is a latticewith negative definite intersection form, so ( g s ) ∗ | H ⊥ and hence g ∗ | Pic( X )are periodic, a contradiction. This proves Lemma 2.20.The key for the ’splitting’ of action below is from Lieberman [33]. Lemma 2.22.
Let X and Y be projective manifolds. Suppose that the secondprojection f Y : V = X × Y → Y is g -equivariant. Then there is an action g | X such that we can write g ( x, y ) = ( g.x, g.y ) for all x ∈ X , y ∈ Y , ifeither Case (1) or (2) below occurs. (1) The irregularity q ( X ) = 0 , and X is non-uniruled (or non-ruled). (2) dim X = dim Y = 1 and rank NS Q ( V ) = 2 . (These hold when one of X, Y is P , or when X, Y are non-isogenius elliptic curves).Proof.
As in Hanamura [22] the proof of Theorem 2.3 there, we express g ( x, y ) = ( ρ g ( y ) .x, g.y ) where ρ g : Y → Aut( X ) is a morphism. We considerCase (1). By [33] Theorem 3.12 and the proof of Theorem 4.9 there, theidentity connected component Aut ( X ) of Aut( X ) is trivial, so Aut( X ) isdiscrete. Thus Im( ρ g ) is a single point, denoted as g | X ∈ Aut( X ). Thelemma is proved in this case.For Case(2), let F be a fibre of f Y . Then g ∗ F = F ′ (another fibre). Let L be a fibre of the projection f X : V → X . Since rank NS Q ( V ) = 2, we haveNS Q ( V ) = Q [ F ] + Q [ L ]. Write g ∗ L = aL + bF . Then 1 = F.L = g ∗ F.g ∗ L = F.g ∗ L = a and 0 = ( g ∗ L ) = 2 ab implies that g ∗ L = L in NS Q ( V ). Thus g ( L ) is a curve with g ( L ) .L = L = 0, whence g ( L ) is another fibre of f X .The result follows. This proves the lemma. (cid:3) We use Lieberman [33] Proposition 2.2 and Kodaira’s lemma to deducethe result below, though it is not needed in this paper (see also Dinh-Sibony[9] the proof of Theorem 4.6 there for a certain case).
Lemma 2.23.
Let X be a projective manifold of dimension n , and H ∈ Nef( X ) a nef and big R -Cartier divisor ( i.e. H is nef and H n > . Then Aut H ( X ) / Aut ( X ) is a finite group. Here Aut ( X ) is the identity compo-nent of Aut( X ) , Aut H ( X ) := { σ ∈ Aut( X ) | σ ∗ H = H in NS R ( X ) } .Proof. By Nakayama [41] II (3.16) and V (2.1), one may write H = A + D inNS R ( X ) with A a Q -ample divisor and D an effective R -divisor. We followthe proof of Lieberman [33] Proposition 2.2. For σ ∈ Aut H ( X ), the volumeof the graph Γ σ is given by:vol(Γ σ ) = ( A + σ ∗ A ) n ≤ ( A + σ ∗ A ) n − ( H + σ ∗ H ) ≤ · · · ≤ ( H + σ ∗ H ) n = 2 n H n . The rest of the proof is the same as [33]. This proves the lemma. (cid:3)
The two results below will be used in the proofs in the next section.
Lemma 2.24.
The following are true. (1) A Q -torus Y does not contain any rational curve. YNAMICS OF AUTOMORPHISMS 19 (2)
Let f : X · · · → Y be a rational map from a normal projective variety X with only rational singularities (resp. log terminal singularities)to an abelian variety (resp. Q -torus) Y . Then f is a well-definedmorphism.Proof. (1) Let T → Y be a finite ´etale cover from a torus T . Suppose thecontrary that P → Y is a non-constant morphism. Then P := T × Y P → P is ´etale and hence P is a disjoint union of P by the simply connectednessof P . So the image in T of P is a union of rational curves, contradictingthe fact that a torus does not contain any rational curve.(2) When Y is an abelian variety, see [27] Lemma 8.1.By Hironaka’s resolution theorem, there is a birational proper morphism σ : Z → X such that the composite τ = f ◦ σ : Z → X · · · → Y is awell defined morphism. By Hacon-McKernan’s solution to the Shokurovconjecture [21] Corollary 1.6, every fibre of σ is rationally chain connectedand is hence mapped to a point in Y , by (1). So for every ample divisor H Y ⊂ Y , we have τ ∗ H Y ∼ Q σ ∗ L X for a Q -Cartier divisor L X ⊂ X ; see [32],page 46, Remark (2). Thus τ factors through σ , and (2) follows. (cid:3) Lemma 2.25.
Let f : X → Y be a g -equivariant surjective morphismbetween projective manifolds and with connected general fibre F . Assumethe following conditions. (1) All of
X, Y and F are of positive dimension. (2) Y is a Q -torus. (3) The Kodaira dimension κ ( X ) = 0 and X has a good (terminal) min-imal model ¯ X , i.e., ¯ X has only terminal singularities and sK ¯ X ∼ for some s > .Then there is a g -equivariant finite ´etale Galois extension ˜ Y → Y from atorus ˜ Y such that the following are true. (1) The composite ¯ X · · · → X → Y extends to a morphism with a generalfibre ¯ F . One has sK ¯ F ∼ , so ¯ F is a good terminal minimal modelof F . (2) X := X × Y ˜ Y is birational to ¯ X := ¯ F × ˜ Y over ˜ Y with sK ¯ X ∼ . (3) Denote by the same G the group Gal( ˜
Y /Y ) and the group id X × Y Gal( ˜
Y /Y ) ≤ Aut( X ) , and by the same g the automorphism ( g | X ) × Y ( g | ˜ Y ) ∈ Aut( X ) . Then g = g | X normalizes G = G | X .In the assertions (4) − (7) below, suppose further that q ( F ) = 0 . (4) g | X ∈ Aut( X ) induces a birational action g on ¯ X with g | ¯ X =( g | ¯ F ) × ( g | ˜ Y ) , where g | ¯ F ∈ Bir( ¯ F ) and g | ˜ Y ∈ Aut( ˜ Y ) .In (5) − (7) below, suppose in addition that ≤ dim F ≤ . (5) Then dim F = 2 and ¯ F is either a K or an Enriques. Further, theinduced birational action of g = ( g | ¯ F ) × ( g | ˜ Y ) on ¯ X = ¯ F × ˜ Y isregular, i.e., g | ¯ F ∈ Aut( ¯ F ) . (6) In (5) , G = G | X ≤ Aut( X ) induces a biregular action by G = G | ¯ X ≤ G ¯ F × G ˜ Y on ¯ X with G ¯ F ≤ Aut( ¯ F ) and G ˜ Y = Gal( ˜ Y /Y ) . (7) g | X is neither rigidly parabolic nor of primitively positive entropy.Proof. (1) follows from Lemma 2.24 and the fact that K ¯ F = K ¯ X | ¯ F . (2) isproved in Nakayama [39] Theorem at page 427. Indeed, for the g -equivalenceof ˜ Y → Y , by [39], (2) is true with an ´etale extension Y ′ → Y . Let T → Y be an ´etale cover of a torus T of minimal degree. Then g | Y lifts to g | T as inBeauville [2] §
3. Now the projection T ′ := T × Y Y ′ → T is ´etale. So there isanother ´etale cover T ′′ → T ′ such that the composite T ′′ → T ′ → T is justthe multiplicative map m T ′′ for some m >
0. In particular, T = m T ′′ ( T ′′ )is isomorphic to T ′′ . Clearly, the natural action g | T ′′ is compactible withthe action g | T via the map m T ′′ . Now the composition ˜ Y := T ′′ → Y is g -equivariant and factors through Y ′ → Y , so that (2) is satisfied. (3) istrue because g | X is the lifting of the action g on X = X /G .We now assume q ( F ) = 0. Assume that a group h h i acts on both X and˜ Y compactibly with the cartesian projection X → ˜ Y . For instance, we maytake h h i to be a subgroup of G | X or h g | X i . This h acts birationally on¯ X . To be precise, for ( x, y ) ∈ ¯ X , we have h. ( x, y ) = ( ρ h ( y ) .x, h.y ), where ρ h : ˜ Y · · · → Bir( ¯ F ) is a rational map. By Hanamura [22] (3.3), (3.10) andpage 135, Bir( ¯ F ) is a disjoint union of abelian varieties of dimension equalto q ( ¯ F ) = q ( F ) = 0 (the first equality is true because the singularities of¯ F are terminal and hence rational). Thus Im( ρ h ) is a single element anddenoted as h | ¯ F ∈ Bir( ¯ F ). So h | ¯ X = ( h | ¯ F ) × ( h | ˜ Y ).(4) follows by applying the arguments above to h = g . For (5), supposedim F = 1 ,
2. Note that κ ( F ) = 0 = q ( F ). So F is birational to ¯ F ,a K S ) = Aut( S ) for smooth minimal surface S ,by the uniqueness of surface minimal model. The argument in the precedingparagraph also shows G = G | ¯ X ≤ G ¯ F × G ˜ Y with G ¯ F ≤ Bir( ¯ F ) = Aut( ¯ F )and G ˜ Y = Gal( ˜ Y /Y ) (= G ) ≤ Aut( ˜ Y ) (so that the two projections from G | ¯ X map onto G ¯ F and G ˜ Y , respectively). This proves (6).(7) Set X ′ := ¯ X /G . Then g acts on X ′ biregularly such that the pairs( X ′ , g ) and ( X, g ) are birationally equivalent, g | X being the lifting of g | X and X being birational to ¯ X . The projections X ′ = ¯ X /G → ¯ F /G ¯ F and X ′ → ˜ Y /G = Y are g -equivariant, since g normalizes G .Suppose the contrary that g | X is either rigidly parabolic, or of primitivelypositive entropy. Then both g | ( ¯ F /G ¯ F ) and g | Y are rigidly parabolic byLemma 2.7 (applied to g -equivariant resolutions of both the source andtargets of the projections). In particular, g | ¯ F is parabolic by Lemma 2.6(applied to g -equivariant resolutions of the source and target of ¯ F → ¯ F /G ¯ F ).By Lemma 2.19, there is a unique g -equivariant surjective morphism τ :¯ F → P (with fibre ¯ F p ) such that a positive power g s | P = id. By (3), g | ¯ F normalizes G ¯ F . So ( g | ¯ F ) ∗ stabilizes the class of the nef divisor L := P h ∈ G ¯ F h ∗ ¯ F p . If L is nef and big, then, by Lemma 2.23, a positive powerof g | ¯ F is in Aut ( ¯ F ) = (1), absurd. Thus L = 0, so ¯ F p .h ∗ ¯ F p = 0 and G ¯ F permutes fibres of τ . Therefore, τ descends to a g -equivariant fibration YNAMICS OF AUTOMORPHISMS 21 ¯ F /G ¯ F → B ∼ = P with g s | B = id, whence g | ( ¯ F /G ) is not rigidly parabolic,absurd. This proves (7) and also the lemma. (cid:3) Results in arbitrary dimension; the proofs
The results in Introduction follow from Theorem 1.1 and three generalresults below in dimension ≥ ≤
3, the good (terminal) minimal model program(as in Kawamata [27], or Mori [38] §
7) has been completed. So in view ofTheorems 1.1, 3.2 and 3.3, we are able to describe the dynamics of (
X, g ) in(3.5) ∼ (3.6). See also Remark 3.4.The result below is parallel to the conjecture (resp. theorem) of Demailly- Peternell - Schneider (resp. Qi Zhang) to the effect that the Albanese mapalb X : X → Alb( X ) is surjective whenever X is a compact K¨ahler (resp.projective) manifold with − K X nef (and hence κ ( X ) = −∞ ). Theorem 3.1.
Let X be a projective complex manifold of dim X ≥ , and g ∈ Aut( X ) . Suppose that the pair ( X, g ) is either rigidly parabolic or ofprimitively positive entropy ( see ( )) . Then we have: (1) The albanese map alb X : X → Alb( X ) is a g -equivariant surjectivemorphism with connected fibres. (2) The irregularity q ( X ) satisfies q ( X ) ≤ dim X . (3) q ( X ) = dim X holds if and only if X is g -equivariantly birational toan abelian variety. (4) alb X : X → Alb( X ) is a smooth surjective morphism if q ( X ) < dim X ; see also ( ) , ( ) . Theorem 3.2.
Let X be a projective complex manifold of dimension n ≥ ,with g ∈ Aut( X ) . Assume the following conditions. (1) The Kodaira dimension κ ( X ) = 0 and the irregularity q ( X ) > . (2) The pair ( X, g ) is either rigidly parabolic or of primitively positiveentropy ( see ( )) . (3) X has a good terminal minimal model ( so (3) is automatic if n ≤ ;see ( ) , ( ) , [27] page , [38] § .Then Case (1) or (2) below occurs. (1) There are a g -equivariantly birational morphism X → X ′ , a pair ( ˜ X ′ , g ) of a torus ˜ X ′ and g ∈ Aut( ˜ X ′ ) , and a g -equivariant ´etaleGalois cover ˜ X ′ → X ′ . In particular, X ′ is a Q -torus. (2) There are a g -equivariant ´etale Galois cover ˜ X → X , a Calabi-Yau variety F with dim F ≥ see ( )) and a birational map ˜ X · · · → F × ˜ A over ˜ A := Alb( ˜ X ) . Further, the biregular action g | ˜ X is conjugate to a birational action ( g | F ) × ( g | ˜ A ) on F × ˜ A , where g | F ∈ Bir( F ) with the first dynamical degree d ( g | F ) = d ( g | X ) , where g | ˜ A ∈ Aut( ˜ A ) is parabolic. In particular, dim X ≥ dim F + q ( X ) ≥ . Theorem 3.3.
Let X ′ be a projective complex manifold of dimension n ≥ ,with g ∈ Aut( X ′ ) . Assume the following conditions ( see ( )) . (1) The Kodaira dimension κ ( X ′ ) = −∞ . (2) The pair ( X ′ , g ) is either rigidly parabolic or of primitively positiveentropy ( see ( )) . (3) The good terminal minimal model program is completed for varietiesof dimension ≤ n ( so (3) is automatic if n ≤ ; see [27] p. , [38] § .Then there is a g -equivariant birational morphism X → X ′ from a pro-jective manifold X such that one of the cases below occurs. (1) X is a rationally connected manifold in the sense of [5] and [31] . (2) q ( X ) = 0 . The maximal rational connected fibration MRC X : X → Z in the sense of [ ibid ] is a well defined g -equivariant surjective mor-phism. Z is a weak Calabi-Yau manifold with dim X > dim Z ≥ . (3) q ( X ) > . There is a g -equivariant ´etale cover ˜ X → X such that thesurjective g -equivariant albanese map alb ˜ X : ˜ X → Alb( ˜ X ) coincideswith the maximal rationally connected fibration MRC ˜ X . (4) q ( X ) > . There is a g -equivariant ´etale Galois cover ˜ X → X suchthat the surjective albanese map alb ˜ X : ˜ X → ˜ A := Alb( ˜ X ) factorsas the g -equivariant MRC ˜ X : ˜ X → ˜ Z and alb ˜ Z : ˜ Z → Alb( ˜ Z ) =˜ A . Further, there are a Calabi-Yau variety F with dim F ≥ see ( )) , and a birational morphism ˜ Z → F × ˜ A over ˜ A , such that thebiregular action g | ˜ Z is conjugate to a birational action ( g | F ) × ( g | ˜ A ) on F × ˜ A , where g | F ∈ Bir( F ) , where g | ˜ A ∈ Aut( ˜ A ) is parabolic.Also dim X > dim F + q ( X ) ≥ . Remark 3.4. (a) By the proof, the condition (3) in Theorem 3.3 can be weakened to:(3)’ X ′ is uniruled. For every projective variety Z dominated by a propersubvariety ( = X ′ ) of X ′ , if the Kodaira dimension κ ( Z ) = −∞ then Z is uniruled, and if κ ( Z ) = 0 then Z has a good terminal minimalmodel Z m (i.e., Z m has only terminal singularities and sK Z m ∼ s > § κ ( X ) = −∞ and the uniruledness of X . It is knownthat the uniruledness of X always implies κ ( X ) = −∞ in any dimension.(c) The birational automorphisms g | F in Theorems 3.2 and 3.3 are indeedisomorphisms in codimenion 1; see [22] (3.4).(d) See [42] Theorem B for a stonger result for the case of K X ≡ ≥ ∼ (3.6) below. YNAMICS OF AUTOMORPHISMS 23
The result below says that the dynamics on an irregular threefold ofKodaira dimension 0, are essentially the dynamics of a torus.
Corollary 3.5.
Let X ′ be a smooth projective complex threefold, with g ∈ Aut( X ′ ) . Assume that the Kodaira dimension κ ( X ′ ) = 0 , irregularity q ( X ′ ) > , and the pair ( X ′ , g ) is either rigidly parabolic or of primitively positiveentropy; see ( ) .Then there are a g -equivariant birational morphism X ′ → X , a pair ( ˜ X, g ) of a torus ˜ X and g ∈ Aut( ˜ X ) , and a g -equivariant ´etale Galois cover ˜ X → X . In particular, X is a Q -torus. The result below shows that the dynamics on a threefold of Kodairadimension −∞ are (or are built up from) the dynamics on a rationallyconnected threefold (or on a rational surface and that on a 1-torus). Theorem 3.6.
Let X be a smooth projective complex threefold, with g ∈ Aut( X ) . Assume that κ ( X ) = −∞ , and the pair ( X, g ) is either rigidlyparabolic or of primitively positive entropy ( see ( )) . Then we have: (1) If q ( X ) = 0 then X is rationally connected in the sense of [5] or [31] . (2) Suppose that q ( X ) ≥ and the pair ( X, g ) is of primitively positiveentropy. Then q ( X ) = 1 and the albanese map alb X : X → Alb( X ) isa smooth surjective morphism with every fibre F a smooth projectiverational surface of Picard number rank Pic( F ) ≥ . . The assertion (1) follows from Lemma 2.12. For (2), in view of (1), we mayassume that (
X, g ) is of imprimitively positive entropy. Then the assertion(2) follows from Lemma 2.10. This proves Theorem 1.1. . We may assume that q ( X ) >
0. By the universal property of A :=Alb( X ), every h ∈ Aut( X ) descends, via the albanese map alb X : X → A ,to some h | A ∈ Aut variety ( A ). By Lemma 2.12, κ (alb X ( X )) ≤
0. Thus, byUeno [52] Lemma 10.1, κ (alb X ( X )) = 0 and alb X ( X ) = A = Alb( X ), i.e.,alb X is surjective. Let X → X → A be the Stein factorization with X → A a finite surjective morphism from a normal variety X , and X → X havingconnected fibres. Note that κ ( X ) ≥ X → A as in Iitaka [25] Theorem5.5. So by Lemma 2.12, κ ( X ) = 0. By the result of Kawamata-Viehwegas in Kawamata [26] Theorem 4, X → A is ´etale, so X is an abelianvariety too. By the universal property of A = Alb( X ), we have X = A .Thus, X → A = X has connected fibres. Theorem 3.1 (1) is proved. NowTheorem 3.1 (2) and (3) follow from (1). If q ( X ) < dim X then g | A is rigidlyparabolic by Lemma 2.7; so Theorem 3.1 (4) follows from Lemma 2.16. Thisproves Theorem 3.1. For a projective variety Z , we denote by A ( Z ) or Alb( Z ) the albanese vari-ety Alb( Z ′ ) with Z ′ → Z a proper resolution. This definition is independentof the choice of Z ′ , and A ( Z ) depends only on the birational equivalence classof Z . If Z is log terminal, then the composition Z · · · → Z ′ → A ( Z ) is awell defined morphism; see Lemma 2.24. By Theorem 3.1, we may assume that q ( X ) < dim X , so g | A ( X ) is rigidlyparabolic by Lemma 2.7. The albanese map alb X : X → A ( X ) has con-nected fibre F and is smooth and surjective; see Theorem 3.1.By the assumption, X has a good terminal minimal model ¯ X with sK ¯ X ∼ s >
0. We apply Lemma 2.25 to alb X : X → Y := A ( X ). Thenthere is a g -equivariant ´etale Galois extension ˜ Y → Y from a torus ˜ Y , suchthat X := X × Y ˜ Y is birational to ¯ X := ¯ F × ˜ Y over ˜ Y , with ¯ F a goodterminal minimal model of F and sK ¯ F ∼
0. Also g | X normalizes G | X ( ∼ = Gal( ˜ Y /Y )).Assume that 0 < q ( F ) < dim F . By [26] Theorem 1, alb X : X → A ( X ) is a surjective morphism with connected smooth general fibre F . Bythe universal property of the albanese map, alb X : X → A ( X ) is h g, G i -equivariant. Both of the natural morphisms X = X /G → Y := A ( X ) /G and A ( X ) → Y are g -equivariant and surjective. Since G acts freely on˜ Y and A ( X ) = A ( ¯ X ) = A ( F ) × ˜ Y , the latter map is ´etale. By the samereason, every general fibre of X → Y can be identified with a fibre F , so itis connected.We apply Lemma 2.25 to X → Y . Then there is a g -equivariant ´etaleGalois extension ˜ Y → Y from a torus ˜ Y , such that X := X × Y ˜ Y isbirational to ¯ X := ¯ F × ˜ Y with ¯ F a good terminal minimal model of F and sK ¯ F ∼
0. Also g | X normalizes G | X ( ∼ = Gal( ˜ Y /Y )).If 0 < q ( F ) < dim F , we can consider X → Y := A ( X ) /G . Continuethis process, we can define X → Y i +1 := A ( X i ) /G i with G i ∼ = Gal( ˜ Y i /Y i )the Galois group of the ´etale Galois extension ˜ Y i → Y i from a torus ˜ Y i , suchthat X i := X × Y i ˜ Y i is birational to ¯ X i := ¯ F i × ˜ Y i with sK ¯ X i ∼
0, where ¯ F i a good terminal model of a general fibre F i of X → Y i (and also of X i → ˜ Y i and X i − → A ( X i − )).Note that q ( F i ) ≤ dim F i because κ ( F i ) = κ ( ¯ F i ) = 0 (see [26] Theorem1). Also dim X ≥ dim Y i +1 = dim Y i + q ( F i ). So there is an m ≥ q ( F m ) equals either 0 or dim F m .Consider the case where q ( F m ) = 0 and dim F m >
0. Then Alb( X m ) =Alb( ¯ X m ) = ˜ Y m , and by Lemma 2.25, Theorem 3.2 Case(2) occurs with˜ X = X m , F = ¯ F m and T = ˜ Y m . Indeed, for the second part of Theorem3.2 (2), since X → Y m is g -equivariant, g | Y m is rigidly parabolic by Lemma2.7 and hence g | T is parabolic by Lemma 2.6 (with d ( g | T ) = 1). The firstdynamical degrees satisfy d ( g | X ) = d ( g | ˜ X ) = d (( g | F ) × ( g | T )) = d ( g | F )by Lemma 2.6, Guedj [19] Proposition 1.2 and the K¨unneth formula for YNAMICS OF AUTOMORPHISMS 25 H as in Griffiths-Harris [17] page 58; see also [8] Proposition 5.7. Alsodim X = dim F + dim ˜ Y m ≥ dim F + dim Y = dim F + q ( X ).Consider the case ( q ( ¯ F m ) =) q ( F m ) = dim F m . Then q ( X m ) = dim X m .By Kawamata [26] Theorem 1, the albanese map X m → ˜ X ′ := A ( X m ) = A ( F m ) × ˜ Y m is a h g, G m i -equivariant birational surjective morphism. Itinduces a g -equivariant birational morphism X = X m /G m → X ′ := ˜ X ′ /G m ,since g normalizes G m as in Lemma 2.25. Also G m acts freely on ˜ Y m , andhence the quotient map ˜ X ′ → X ′ is ´etale. Note that X ′ × Y m ˜ Y m ∼ = ˜ X ′ over˜ Y m , since both sides are finite (´etale) over X ′ and birational to each other,by the construction of X m . Thus Case(1) of Theorem 3.2 occurs with the´etale Galois cover ˜ X ′ → X ′ . This proves Theorem 3.2. Let MRC X ′ : X ′ · · · → Z be a maximal rationally connected fibration;see [5], or [30] IV Theorem 5.2. The construction there, is in terms ofan equivalence relation, which is preserved by g | X ′ . So we can replace( X ′ , g ) by a g -equivariant blowup ( X, g ) such that MRC X : X → Z is awell defined g -equivariant surjective morphism with general fibre rationallyconnected, g | X ∈ Aut( X ), and X , Z projective manifolds; see Hironaka[24]. Further, Z is non-uniruled by Graber-Harris-Starr [16] (1.4). Thenatural homomorphism π ( X ) → π ( Z ) is an isomorphism; see Campana[5] or Koll´ar [30]. So q ( X ) = q ( Z ). If dim Z = 0, then Case(1) of thetheorem occurs.Consider the case dim Z >
0. Since X ′ is uniruled by the assumptions ofthe theorem (see Remark 3.4), dim Z < dim X ′ . Since Z is non-uniruled,we have κ ( Z ) ≥ κ ( Z ) = 0 by Lemma 2.12.Now g | Z is rigidly parabolic by Lemma 2.7. If q ( Z ) = 0 then dim Z ≥ κ ( Z ) = 0 and by Lemma 2.20. So Case(2) of the theorem occurs.Suppose q ( Z ) >
0. Since an abelian variety contains no rational curves,alb X : X → A := Alb( X ) factors as MRC X : X → Z and alb Z : Z → Alb( Z )= A ; see Lemma 2.24 and [26] Lemma 14. By Lemma 2.7, g | A is rigidlyparabolic. Also alb X and alb Z are smooth and surjective with connectedfibres by Theorem 3.1.We apply Theorem 3.2 to ( Z, g ), so two cases there occur; in the firstcase there, we may assume that K Z is torsion after replacing Z by its g -equivariant blowdown. Let ˜ Z → Z be the g -equivariant ´etale Galois exten-sion as there. So either ˜ Z is a torus, or ˜ Z → F × ˜ A is a well defined birationalmorphism over ˜ A := Alb( ˜ Z ) (after replacing Z and X by their g -equivariantblowups), with q ( F ) = 0 etc as described there. Set ˜ X := X × Z ˜ Z . Thenthe projection ˜ X → ˜ Z coincides with MRC ˜ X . So alb ˜ X : ˜ X → Alb( ˜ X ) = ˜ A factors as ˜ X → ˜ Z and alb ˜ Z : ˜ Z → ˜ A . If ˜ Z is a torus, then Case(3) ofTheorem 3.3 occurs. In the situation ˜ Z → F × ˜ A , Case(4) of Theorem 3.3occurs in view of Theorem 3.2. This proves Theorem 3.3. ∼ Corollary 3.5 and Theorem 3.6 (1) follow respectively from Theorems 3.2and 3.3, while Theorem 1.2 follows from Lemma 2.10, Corollary 3.5, Theo-rem 3.6, and Lemma 2.8 applied to alb X . Theorem 1.3 follows from Theorem1.2 (and its proof). Theorem 1.5 follows from Lemma 2.10, Corollary 3.5,Theorem 3.6, the proof of Lemma 2.8, and Lemma 2.6. See Remark 3.4.For Theorem 1.4, by Theorems 1.1 and 3.6, we have only to considerthe case in Theorem 3.6 (2). But then π ( X ) = π (Alb( X )) = Z ⊕ sincegeneral (indeed all) fibres of alb X : X → Alb( X ) are smooth projectiverational surfaces (see [5] or [31]).We now prove Theorem 3.6 (2) directly. We follow the proof of Theorem3.3. Let MRC X : X · · · → Y be a maximal rationally connected fibration,where κ ( Y ) ≥
0. Replacing Y by a g -equivariant modification, we mayassume that Y is smooth and minimal. Since κ ( X ) = −∞ , our X is uniruled(see Remark 3.4). So dim Y < dim X . Our alb X : X → A := Alb( X ) issmooth and surjective (with connected fibre) and factors as MRC X : X · · · → Y and alb Y : Y → Alb( Y ) = A ; also 3 = dim X > dim Y ≥ q ( Y ) = q ( X ) > κ ( Y ) = 0 and g | Y and g | A arerigidly parabolic; see the proof of Theorem 3.3. Theorem 3.6 (2) followsfrom the two claims below. Claim 3.13.
In Theorem 3.6 (2), q ( X ) = 1. Proof.
Suppose the contrary that q ( X ) ≥
2. Then dim Y = q ( Y ) = q ( X ) =2, so Y = A and alb X = MRC X , by Theorem 3.1 and the proof of Theorem3.3. By Theorem 3.1, alb X : X → A is surjective with every fibre F asmooth projective curve. F is a rational curve by the definition of MRC X : X → Y = A . Take a nef L = v ( g | X ) ∈ Nef( X ) as in Lemma 2.4 suchthat g ∗ L = d L with d = d ( g | X ) >
1. By Lemma 2.15, either g s | A is atranslation and we let C be a very ample divisor on A , or g s ( C ) = C + t for an elliptic curve C and t ∈ A . Rewrite g s as g and we always have g ∗ C ≡ C in N ( A ) = NS R ( A ).Let X C ⊂ X be the inverse of C , a Hirzebruch surface. Then the restric-tion alb X | X C : X C → C is a ruling. Note that g ∗ F = F in N ( X ). Also g ∗ =id on H ( X, R ) ∼ = R . So L.F = g ∗ L.g ∗ F = d L.F , whence
L.F = 0. By theadjunction formula, K X C = ( K X + X C ) | X C = K X | X C + eF with the scalar e = C ≥
0. Note that R ∼ = H ( X, R ) ∋ L.K X .X C = g ∗ L.g ∗ K X .g ∗ X C = d L.K X .X C , whence 0 = L.K X .X C = ( L | X C ) . ( K X | X C ) = ( L | X C ) .K X C since L.F = 0. Now ( L | X C ) .F = L.F = 0 and ( L | X C ) .K X C = 0 imply that L.X C = L | X C = 0 in the lattice NS R ( X C ) because the fibre F and K X C span this lattice. Hence X C and L are proportional in NS R ( X ) by Lemma2.3 (1), noting that any curve like C in the abelian surface A is nef andhence X C is nef. But g ∗ X C = X C while g ∗ L = d L with d >
1, so X C and L are not proportional in NS R ( X ). Thus, the claim is true. (cid:3) Claim 3.14.
In Theorem 3.6 (2), every fibre X a of alb X : X → A = Alb( X )is a smooth projective rational surface (so NS( X ) = Pic( X )) with non-big − K X a . Further, rank Pic( X a ) ≥ YNAMICS OF AUTOMORPHISMS 27
Proof.
If dim
Y > dim A = q ( X ) = 1, then Y is a surface with κ ( Y ) = 0, q ( Y ) = 1 and g | Y rigidly parabolic, which contradicts Lemma 2.20. Thusdim Y = 1 = q ( Y ), so Y = Alb( Y ) = A and alb X = MRC X by Theorem3.1. Therefore, every fibre F = X a of X → A is a smooth projective rationalsurface; see Koll´ar [30] IV Theorem 3.11.Assume the contrary that − K F is big or rank Pic( F ) ≤
10 (i.e., K F ≥ K F ≥
1, then − K F is big by theRiemann-Roch theorem applied to − nK F . Thus we may assume that either − K F is big or K F = 0 and shall get a contradiction.As in the proof of the previous claim, for K F = K X | F and L := v ( g | X ) ,we have 0 = L.K X .F = ( L | F ) .K F , so L | F ≡ cK F = cK X | F for somescalar c by the Hodge index theory; see the proof of [1] IV (7.2). If c = 0,applying g ∗ , we get d ( g ) = 1, absurd. Hence c = 0 and L.F = 0. Then L ≡ eF for some scalar e > g ∗ , we get the samecontradiction. This proves the claim and also Theorem 3.6. (cid:3) References [1] W. Barth, K. Hulek, C. Peters and A. Van de Ven, Compact complex surfaces, 2nd ed.,Springer-Verlag 2004, MR2030225, Zbl 1036.14016.[2] A. Beauville, Some remarks on K¨ahler manifolds with c = 0, in : Classification of alge-braic and analytic manifolds (Katata, 1982), 1–26, Progr. Math., Vol. , MR0728605, Zbl0537.53057.[3] C. Birkar, P. Cascini, C. D. Hacon and J. McKernan, Existence of minimal models for varietiesof log general type, math.AG/ .[4] C. Birkenhake and H. Lange, Complex abelian varieties, 2nd ed., Springer-Verlag, Berlin,2004, MR2062673, Zbl 1056.14063.[5] F. Campana, Connexit´e rationnelle des vari´et´es de Fano, Ann. Sci. ´ecole Norm. Sup. (4) (1992), no. 5, 539–545, MR1191735, Zbl 0783.14022.[6] S. Cantat, Dynamique des automorphismes des surfaces projectives complexes, C. R. Acad.Sci. Paris Sr. I Math. (1999), no. 10, 901–906, MR1689873, Zbl 0943.37021.[7] S. Cantat, Dynamique des automorphismes des surfaces K
3, Acta Math. (2001), no. 1,1–57, MR1864630, Zbl 1045.37007.[8] T. -C. Dinh, Suites d’applications m´eromorphes multivalu´ees et courants laminaires, J. Geom.Anal. (2005) 207–227, MR2152480, Zbl 1085.37039.[9] T. -C. Dinh and N. Sibony, Groupes commutatifs d’automorphismes d’une vari´et´e k¨ahl´eriennecompacte, Duke Math. J. (2004), no. 2, 311–328, MR2066940, Zbl 1065.32012.[10] T. -C. Dinh and N. Sibony, Green currents for holomorphic automorphisms of compact K¨ahlermanifolds, J. Amer. Math. Soc. (2005), no. 2, 291–312, MR2137979, Zbl 1066.32024.[11] T. -C. Dinh and N. Sibony, Regularization of currents and entropy, Ann. Sci. ´ecole Norm.Sup. (4) (2004), no. 6, 959–971, MR2119243, Zbl 1074.53058.[12] S. Friedland, Entropy of algebraic maps, in : Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, 1993), J. Fourier Anal. Appl. 1995, Special Issue, 215–228, MR1364887,Zbl 0890.54018.[13] S. Friedland, Entropy of holomorphic and rational maps: a survey, math.DS/ .[14] R. Friedman and J. Morgan, Smooth four-manifolds and complex surfaces, Ergebnisseder Mathematik und ihrer Grenzgebiete. 3. Folge. Vol. , Springer-Verlag, Berlin, 1994,MR1288304, Zbl 0817.14017.[15] W. Fulton, Intersection theory, Second edition. Springer-Verlag, Berlin, 1998, MR1644323,Zbl 0885.14002.[16] T. Graber, J. Harris and J. Starr, Families of rationally connected varieties, J. Amer. Math.Soc. (2003), no. 1, 57–67, MR1937199, Zbl 1092.14063.[17] P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley-Interscience, New York,1978, MR1288523, Zbl 0836.14001. [18] M. Gromov, On the entropy of holomorphic maps (Preprint 1977), Enseign. Math. (2) (2003), no. 3-4, 217–235, MR2026895, Zbl 1080.37051.[19] V. Guedj, Ergodic properties of rational mappings with large topological degree, Ann. ofMath. (2) (2005), no. 3, 1589–1607, MR2179389, Zbl 1088.37020.[20] V. Guedj, Propri´et´es ergodiques des applications rationnelles, math.CV/ .[21] C. Hacon and J. McKernan, On Shokurov’s rational connectedness conjecture, Duke Math.J. (2007), no. 1, 119–136, MR2309156, Zbl 1128.14028.[22] M. Hanamura, On the birational automorphism groups of algebraic varieties, CompositioMath. (1987), 123 – 142, MR0906382, Zbl 0655.14007.[23] B. Harbourne, Rational surfaces with infinite automorphism group and no antipluricanonicalcurve, Proc. Amer. Math. Soc. (1987), no. 3, 409–414, MR0875372, Zbl 0643.14019.[24] H. Hironaka, Bimeromorphic smoothing of a complex-analytic space (Univ. of WarwickPreprint 1971), Acta Math. Vietnam. (1977), no. 2, 103–168, MR0499299, Zbl 0407.32006.[25] S. Iitaka, Algebraic geometry, Graduate Texts in Mathematics, Vol. , Springer-Verlag, NewYork-Berlin, 1982, MR0637060, Zbl 0491.14006.[26] Y. Kawamata, Characterization of abelian varieties, Compositio Math. (1981), no. 2,253–276, MR0622451, Zbl 0471.14022.[27] Y. Kawamata, Minimal models and the Kodaira dimension of algebraic fiber spaces, J. ReineAngew. Math. (1985), 1–46.[28] Y. Kawamata, K. Matsuda and K. Matsuki, Introduction to the minimal model problem,in: Algebraic geometry, Sendai, 1985, 283–360, Adv. Stud. Pure Math., Vol. , 1987,MR0946243, Zbl 0672.14006.[29] J. Keum, K. Oguiso and D. -Q. Zhang, Conjecture of Tits type for complex varieties and Theo-rem of Lie-Kolchin type for a cone, Math. Res. Lett. (to appear); also: arXiv:math/ .[30] J. Koll´ar, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Gren-zgebiete. 3. Folge. Vol. , Springer-Verlag, Berlin, 1996, MR1440180, Zbl 0877.14012.[31] J. Koll´ar, Y. Miyaoka and S. Mori, Rational connectedness and boundedness of Fano mani-folds, J. Differential Geom. (1992), no. 3, 765–779, MR1189503, Zbl 0759.14032.[32] J. Koll´ar and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Math-ematics, Vol. , Cambridge University Press, Cambridge, 1998.[33] D. I. Lieberman, Compactness of the Chow scheme: applications to automorphisms anddeformations of K¨ahler manifolds, Lecture Notes in Math. , pp. 140–186, Springer, 1978,MR0521918, Zbl 0391.32018.[34] K. Matsuki, Lectures on Factorization of Birational Maps, math.AG/ .[35] C. McMullen, Coxeter groups, Salem numbers and the Hilbert metric, Publ. Math. Inst.Hautes ´etudes Sci. No. , (2002), 151–183, MR1953192, Zbl pre01874454.[36] C. McMullen, Dynamics on K (2002), 201–233, MR1896103, Zbl 1054.37026.[37] C. McMullen, Dynamics on blowups of the projective plane, Publ. Math. Inst. Hautes ´etudesSci. No. (2007), 49–89,[38] S. Mori, Classification of higher-dimensional varieties, Algebraic geometry, Bowdoin, 1985(Brunswick, Maine, 1985), 269–331, Proc. Sympos. Pure Math., Vol. , Part 1, Amer.Math. Soc., Providence, RI, 1987, MR0927961, Zbl 0656.14022.[39] N. Nakayama, On Weierstrass models, in: Algebraic geometry and commutative algebra, Vol. II , 405–431, Kinokuniya, Tokyo, 1988, MR0977771, Zbl 0699.14049.[40] N. Nakayama, Compact K¨ahler manifolds whose universal covering spaces are biholomorphicto C n , (a modified version, but in preparation); the original is RIMS preprint , Res.Inst. Math. Sci. Kyoto Univ. 1999.[41] N. Nakayama, Zariski-decomposition and abundance, MSJ Memoirs, , Mathematical So-ciety of Japan, Tokyo, 2004, MR2104208, Zbl 1061.14018.[42] N. Nakayama and D. -Q. Zhang, Building blocks of ´etale endomorphisms of complexprojective manifolds, RIMS preprint .[44] Y. Namikawa and J. H. M. Steenbrink, Global smoothing of Calabi-Yau threefolds, Invent.Math., (1995), 403–419, MR1358982, Zbl 0861.14036. YNAMICS OF AUTOMORPHISMS 29 [45] K. Oguiso, Bimeromorphic automorphism groups of non-projective hyperk¨ahler manifolds -a note inspired by C.T. McMullen, J. Differential Geom. (2008), 163–191, MR2406267,Zbl 1141.14021.[46] K. Oguiso, Automorphisms of hyperk¨ahler manifolds in the view of topological entropy, in :Algebraic geometry, 173–185, Contemp. Math., Vol. , Amer. Math. Soc., Providence, RI,2007.[47] K. Oguiso, Tits alternative in hyperk¨ahler manifolds, Math. Res. Lett. (2006), 307 – 316,MR2231119, Zbl 1107.14013.[48] K. Oguiso and J. Sakurai, Calabi-Yau threefolds of quotient type, Asian J. Math. (2001),no. 1, 43–77, MR1868164, Zbl 1031.14022.[49] M. Reid, Chapters on algebraic surfaces, in : Complex algebraic geometry (Park City,UT, 1993), 3–159, IAS/Park City Math. Ser., , Amer. Math. Soc., Providence, RI, 1997,MR1442522, Zbl 0910.14016.[50] R. Salem, Algebraic numbers and Fourier analysis, Selected reprints, Wadsworth Math. Ser.,Wadsworth, Belmont, CA, 1983, MR0732447, Zbl 0505.00033.[51] H. Schneider and B.S. Tam, Matrices leaving a cone invariant, in: Handbook for LinearAlgebra, ed. L. Hogben, Chapman & Hall (2006).[52] K. Ueno, Classification theory of algebraic varieties and compact complex spaces, LectureNotes in Mathematics, Vol. , Springer-Verlag, Berlin-New York, 1975, MR0506253, Zbl0299.14007.[53] E. Viehweg, Weak positivity and the additivity of the Kodaira dimension for certain fibrespaces, in: Algebraic varieties and analytic varieties (Tokyo, 1981), 329–353, Adv. Stud. PureMath., Vol. , North-Holland, Amsterdam, 1983, MR0715656, Zbl 0513.14019.[54] E. Viehweg and K. Zuo, On the isotriviality of families of projective manifolds over curves,J. Algebraic Geom. (2001), no. 4, 781–799, MR1838979, Zbl 1079.14503.[55] Y. Yomdin, Volume growth and entropy, Israel J. Math. (1987), no. 3, 285–300,MR0889979, Zbl 0641.54036.[56] D. -Q. Zhang, Dynamics of automorphisms of compact complex manifolds, Proceedings of TheFourth International Congress of Chinese Mathematicians (ICCM2007), 17 - 22 December2007, HangZhou, China, Vol II , pp. 678 - 689; also: arXiv: [57] D. -Q. Zhang, Automorphism groups and anti-pluricanonical curves, Math. Res. Lett. (2008), no. 1, 163–183, MR2367182, Zbl pre05302022.[58] D. -Q. Zhang, Cohomologically hyperbolic endomorphisms of complex manifolds, Intern. J.Math. to appear; also: arXiv: .[59] S. -W. Zhang, Distributions in Algebraic Dynamics, A tribute to Professor S. S. Chern, Surveyin Differential Geometry, Vol. , 381-430, International Press 2006, Zbl pre05223182. Department of MathematicsNational University of Singapore, 2 Science Drive 2, Singapore 117543, SingaporeandMax-Planck-Institut f¨ur Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
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