The Geometric Dynamical Northcott and Bogomolov Properties
aa r X i v : . [ m a t h . D S ] O c t THE GEOMETRIC DYNAMICAL NORTHCOTT AND BOGOMOLOVPROPERTIES
THOMAS GAUTHIER AND GABRIEL VIGNY
Abstract.
We establish the Geometric Dynamical Northcott Property for polarizedendomorphisms of a projective normal variety over a function field K of characteristiczero, and we relate this property to the stability of dynamical pairs in complex dynam-ics. This extends previous results of Benedetto, Baker and DeMarco in dimension 1,and of Chatzidakis-Hrushovski in higher dimension. Our proof uses complex dynamicsarguments and does not rely on the previous ones. We first show that, when K is thefield of rational functions of a normal complex projective variety, the canonical height ofa subvariety is the mass of the appropriate bifurcation current and that a marked pointis stable if and only if its canonical height is zero. We then establish the GeometricDynamical Northcott Property using a similarity argument.Moving from points to subvarieties, we propose, for polarized endomorphisms, a dy-namical version of the Geometric Bogomolov Conjecture, recently proved by Cantat, Gao,Habegger and Xie. We establish several cases of this conjecture notably non-isotrivialpolynomial skew-product with an isotrivial first coordinate. Introduction
Let k be a field of characteristic zero, X be a normal projective variety and L be anample linebundle on X . An endomorphism f : X → X , defined over k , is polarized if thereexists an integer d ≥ f ∗ L ≃ L ⊗ d . We call such a data ( X, f, L ) a polarizedendomorphism over k . A typical example is an endomorphism f of a projective space ofdegree d ≥
2, since it satisfies f ∗ O (1) ≃ O ( d ) ≃ O (1) ⊗ d . For a polarized endomorphism( X, f, L ) defined over a global field K and h X,L a Weil height function on X ( ¯ K ), relative to L , by the functorial properties of Weil height functions, we have h X,L ◦ f = dh X,L + O (1),and Call and Silverman [CS] defined the canonical height b h f : X ( ¯ K ) → R + of f as b h f = lim n →∞ d n h X,L ◦ f n . When K is a number field, using b h f ≤ h X,L + O (1) and Northcott’s Theorem [No] for thenaive height, this implies that the set of points x ∈ X ( K ) with b h f ( x ) ≤ ε is finite andconsists only of preperiodic points for any ε > K can be written as K = k ( B ) where k is a field of characteristic zero and B is anormal projective k -variety. In other words, K is the field of rational functions on B . Inthat setting, Baker proved the following result (see definition 6 of isotriviality below whichbasically means that the specializations of f are conjugated as polarized endomorphisms). Both authors are partially supported by the ANR grant Fatou ANR-17-CE40-0002-01.Keywords: Polarized endomorphism, canonical height, algebraic family of rational maps, arithmeticcharacterizations of stability.Mathematics Subject Classification (2010): 37P15, 37P30, 11G50, 37P35, 37F45.
Theorem 1 (Northcott Property [Ba]) . Pick any function field K and assume f ∈ K ( x ) is non-isotrivial. Then, there exists ε > such that { z ∈ P ( K ) , b h f ( z ) ≤ ε } < ∞ . In particular, any z ∈ P ( K ) with b h f ( z ) = 0 is preperiodic under iteration of f , i.e. thereexist n > m ≥ such that f n ( z ) = f m ( z ) . This result was already proved by Benedetto [Ben] for polynomials. In the case ofabelian varieties over a global function field endowed with the multiplication by an integer,the analogue of Baker Theorem is due to N´eron and Tate [Ne, L].In higher dimension, Chatzidakis and Hrushovski proved a model-theoretic version ofTheorem 1 in [CH] with ε = 0 with the additional assumption that the dynamics isa primitive algebraic dynamics which basically means that f neither has an invariantsubvariety nor admits an invariant fibration.Consider ( X, f, L ) as above, defined over the field of rational function K = k ( B ) of anormal projective k -variety. As k is a field of characteristic zero, up to replacing it withan algebraic extension, we can assume there exists an algebraically closed subfield k ′ of k such that X , f and L are defined over k ′ and the transcendental degree of k ′ over Q is finite. In particular, k ′ can be embedded in the field C ( A ) of rational functions of anormal projective complex variety A , so that we can assume K = C ( A × B ). Hence, wecan assume ( X, f, L ) is defined over C ( B ). We thus restrict to the case k = C in the restof the paper. This will enable the use of complex methods.To the polarized endomorphism ( X, f, L ) we associate a model ( X , f , L ) over B , i.e. afamily π : X → B of complex projective varieties which is normal and flat over a Zariskiopen subset Λ of B , a rational map f : X X and a relatively ample linebundle L on X , such that for each λ ∈ Λ, if X λ := π − { λ } is the fiber of π over λ , L λ := L | X λ and f λ := f | X λ , then ( X λ , f λ , L λ ) is a polarized endomorphism, and f is the restriction of f tothe generic fiber of π . We let X Λ := π − (Λ).To an algebraic family ( X , f , L ) of polarized endomorphisms, we can associate its fiberedGreen current b T f . It is a closed, positive current on X of bidegree (1 ,
1) which detectsthe measurable dynamics of f : it satisfies b T dim X +1 f = 0 and, for each λ ∈ Λ, the slice µ f λ of b T dim X f along X λ is the unique maximal entropy measure of f λ and it equidistributesperiodic points of f λ .A marked point is a rational section a : B X . In particular, a defines a regularmap a : Λ → X such that a ( λ ) ∈ X λ for all λ ∈ Λ. Note that, to any subvariety Z of X which is defined over K , we can associate a subvariety Z such that π | Z : Z B isflat over Λ. Hence, points of X ( K ) correspond to such marked points a : B X . Incomplex dynamics, the central notion is that of stability: a marked point a is stable if thesequence { λ f nλ ( a ( λ )) } n is equicontinuous on Λ. By Bishop Theorem, this means thatthe volume of the graph Γ n of λ f nλ ( a ( λ )) seen as a subvariety of X Λ is locally boundedindependently of n . This notion is, by nature, a local notion. Also, standard Weil heightarguments show that b h f (a) = 0 if and only if the volume of the Zariski closure of thegraph of Γ n in X is bounded (in a given polarization). Having height 0 is thus a globalproperty that implies dynamical stability.Let us come back to the case of a family of endomorphisms of degree d of P , parametrizedby a smooth complex quasi-projective curve Λ, i.e. a morphism f : ( z, λ ) ∈ P × Λ HE GEOMETRIC DYNAMICAL NORTHCOTT AND BOGOMOLOV PROPERTIES 3 ( f λ ( z ) , λ ) ∈ P × Λ such that for any λ ∈ Λ, f λ is a rational map of degree d . Let Crit( f λ )denote the critical set of f . A fundamental result of McMullen [Mc] states that, if thesequence { λ f nλ (Crit( f λ )) } n is an equicontinuous family of correspondences, then either f is isotrivial (which means here that there exist a finite branched cover ρ : Λ ′ → Λ anda family φ : P × Λ ′ → P × Λ ′ of M¨obius transformations such that φ − λ ◦ f ρ ( λ ) ◦ φ λ isindependent of λ ), or f is a family of Latt`es maps. Dujardin and Favre [DF] extendedthe result to the case of of a given marked critical point showing that stability impliespreperiodicity or isotriviality. Finally, DeMarco [D2] proved such a statement for anymarked point: the stability of a implies that the corresponding point a ∈ P ( K ), where K = C (Λ), satisfies b h f (a) = 0 and gave another proof of Baker’s result.Our first purpose here is to give a new proof of the Northcott Property on function fieldsof characteristic zero in higher dimension with no assumption on f . When K = C ( B ),we also extend DeMarco results and show that a marked point is stable if and only if itscanonical height is 0.1.1. The Geometric Dynamical Northcott Property.
Our first result is what wecall the Geometric Dynamical Northcott Property for polarized endomorphisms over afunction field of characteristic zero.
Theorem A.
Let ( X , f , L ) be a non-isotrivial complex algebraic family of polarized en-domorphisms over a normal complex projectvie variety B , with regular part Λ and let ( X, f, L ) be the induced polarized endomorphism over the field K of rational functions on B . Let N be a very ample linebundle on B such that M := L ⊗ π ∗ ( N ) is ample on X . (1) Let a : B X be a rational section of π with corresponding point a ∈ X ( C ( B )) ,and let C n be the Zariski closure of f n (a) , then ( ⋆ ) a is stable ⇐⇒ sup n ≥ deg M ( C n ) < ∞ ⇐⇒ b h f (a) = 0 . (2) Furthermore, there exist a proper subvariety Y of X which is flat over B andintegers N ≥ and D ≥ such that (a) f ( Y ) = Y and for any periodic irreducible component V of Y with f n ( V ) = V ,the family of polarized endomorphisms ( V , f n | V , L | V ) is isotrivial, (b) For any rational section a : B X of π such that (( X , f , L ) , a ) is stable,the following properties hold: (i) if a N is the rational section of π defined by a N ( λ ) := f Nλ ( a ( λ )) , then a N is in fact a section of π | Y : Y → B , (ii) if C a is the Zariski closure of a (Λ) , we have deg M ( C a ) ≤ D . As explained above, we can deduce the following corollary which is what we call the
Geometric Dynamical Northcott Property on arbitrary function fields of characteristic zero.
Corollary 2.
Let K be a function field of characteristic zero. Let ( X, f, L ) be a polarizedendomorphism on a normal projective variety defined over K . Assume ( X, f, L ) is non-isotrivial. Then, there exist a (possibly reducible) subvariety Y ⊂ X K with f ( Y ) = Y and N ≥ such that (1) for any irreducible component V of Y with f n ( V ) = V , ( V, f n | V , L | V ) is isotrivial, (2) for any point z ∈ X ( K ) with b h f ( z ) = 0 , we have f N ( z ) ∈ Y .In particular, if X K contains no periodic isotrivial subvariety of positive dimension, then { z ∈ X ( K ) , b h f ( z ) = 0 } is finite and consists only of preperiodic points. THOMAS GAUTHIER AND GABRIEL VIGNY
Remark. (1) ( ⋆ ) is new for families of endomorphisms of varieties of dimension atleast 2. As explained below, our strategy is different from DeMarco’s strategy [D2].(2) The case where Y has an irreducible component of positive dimension over K cannot be excluded. Consider, for example, an endomorphism of P given by anon-isotrivial polynomial map of A , whose restriction to the line at infinity istrivial (e.g. ( z d , z d + λz ) defined over C ( λ )). Then any non periodic constantwith value on the line at infinity has canonical height zero.(3) As mentioned above, Chatzidakis and Hrushovski already proved a version ofCorollary 2 in [CH] (see also [CL] for a detailed exposition on this work). Ourproof is different and based on complex dynamics ideas. We do not make addi-tional assumptions as they did so that we clarify the issue of isotrivial subvariety.(4) In the course of the proof, periodic points of a complex polarized endomorphismplay an important part and as such, we often need to replace f by f n to reduce tofixed points. We need for that Lemma 12, which is based on results of Fakhrud-din and Petsche-Szpiro-Tepper [Fak, PST], and states that a map is isotrivial ifand only if one of its iterates is. Isotriviality is therefore a dynamically relevantproperty.(5) The classical Northcott property, on number fields, implies that the set of rationalpoints of height ≤ ε is finite for any ε >
0. However, in the function field caseover a field of characteristic zero, Baker observed that they are infinitely manyelements of degree 0 (which corresponds to constant rational functions) and allof them have uniformly bounded canonical height [Ba]. In particular, one cannotreplace in Theorem 1 ” ∃ ε >
0” by ” ∀ ε > ε = 0 and still use the terminology of ”Northcott Property”. Nevertheless,an interesting problem is to show that sup { ε ≥ , ∀ x ∈ X, h f ( x ) ≤ ε = ⇒ h f ( x ) = 0 } is positive.We now describe our new strategy of the proof. Recall that K = C ( B ) is the field offunctions over the complex projective variety B . Let ( X, f, L ) be a polarized endomor-phism and ( X , f , L ) over B an associated model. Let N be a very ample linebundle on B such that M := L ⊗ π ∗ ( N ) is ample on X , and let ω B be a K¨ahler form on B which iscohomologous to c ( N ). Similarly, let b ω be a continuous closed positive (1 , X cohomologous c ( L ).Take any irreducible subvariety Z of dimension 0 ≤ ℓ ≤ k = dim X of X defined over K and let Z be the associated subvariety of X . By [G1, G3, Fab, CGHX]), the notion ofheight h X,L can be extended to the irreducible subvariety Z as the intersection number h X,L ( Z ) := (cid:16) Z · c ( L ) ℓ +1 · c ( π ∗ N ) dim B − (cid:17) , and it can be computed as h X,L ( Z ) = Z X Λ b ω ℓ +1 ∧ ( π ∗ ω B ) dim B − ∧ [ Z ] . We can also defined the canonical height ˆ h f ( Z ) of Z asˆ h f ( Z ) = lim n d ( ℓ +1) n h X,L (( f ∗ ) n ( Z )) . HE GEOMETRIC DYNAMICAL NORTHCOTT AND BOGOMOLOV PROPERTIES 5
So, one could hope thatˆ h f ( Z ) = lim n → + ∞ d ( ℓ +1) n Z X Λ b ω ℓ +1 ∧ ( π ∗ ω B ) dim B − ∧ ( f n ) ∗ [ Z ]= Z X Λ lim n → + ∞ d ( ℓ +1) n ( f n ) ∗ ( b ω ℓ +1 ) ∧ ( π ∗ ω B ) dim B − ∧ [ Z ] , and use the convergence d − ( ℓ +1) n ( f n ) ∗ ( b ω ℓ +1 ) → b T ℓ +1 f in the sense of currents.This has no reason to be true since some of the mass is ”lost” on the boundary π − ( B \ Λ).This corresponds to the places of bad reduction and DeMarco’s idea in the case where X = P is to do analysis at those places in a delicate way [D2, Sections 3 & 4]. In here, welook instead at what happens away from π − ( B \ Λ) using an appropriate cut-off functionbuilt with a naive degeneration’s estimate of potentials of the current b T f (such estimateis present in DeMarco’s article). We are able to compute the canonical height of anyirreducible subvariety Z of X in term of the fibered Green current, showing that stability(a current is zero) is in fact a global notion (the canonical height is zero). The cut-offfunction we use is a DSH function, the strength of such objects, introduced by Dinh andSibony [DS], is that their definition involves the complex structure of the space (which isnot the case for C α -functions). This is the content of our second result: Theorem B.
Let ( X, f, L ) be a polarized endomorphism over the field of rational functions K of a normal complex projective variety B and let ( X , f , L ) be an associated model withregular part Λ . Let N be an ample linebundle on B such that L ⊗ π ∗ ( N ) is ample. Let Z be any irreducible subvariety of dimension ≤ ℓ ≤ k = dim X of X defined over K andlet Z be the corresponding subvariety of X . Then b h f ( Z ) = Z X Λ b T ℓ +1 f ∧ [ Z ] ∧ ( π ∗ ω B ) dim B − , where ω B is a K¨ahler form on B which is cohomologous to c ( N ) . Such formula is probably known in special cases, especially for multiplication by n ≥ ⋆ ) inTheorem A. Corollary 3.
Under the hypothesis of Theorem B, the pair (( X , f , L ) , [ Z ]) is stable if andonly if b h f ( Z ) = 0 if and only if deg M ( f n ( Z )) · deg( f n | Z ) ≍ d ℓn . Pick now a point z ∈ X ( K ) and let Z n be the irreducible subvariety of X inducedby f n ( z ). Assume Z is stable so that (deg M ( Z n )) n is bounded by some constant D > n . We note that, in particular, f induces an endomorphism on the Zariskiclosure W of { Z n : n ≥ } in the appropriate Hilbert scheme of X . Showing thatthis variety has dimension 0 automatically gives that z is preperiodic. Set b W := { ( C , x ) ∈ W × X : x ∈ C } . Using a similarity argument between the phase space and the parameterspace at a repelling periodic point `a la Tan Lei (Lemma 18), we deduce that, if the image Y of the canonical projection b W → X has positive dimension, then all fibers of π | Y are isomorphic to W and the action of f on Y is isotrivial. Finally, we prove that Y isindependent of z using the functoriality properties of height functions, see § THOMAS GAUTHIER AND GABRIEL VIGNY
The Geometric Dynamical Bogomolov Property.
As above, the GeometricBogomolov Conjecture (proved by Cantat, Gao, Habegger and Xie [CGHX]) can be in-terpreted as a dynamical statement, we propose a Dynamical version of the GeometricBogomolov Conjecture:
Conjecture 1 (Geometric Dynamical Bogomolov) . Let K be a function field of character-istic zero. Let ( X, f, L ) be a non-isotrivial polarized endomorphism defined over K , where X is normal. Let Z ⊂ X ¯ K be an irreducible subvariety and assume that, for any ε > ,the set Z ε := { x ∈ Z ( ¯ K ) : b h f ( x ) < ε } is Zariski dense in Z . Then, one of the followingholds: (1) either Z is preperiodic under iteration of f , (2) or there exist an integer k ≥ and an isotrivial polarized endomorphism ( Y, g, E ) ,a subvariety V ⊂ X ¯ K with f k ( V ) = V , and a dominant rational map p : V Y such that p ◦ ( f k | V ) = g ◦ p , and an integer N ≥ and an isotrivial subvariety W ⊂ Y , such that f N ( Z ) = p − ( W ) . Note that, if X = A is a non-isotrivial abelian variety and if f is the multiplicationby an integer n ≥
2, this reduces to the Geometric Bogomolov Conjecture proposed byYamaki [Y, Conjecture 0.3], where V is the K / k -Trace of A , whence it is known to hold.Inspecting the proof of Corollary 23 this implies that, if f is a non-isotrivial Latt`es map,then b h f ( Z ) = 0 implies Z is preperiodic. Note also that the second case actually occurs,i.e. if f preserves a fibration of which Z is a fiber, we actually have b h f ( Z ) = 0, eventhough it may not be preperiodic under iteration of f , see Section 5 for more details.We prove the Geometric Dynamical Bogomolov Conjecture holds in certain situations:(1) when f : P → P is non-isotrivial polynomial skew-product with an isotrivial firstcoordinate using properness of the composition maps on moduli spaces of complexpolynomials in one variable, see Theorem 29,(2) when f : P → P is a non-isotrivial endomorphism and deg( f | Z ) ≥ Cd n for all n ≥
1, where C is a constant, using a strategy similar to that employed to proveTheorem A, see Theorem 31.Let f : P × Λ → P × Λ be an algebraic family of complex polynomial skew-products andlet f : P K → P K be the induced endomorphism defined over K = C (Λ). The fundamentalwork of Berteloot, Bianchi and Dupont [BBD] sets up a good notion of stability, J -stability ,for families of endomorphisms of P : one way to formulate the J -stability of the family f is to require that b h f (Crit( f )) = 0. We can come back to the initial question studied byMcMullen: can we prove a rigidity property for J -stable families? As an easy applicationof the above, we prove Theorem C. (1)
Let f : P × Λ → P × Λ be a non-isotrivial complex algebraic familyof polynomial skew-product. Then f is not J -stable. (2) Let f : P × Λ → P × Λ be a non-isotrivial complex algebraic family of endomor-phisms of P of degree d ≥ which is J -stable. Assume (a) no iterate f n of f preserves a fibration over an isotrivial family of rationalmaps g : P × Λ → P × Λ , (b) for a general parameter λ ∈ Λ and any irreducible component C of Crit( f λ ) ,there exists M > such that deg( f nλ | C ) ≥ M · d n .Then f is postcritically finite. HE GEOMETRIC DYNAMICAL NORTHCOTT AND BOGOMOLOV PROPERTIES 7
Organization of the article.
Section 2 is devoted to analytic aspects of families ofpolarized endomorphisms. In section 3, we study analytic aspects of algebraic familiesof polarized endomorphisms and prove Theorem B. Section 4 is dedicated to the proofof Theorem A and applications. In Section 5, we reformulate the Geometric BogomolovConjecture to justify the above dynamical variant of this problem and we prove somerelated basic properties. In Section 6, we focus on the case of endomorphisms of P . Inparticular, we prove Theorem C. In Sections 4 and 6, we illustrate our results with anexplicit family of endomorphisms of P : the elementary Desboves family, see also 3.2. Analytic families of polarized endomorphisms
Dynamics of polarized endomorphisms of projective varieties.
Fix a field K of characteristic zero. Definition 1.
The triple ( X, f, L ) is called a polarized endomorphism over K if (1) X is an irreducible projective variety defined over K , (2) L is an ample divisor of X which is defined over K and (3) f : X → X is an endomorphism defined over K which is polarized , i.e. thereexists an integer d ≥ such that f ∗ L ≃ L ⊗ d .Then integer d is the degree of ( X, f, L ) . We now focus on the complex setting. When X is a complex projective variety, denoteby X reg its regular part. Recall that a very ample divisor L on X corresponds to anembedding ι : X ֒ → P N and may be defined as the pullback ι ∗ O (1) of a hyperplane of P N .We thus may also define a (degenerate) hermitian metric on X by setting ˜ ω X := ι ∗ ( ω FS )on X reg , where ω FS is the Fubini-Study form on P N , and extending ˜ ω X trivially to X . Thisis what we call a K¨ahler form on X . We then let ω X := α · ˜ ω X where α := (cid:0)R X ˜ ω kX (cid:1) /k .If f is polarized with respect to L , we have f ∗ ω X = d · ω X + dd c φ, where φ is a continuous function on X . In the rest of the article, a ( p, p )-current means acurrent of bidegree ( p, p ). Building on recent results, we prove the following: Proposition 4.
Let ( X, f, L ) be a polarized endomorphism over C of degree d . Then thesequence ( d − n ( f n ) ∗ ω X ) converges in the sense of currents to a closed positive (1 , -current T f . Moreover, (1) T f = ω X + dd c g f where g f is continuous on X and f ∗ T f = d · T f , (2) µ f := T kf is a probability measure of maximal entropy k log d which does not givemass to pluripolar sets and with f ∗ µ f = d k µ f , (3) J -repelling periodic points of f are equidistributed with respect to µ f , i.e. if R n isthe set of repelling periodic points z ∈ supp( µ f ) with f n ( z ) = z , then d nk X z ∈ R n δ z −→ µ f , in the weak sense of measures on X . The proof relies on the next lemma, which is a direct consequence of [NZ, Lemma 2.1].
THOMAS GAUTHIER AND GABRIEL VIGNY
Lemma 5.
Let X be a normal complex projective variety of dimension k , let L be a bigand nef divisor on X and let f : X → X be an endomorphism. Assume there exists aninteger d ≥ such that f ∗ L ≃ L ⊗ d . Then f is surjective and has large topological degree d k .Proof. Under the assumption of the lemma, [NZ, Lemma 2.1] implies that f is surjectiveand has topological degree d k . Moreover, for any 1 ≤ j ≤ k and any n ≥
1, as L is bigand nef, deg j,L ( f n ) ≤ deg j,L ( f ) n = (cid:16) ( f ∗ L ) j · L k − j (cid:17) n = d jn . In particular, the j -th dynamical degree of f satisfies λ j ( f ) := lim n →∞ (cid:0) deg j,L ( f n ) (cid:1) /n ≤ d j . As λ k ( f ) is the topological degree of f and d >
1, we have λ j ( f ) ≤ d j < d k = λ k ( f ) forall 1 ≤ j < k , as required. (cid:3) Proof of Proposition 4.
Recall that f ∗ ω X = d ( ω X + dd c u f ), where u f is a continuousfunction on X . By an immediate induction, we find1 d n ( f n ) ∗ ω X = ω X + dd c n − X j =0 u f ◦ f j d j . The sequence u n := P n − j =0 u f ◦ f j d j of continuous functions converges uniformly on X to a con-tinuous function g f and, by construction, if T f := ω X + dd c g f , then T f = lim n d − n ( f n ) ∗ ω X and f ∗ T f = d · T f .To prove points 2 and 3, we let p : ˜ X → X be a resolution of the singularities of X .Then ˜ X is a smooth whence normal projective variety of dimension k and the morphism f ◦ p : ˜ X → X lifts as a morphism ˜ f : ˜ X → ˜ X . Let ˜ L := p ∗ ( L ). As p is generically finiteand L is ample, ˜ L is big and nef, and ˜ f ∗ ˜ L ≃ ˜ L ⊗ d by construction. We can thus applyLemma 5: the morphism ˜ f is dominant with large topological degree. Moreover, T ˜ f is thestrict transform p ∗ ( T f ) of T f by p , i.e. the trivial extension of p ∗ ( T f ) | p − ( X reg ) to ˜ X . By[DNT] (see also [BD2]), the probability measure p ∗ ( µ f ) = T k ˜ f is invariant by ˜ f : ˜ X → ˜ X and satisfies points 2 and 3 of the proposition.To conclude, we remark that, since T k ˜ f does not give mass to pluripolar sets, and since µ f is the natural extension to X of the restriction of p ∗ ( T k ˜ f ) to X reg , items 2 and 3 areproved. (cid:3) An immediate consequence is the following:
Corollary 6.
Let ( X, f, L ) be a polarized endomorphism over C . Then J -repelling periodicpoints of f contained in X reg are Zariski dense in X . We will also use the next lemma in the sequel.
Lemma 7.
Let ( X, f, L ) be a polarized endomorphism over C . For any n ≥ , the set { z ∈ X : f n ( z ) = z } is finite.Proof. Fix n ≥ Y := { z ∈ X : f n ( z ) = z } of X has anirreducible component W of dimension at least 1. Then W is periodic under iteration,whence, up to replacing f with f ℓ for some ℓ ≥
1, we may assume f ( W ) = W . Now, f HE GEOMETRIC DYNAMICAL NORTHCOTT AND BOGOMOLOV PROPERTIES 9 restricts to W as an endomorphism and, if f ∗ L = L ⊗ d with L ample, L | W is still ampleand f ∗ ( L | W ) = ( L | W ) ⊗ d so that f | W is also polarized, and so is f n | W . In particular, thetopological degree of f n | W is at least 2. This contradicts the fact that f n | W = id W . (cid:3) We will also rely on [Fak, Corollary 2.2]:
Proposition 8.
Let ( X, f, L ) be a polarized endomorphism over an infinite field k . Thenthere exist N ≥ , an embedding ι : X ֒ → P Nk and an endomorphism F : P N k → P N k suchthat ι ◦ f = F ◦ ι on X. The fibered Green current of a family of endomorphisms.
We extend herethe definitions of [DF] to the case of families of endomorphisms of complex projectivevarieties and we inspire from their work.Fix first a complex K¨ahler manifold Λ and let π : X Λ → Λ be a family of irreduciblecomplex projective varieties of dimension k ≥
1, i.e. π is an analytic submersion and X λ = π − { λ } is an irreducible complex projective variety of dimension k for all λ ∈ Λ.Fix a relatively ample divisor L on X Λ , i.e. L λ := L | X λ is an ample divisor on X λ for all λ . We let ( X Λ , f , L ) be a (complex) family of endomorphisms (or holomorphic family ofendomorphisms ), i.e. f : X Λ → X Λ is analytic and for all λ ∈ Λ, f λ := f | X λ : X λ → X λ is a morphism of varieties. We assume f is polarized with respect to L in the sense that f ∗ L ≃ L ⊗ d . Up to removing an analytic subvariety from Λ, we can assume ( f λ ) ∗ L λ islinearly equivalent to L ⊗ dλ for all λ ∈ Λ.We let b ω be a continuous closed positive (1 , X Λ cohomologous to a multipleof L such that ω λ := b ω | X λ is a K¨ahler form on ( X λ ) reg . We assume b ω is normalized sothat R X λ ω kλ = 1. Proposition 9.
Let ( X Λ , f , L ) be a family of endomorphisms. The sequence d − n ( f n ) ∗ ( b ω ) converges in the sense of currents towards a closed positive (1 , -current b T f with contin-uous potentials on X Λ . Moreover, (1) f ∗ b T f = d · b T f , (2) for all λ , the slice T λ := b T f | X λ is well-defined and ( f λ ) ∗ T λ = d · T λ .Proof. Since L λ is cohomologous to ω λ for any λ , we can write f ∗ b ω = d · ( b ω + dd c u ), where u is continuous on X Λ . An easy induction then gives1 d n ( f n ) ∗ b ω = b ω + dd c n − X j =0 u ◦ f j d j , and d − n ( f n ) ∗ b ω converges towards the closed positive (1 , b T f := b ω + dd c g with g := P j d − j u ◦ f j . By construction, it has continuous potentials and satisfies f ∗ b T f = d · b T f and the slices of b T f are well-defined and satisfy b T f | X λ = ω λ + dd c ( g | X λ ). We thushave proved the proposition. (cid:3) Definition 2.
The current b T f is the fibered Green current of the family ( X Λ , f , L ) . The bifurcation current of a horizontal current.
Using the notations of theprevious subsection, we say that an analytic set V ⊂ X Λ of codimension p is horizontal iffor any λ ∈ Λ, codim ( X λ ∩ V ) = p. A closed positive ( p, p )-current S on X Λ is horizontal if for any λ ∈ Λ, the slice S | X λ isa well-defined closed positive ( p, p )-current on X λ . We may use the following terminologyin the sequel. Definition 3. A p - measurable dynamical pair (( X Λ , f , L ) , S ) on X Λ parametrized bya complex manifold Λ is the datum of a holomorphic family ( X Λ , f , L ) as above and ahorizontal positive closed ( p, p ) -current S on X Λ . The bifurcation current of a p -measurable dynamical pair (( X Λ , f , L ) , S ) is defined as T f ,S := π ∗ (cid:16) b T k +1 − p f ∧ S (cid:17) . The idea to study bifurcations using currents goes back to the work of DeMarco [D1] andhas been intensively used since then, see e.g. [Ber] and references therein.We also endow Λ with a K¨ahler form ω Λ , and let ˇ ω Λ := ( π ) ∗ ω Λ . In this case, ˇ ω Λ + b ω isa K¨ahler form on ( X Λ ) reg and we can define the mass of a closed positive ( p, p )-current S on a compact subset K of X Λ as k S k K := Z K S ∧ ( b ω + π ∗ ω Λ ) k +1 − p . Definition 4.
We say that the dynamical pair (( X Λ , f , L ) , S ) is stable if the sequence { ( f n ) ∗ ( S ) } n ≥ of horizontal closed positive ( p, p ) -currents satisfies k ( f n ) ∗ ( S ) ∧ ( π ∗ ω Λ ) dim Λ − k K = o ( d n ( k +1 − p ) ) in any compact subset K of X Λ . Following the proof of [DF, Proposition-Definition 3.1] and of [BBD, Lemma 3.13], wesee that (( X Λ , f , L ) , S ) is stable if and only if T f ,S = 0 as a current on Λ. We give theproof below. Proposition 10.
Let ( X Λ , f , L ) be any holomorphic family of polarized endomorphisms asabove and let S be any horizontal positive closed ( p, p ) -current of X Λ . Then, the followingassertions are equivalent: (1) the pair (( X Λ , f , L ) , S ) is stable, (2) for any compact set K ⊂ X Λ , we have k ( f n ) ∗ ( S ) ∧ ( π ∗ ω Λ ) dim Λ − k K = O ( d n ( k − p ) ) , (3) as a current on Λ , we have T f ,S = 0 . In fact, Proposition 10 is a direct consequence of the next lemma.
Lemma 11.
For any compact sets K ⋐ K ⋐ Λ , there exists a constant C ( K , K ) > depending only on K , K and on ( X Λ , f , L ) , such that for any n ≥ , the quantity (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k ( f n ) ∗ ( S ) ∧ ( π ∗ ω Λ ) dim Λ − k π − ( K ) − d ( k +1 − p ) n Z π − ( K ) b T k +1 − p f ∧ S ∧ ( π ∗ ω Λ ) dim Λ − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) is bounded above by C ( K , K ) · d ( k − p ) n k S ∧ ( π ∗ ω Λ ) dim Λ − k π − ( K ) . HE GEOMETRIC DYNAMICAL NORTHCOTT AND BOGOMOLOV PROPERTIES 11
Proof.
Let K ⋐ K ⋐ Λ be any compact subsets and write S n := ( f n ) ∗ ( S ) ∧ ( π ∗ ω Λ ) dim Λ − ,for any integer n ≥
0. Since ( π ◦ f n ) ∗ ω Λ = π ∗ ω Λ , we have k S n k π − ( K ) = Z π − ( K ) S ∧ ( π ∗ ω Λ + ( f n ) ∗ b ω ) k +1 − p . Now we use that S = S ∧ ( π ∗ ω dim Λ − ), ω dim Λ+1Λ = 0 and d n b T f = ( f n ) ∗ b ω + dd c ( g ◦ f n ),where g is a continuous b ω -psh function on X Λ . So k S n k π − ( K ) = Z π − ( K ) S ∧ ( π ∗ ω Λ ) dim Λ − ∧ (cid:16) d n b T f − dd c ( g ◦ f n ) (cid:17) k +1 − p + ( k + 1 − p ) Z π − ( K ) S ∧ ( π ∗ ω Λ ) dim Λ ∧ (cid:16) d n b T f − dd c ( g ◦ f n ) (cid:17) k − p (1)Now, (cid:16) d n b T f − dd c ( g ◦ f n ) (cid:17) k +1 − p = k +1 − p X j =0 (cid:18) k + 1 − pi (cid:19) ( − k +1 − p − i d ni b T i f ∧ ( dd c ( g ◦ f n )) k +1 − p − i , (cid:16) d n b T f − dd c ( g ◦ f n ) (cid:17) k − p = k − p X j =0 (cid:18) k − pi (cid:19) ( − k − p − i d ni b T i f ∧ ( dd c ( g ◦ f n )) k − p − i . Take i ∈ { , . . . , k − p } . By the Chern-Levine-Nirenberg inequality, there exists a constant C > K and K , k b T f k π − ( K ) and k g k L ∞ ( π − ( K )) such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z π − ( K ) S ∧ ( π ∗ ω Λ ) dim Λ − ∧ b T i f ∧ ( dd c ( g ◦ f n )) k +1 − p − i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C k S k π − ( K ) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z π − ( K ) S ∧ ( π ∗ ω Λ ) dim Λ ∧ b T i f ∧ ( dd c ( g ◦ f n )) k − p − i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C k S k π − ( K ) . Now, subtracting d ( k +1 − p ) n R π − ( K ) b T k f ∧ S to both sides of (1), we have, up to enlarging C : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k S n k π − ( K ) − d ( k +1 − p ) n Z π − ( K ) b T k f ∧ S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C k S k π − ( K ) d ( k − p ) n which ends the proof. (cid:3) Proof of Proposition 10.
Let K be any compact subset of X Λ . Remark that it is alwayscontained in a compact subset of the form π − ( K ), where K is compact in Λ, since thefibers of π are compact. By Lemma 11, • if k ( f n ) ∗ ( S ) ∧ ( π ∗ ω Λ ) dim Λ − k π − ( K ) = O ( d n ( k − p ) ), then T f,S ∧ ω dim Λ − is zero on K , • if T f ,S ∧ ω dim Λ − is zero on K , then k ( f n ) ∗ ( S ) ∧ ( π ∗ ω Λ ) dim Λ − k π − ( K ) = O ( d n ( k − p ) ).As this holds for any compact set K ⋐ Λ, this proves the equivalence between points 2and 3, which obviously imply point 1. The proof that point 1 implies point 2 follows alsofrom the above. (cid:3)
Remark.
When dim Λ = 1 and S = [ C ] is the integration current on a horizontal curve C ⊂ X Λ , the definition of stability we give here says in particular that, for any compact set K , if C n := f n ( C ), the mass k [ C n ] k K is bounded. By a famous Theorem of Bishop, there exists a subsequence ( C n k ) which converges in Hausdorff topology towards an analytic set C ∞ . In particular, if C is the graph of a holomorphic section σ : Λ → X Λ of π , this isequivalent to the local uniform convergence of the sequence σ k := f n k ◦ σ of sections of π to a holomorphic section. In other words, ( f n ◦ σ ) is a normal family.3. Algebraic families of polarized endomorphisms
Polarized endomorphisms over a function field versus algebraic families.
Let B be a normal complex projective variety and let K := C ( B ) be its field of rationalfunctions. Let ( X, f, L ) be a polarized endomorphism over K . Assume in addition that X is normal. Such a variety X endowed with the ample divisor L gives rise to a normalmodel ( X , L ) of ( X, L ), i.e. a morphism π : X −→ B between projective varieties, where X is normal and L is a relatively ample divisor, andsuch that(1) the generic fiber of X is isomorphic to X ,(2) the linebundle L is isomorphic to the restriction of L to the generic fiber,(3) there exists a Zariski open set of B whose complement has codimension at least 2,and over which π is flat.Note that, up to replacing L with L ⊗ e for some integer e >
1, we can assume it isvery ample, so that L defines an embedding i : X ֒ → P N K with L = i ∗ O P N (1). Theabove construction amounts to taking the Zariski closure of X in P N B , and then taking itsnormalization. We may also regard this as an embedding ι : X ֒ → P N C × B ֒ → P M C and L is simply O P N C (1) | X . For any λ lying in a non-empty Zariski open subset Λ of B , the fiber X λ := π − { λ } is a normal complex projective variety of dimension dim X and L λ is anample divisor of X λ . In particular, denoting X Λ := π − (Λ), then X Λ → Λ gives rise to ananalytic family of projective varieties as in Section 2.2.Up to replacing Λ with a Zariski open subset Λ ′ ⊂ Λ, the morphism f induces a familyof polarized endomorphism on X Λ , denoted by f as follows: it induces a dominant rationalmap f : X X such that:(1) the following diagram commutes X f / / ❴❴❴❴❴❴❴ π ❆❆❆❆❆❆❆❆ X π ~ ~ ⑥⑥⑥⑥⑥⑥⑥⑥ B (2) f | X Λ : X Λ → X Λ is a morphism,(3) for any λ ∈ Λ, if we set f λ := f | X λ and L λ := L | λ , then ( X λ , f λ , L λ ) is a polarizedendomorphism over C , as in Section 2.1. Definition 5.
A triple ( X , f , L ) as above is called an algebraic family of polarized endo-morphisms, which is a model of ( X, f, L ) . The open set Λ is the regular part of ( X , f , L ) . We note that the previous discussion implies that a polarized endomorphism (
X, f, L )over the field K = C ( B ) of rational functions of a normal projective complex variety B gives rise to an algebraic family of polarized endomorphisms. The converse is also true:let X be the generic fiber of X , and let L and f be the respective restrictions of L and f to the generic fiber, then ( X, f, L ) is a polarized endomorphism over K . HE GEOMETRIC DYNAMICAL NORTHCOTT AND BOGOMOLOV PROPERTIES 13
Definition 6.
Let ( X, f, L ) be a polarized endomorphism over K and let ( X , f , L ) be amodel of ( X, f, L ) with regular part Λ . We say ( X, f, L ) , (or equivalently ( X , f , L ) ), is isotrivial if for any λ, λ ′ ∈ Λ , there exists an isomorphism φ : X λ −→ X λ ′ such that φ ◦ f λ = f ′ λ ◦ φ and φ ∗ L λ ′ ≃ L λ . Remark. (1) In particular, if (
X, f, L ) is isotrivial, so is the polarized variety (
X, L ).(2) When X = P k (or equivalently when X = P k × B ), the last condition φ ∗ L λ ′ ≃ L λ is automatically satisfied. Indeed, we can assume L = O P k (1), so that for any λ , we have L λ = O P k (1). In this case, the isotriviality of ( X , f , L ) reduces tothe existence, for any λ, λ ′ ∈ Λ, of linear automorphisms φ : P k → P k such that φ ◦ f λ = f λ ′ ◦ φ . Example (The elementary Desboves family) . As already used by e.g. [BD1, BDM, BT],for any λ ∈ C ∗ let f λ : P → P be the endomorphism of degree 4 given by f λ ([ x : y : z ]) := [ − x ( x + 2 z ) : y ( z − x + λ ( x + y + z ) : z (2 x + z )] . This defines an algebraic family ( P × P , f , O P (1)) with regular part C ∗ ⊂ P . This familyinduces an endomorphism of degree 4 of P C ( z ) .Let us now describe more precisely some parts of the dynamics of this family. It isknown that for any λ ∈ C ∗ , the point ρ := [0 : 1 : 0] is totally invariant by f λ , i.e. f − λ { ρ } = { ρ } and that f λ preserves the pencil P of lines of P passing through ρ .Furthermore, f λ preserves the lines X = { x = 0 } and Z = { z = 0 } which belong to P andthe line Y = { y = 0 } and the Fermat curve C := { [ x : y : z ] ∈ P : x + y + z = 0 } . Moreover, for any λ ∈ C ∗ , the restriction of f λ to each of those curves can be described: • the restriction of f λ to the line X is the degree 4 polynomial p λ ( z ) = λz +(1+ λ ) z , • the restriction of f λ to the line Z is the degree 4 polynomial p − λ , • the restriction of f λ to the line Y is the Latt`es map g : [ x : y ] ∈ P [ − x ( x − y ) : y (2 x + y )] ∈ P . • Finally, the restriction of f λ to the elliptic curve C is the isogeny u
7→ − u + β forsome β independent of λ .Note that the family ( P × P , f , O P (1)) is obviously non-isotrivial, since the restriction p λ of f λ to X has a fixed point with multiplier 1 + λ , which is a non-constant rationalfunction of λ . This example will be investigated again in the rest of the article to illustrateour main results, with a concrete example. † In the rest of the article, we will use the following, which in turn says that isotrivialitycan be read on the iterates of a family.
Lemma 12.
Let ( X , f, L ) be an algebraic family of polarized endomorphisms of degree d . Then ( X , f , L ) is isotrivial if and only if there exists n ≥ such that ( X , f n , L ) isisotrivial.Proof. If ( X , f , L ) is isotrivial, obviously ( X , f n , L ) is isotrivial for any n ≥
1. We thusprove the converse implication. Let X be the generic fiber of π , L := L | X and f := f | X .Then ( X, f, L ) is a polarized endomorphism over C ( B ) and Proposition 8 implies it inducesa polarized endomorphism ( P k , F, O P k (1)) over C ( B ), for some suitable k and F . In particular, the family ( X , f , L ) induces a family ( P k × B , F , O P k (1)) of endomorphisms of P k .If ( X , f n , L ) is isotrivial, by construction of the induced map ( P k , F, O P k (1)), the iter-ated family ( P k × B , F n , O P k (1)) is also isotrivial. Let us explain briefly the constructionof F to see why it is true: up to replacing L with L ⊗ e we may assume L is very am-ple. In this case, L induces an embedding ι : X ֒ → P ( H ( X, L )) and, if ( s , . . . , s k )is a basis of H ( X, L ) with no common zeros, we then have f j := f ∗ ( s j ) which is adegree d polynomial in the s i ’s. We may consider ( s , . . . , s k ) as affine coordinates on A k +1 , so that the pullback map f ∗ induces an endomorphism F : P k → P k , whichmay be defined by F ([ s : · · · : s k ]) = [ f : · · · : f k ]. In particular, F n is induced by( f n ) ∗ : H ( X, L ) → H ( X, L ⊗ d n ) and is thus isotrivial.We conclude using Corollary 16 from [PST], which states that this is equivalent to thefact that ( P k × B , F , O P k (1)) is isotrivial. (cid:3) Remark.
Corollary 16 from [PST] is proved in the case when K is the field of rationalfunctions of a curve. However, their proof only uses the two following facts that remaintrue if dim B > • The field K is a product formula field with only non-archimedean places. Thisis also the case for when K is the field of rational functions of a normal complexprojective variety, see e.g. [BG], • The set of
P GL ( k + 1)-conjugacy classes of endomorphisms of P k is an affinevariety over K . This follows from the fact that the quotient is geometric and thusit remains true over any field.To finish the present discussion, we recall that to any subvariety Z of X , which isdefined over K , we can associate a subvariety Z of X (which can be defined as the Zariskiclosure of Z in X ) such that the restriction of π to Z is flat over Λ. In particular, toany point x ∈ X ( K ) with corresponding subvariety x , we can associate a rational section σ : B X of π , i.e. a rational map such that(1) π ◦ σ = id B ,(2) σ is regular over Λ,(3) the Zariski closure of σ (Λ) in X is x .3.2. Global properties of the bifurcation current.
Pick a polarized endomorphism(
X, f, L ) over the field of rational functions K of a normal complex projective variety B .Let ( X , f , L ) be a normal model of ( X, f, L ) as above.Fix now a very ample linebundle N on B , i.e. N := O P M (1) | B for some embedding B ֒ → P M C , the linebundle M := L ⊗ π ∗ ( N )is ample on X . Indeed, we also may embed X in P N × B : X ֒ → P N × B ֒ → P N × P M ⊂ P R C , with L = O P R (1) | X . We then let ω B be the restriction of the Fubini-Study form of P M on B such that deg N ( B ) = R B ω dim BB , i.e. with { ω B } = N and b ω be the restriction ofthe Fubini-Study form of P N on X so that { b ω } = L . The closed positive (1 , b ω + π ∗ ( ω B ) is a K¨ahler form on X . HE GEOMETRIC DYNAMICAL NORTHCOTT AND BOGOMOLOV PROPERTIES 15
Again, for any Borel subset Ω of X , we let k S k Ω := D S, Ω ( π ∗ ( ω B ) + b ω ) k +dim B − p E . When k S k := k S k X Λ < + ∞ , the current S extends trivially to the Zariski closure X of X Λ as a closed positive ( p, p )-current on X we still denote by S .We use the notations of Section 3.1. We want here to prove the following, which is aglobal version of Lemma 11: Proposition 13.
Let ( X , f , L ) be an algebraic family of polarized endomorphisms withregular part Λ and let k := dim X λ for any λ ∈ B . Then there exists a constant C > depending only on ( X , f , L ) such that (1) For any ≤ ℓ ≤ dim B , the current b T f ∧ ( π ∗ ω B ) ℓ − has bounded mass on X Λ .Moreover, for any closed positive ( ℓ, ℓ ) -current S on X with ≤ ℓ ≤ dim X − and any n ≥ , (cid:12)(cid:12)(cid:12) k b T f ∧ S k − d − n k ( f n ) ∗ ( b ω ) ∧ S k (cid:12)(cid:12)(cid:12) ≤ Cd n k S k . (2) For any ≤ p ≤ k , any ≤ ℓ ≤ dim B and any horizontal closed positive ( p, p ) -current S on X and any n ≥ , if q := k + dim B − ℓ − p , the quantity (cid:12)(cid:12)(cid:12)(cid:12)Z X Λ ( f n ) ∗ ( S ) ∧ ( π ∗ ω B ) ℓ ∧ b ω q − d nq Z X Λ b T q f ∧ S ∧ ( π ∗ ω B ) ℓ (cid:12)(cid:12)(cid:12)(cid:12) is bounded from above by C P j ≤ q − d nj k b T j f ∧ S ∧ ( f n ) ∗ ( b ω ) q − j − ∧ ( π ∗ ω B ) ℓ k .Proof. Let us first justify that we may view Λ as an affine variety. Let X be the genericfiber of π , L := L | X and f := f | X . Then ( X, f, L ) is a polarized endomorphism over C ( B )and Proposition 8 implies it induces a polarized endomorphism ( P k , f, O P k (1)) over C ( B ).In particular, the family ( X , f , L ) induces a family ( P k × B , F, O P k (1)) of endomorphisms of P k . Since, when d is fixed, the space Hom d ( P k ) of such endomorphisms is the complementin some projective space of an irreducible hypersurface by, e.g., [Mac], it is an affine variety.Whence Λ has to be affine.We thus can define an embedding ι : B ֒ → P M with ι − ( A M ( C )) = Λ. We also mayembed X in P N × B with L = ι ∗ ( O P N (1)): ι : X ι ֒ → P N × B ι ֒ → P N × P M , We now write ( z, t ) for a point of P N × P M and let k · k be the standard Hermitian normon A M ( C ). We also identify λ = π ( z ) with the second coordinate t of ι ◦ ι ( z ) in theproof. We rely on the next lemma, which is an application of Hilbert’s Nullstellensatzt: Lemma 14.
There exist
C, C ′ > such that for all n ≥ , we can write d n ( f n ) ∗ ( b ω ) − b T f = dd c φ n on X Λ , where | φ n ( z ) | ≤ d − n ( C log + k λ k + C ′ ) , for all λ ∈ Λ and all z ∈ X λ . We take the lemma for granted and continue the proof. We follow ideas of [GOV]. Forany
A >
0, we pick the following test functionΨ A ( λ ) := log max( k λ k , e A ) − log max( k λ k , e A ) A .
Then, Ψ A is continuous and DSH on Λ with dd c Ψ A = T + A − T − A where T ± A are somepositive closed (1 , k T ± A k ≤ C ′ /A for some C ′ > A nor on T ± A . Observe also that Ψ A is equal to − B (0 , e A ), and0 outside B (0 , e A ). We first prove point 1: pick 0 ≤ ℓ ≤ dim B − S a closed positive( ℓ, ℓ )-current on X and n ≥
1, then by Stokes J An := (cid:28)(cid:18) b T f − d n ( f n ) ∗ ( b ω ) (cid:19) ∧ ( b ω + π ∗ ω B ) k +dim B − ℓ ∧ S, Ψ A ◦ π (cid:29) = D φ n · ( b ω + π ∗ ω B ) k +dim B − ℓ , dd c (Ψ A ◦ π ) ∧ S E and the definition of Ψ A implies | J An | ≤ Z ι − ( P N × B (0 ,e A )) | φ n | ( b ω + π ∗ ω B ) k +dim B − ℓ ∧ ( T + A + T − A ) ∧ S ≤ C A sup P N × B (0 ,e A ) | φ n | · k S k , for some constant C >
0. Lemma 14 then implies | J An | ≤ C d − n k S k for some constant C > A . Making A → ∞ gives the first point of the Proposition.We now prove the second estimate. Let q := k + dim B − ℓ − p , since π ◦ f n = π , I An := (cid:28) d qn ( f n ) ∗ ( S ) ∧ (cid:16) b T q f − b ω q (cid:17) ∧ ( π ∗ ω B ) ℓ , Ψ A ◦ π (cid:29) = (cid:28) S ∧ (cid:18) b T q f − (cid:18) d n ( f n ) ∗ b ω (cid:19) q (cid:19) , Ψ A ◦ π ◦ f n · (( π ◦ f n ) ∗ ω B ) ℓ (cid:29) = q − X j =0 ( − d ) − n ( q − − j ) (cid:18) q − j (cid:19) · D S ∧ ( dd c φ n ) ∧ b T j f ∧ ( f n ) ∗ ( b ω ) q − − j , Ψ A ◦ π · ( π ∗ ω B ) ℓ E = q − X j =0 ( − d ) − n ( q − − j ) (cid:18) q − j (cid:19) · Z φ n · S ∧ b T j f ∧ ( f n ) ∗ ( b ω ) q − − j ∧ dd c (Ψ A ◦ π ) ∧ ( π ∗ ω B ) ℓ , where we used f ∗ b T f = d · b T f , b T f = d n ( f n ) ∗ ( b ω ) − dd c φ n and Stokes formula.We now let S j := S ∧ b T j f ∧ ( π ∗ ω B ) ℓ for any 0 ≤ j ≤ q −
1. The above implies | I An | ≤ q − X j =0 d − n ( q − − j ) (cid:18) q − j (cid:19) · Z | φ n | · S j ∧ ( f n ) ∗ ( b ω ) q − − j ∧ ( T + A + T − A ) ≤ q − X j =0 d − n ( q − − j ) (cid:18) q − j (cid:19) C A · k S j ∧ (( f n ) ∗ ( b ω )) q − − j k · sup P N × B (0 ,e A ) | φ n | , for some universal constant C >
0. By the first point, b T f has bounded mass and d − n ( f n ) ∗ b ω has uniformly bounded mass. In particular, using again Lemma 14, we find constants HE GEOMETRIC DYNAMICAL NORTHCOTT AND BOGOMOLOV PROPERTIES 17 C , C > | I An | ≤ q − X j =0 C A d − n ( q − − j ) (cid:18) q − j (cid:19) · k S j ∧ (( f n ) ∗ ( b ω )) q − − j k · sup P N × B (0 ,e A ) | φ n |≤ C q − X j =0 d − n ( q − j ) (cid:18) q − j (cid:19) · k S j ∧ (( f n ) ∗ ( b ω )) q − j − k . Since the constants do not depend on A , we can make A → ∞ and multiply by d nq tocomplete the proof. (cid:3) We now give the proof of Lemma 14.
Proof of Lemma 14.
Write F λ ( z ) = [ P ,λ ( z ) : · · · : P N,λ ( z )] , z ∈ P N , where P i,λ ∈ C ( X )[ z , . . . , z N ] are homogeneous polynomials and Res( P ,λ , . . . , P N,λ ) = 0 if and only if λ ∈ B \ Λ. Write ˜ F λ = ( P ,λ , . . . , P N,λ ) and recall that d − ˜ F ∗ ˆ ω FS − ˆ ω FS = dd c g , where g ( z, λ ) := d log k ˜ F λ ( p ) k − log k p k , for any λ ∈ Λ and any p = ( p , . . . , p N ) ∈ C N +1 − { } such that z = [ p : · · · : p N ]. By construction, we have b T f − d n ( ˜ F n ) ∗ ˆ ω FS = dd c ∞ X j = n d j g ◦ ˜ F j . (2)Note that the coefficient of ˜ F λ in a given system of homogeneous coordinates are regularfunctions on Λ and that also Res( ˜ F λ ) ∈ C [Λ], so that (cid:12)(cid:12)(cid:12) log | Res( ˜ F λ ) | (cid:12)(cid:12)(cid:12) ≤ C log + k λ k + C . An application of the homogeneous Hilbert’s Nullstellensatzt (see, e.g., [GOV, proof ofLemma 6.5]) implies there exist a constant
C > N ≥ C − | Res( ˜ F λ ) | ≤ k ˜ F λ ( p ) kk p k d ≤ C max ( k λ k , N for any p ∈ C N +1 \ { } and any λ ∈ Λ . In particular, since g ( z, λ ) = log k ˜ F λ ( p ) k / k p k d , this gives easily | g ( z, λ ) | ≤ C ′ log + k λ k + C ′′ for any z ∈ P k and any λ ∈ Λ.Using that ι ◦ f ( z ) = ( F λ ◦ ι ( z ) , λ ), for all λ ∈ Λ and all z ∈ X λ , we deduce the lemmafrom the above. (cid:3) Global height function versus mass of a current.
Again, pick a polarized endo-morphism (
X, f, L ) over the field of rational functions K of a normal complex projectivevariety B . By the classical functorial properties of Weil heights on varieties over functionfields, and since f ∗ L ≃ L ⊗ d , we have h X,L ◦ f = d · h X,L + O (1), see, e.g. [HS, Theo-rem B.10.4]. Recall that, following [CS], we define the canonical height of f as b h f :=lim n →∞ d n h X,L ◦ f n . By [CS, Theorem 1.1], it is the unique function ˆ h f : X ( ¯ K ) → R + satisfying(1) b h f ◦ f = d · b h f ,(2) b h f = h X,L + O (1).For any irreducible subvariety Z of dimension 0 ≤ ℓ ≤ k = dim X of X defined over K ,the height h X,L ( Z ) can be defined as the intersection number h X,L ( Z ) := (cid:16) Z · c ( L ) ℓ +1 · c ( π ∗ N ) dim B − (cid:17) , where Z be the Zariski closure of Z in X , see, e.g., [G1, G3, Fab, CGHX], and the height h X,L ( Z ) of Z can be computed as h X,L ( Z ) = Z X Λ b ω ℓ +1 ∧ ( π ∗ ω B ) dim B − ∧ [ Z ] , by Skoda Theorem, since π ∗ ω B and b ω have continuous potentials. When Z is definedover a finite extension K ′ of K , we let ρ ′ : B ′ → B be the normalization of B in K ′ . If X ′ := X × B B ′ , the projection ρ : X ′ → X onto the first factor is a finite branched coverand, if Z ′ is the Zariski closure of Z in X ′ , we can set h X,L ( Z ) = 1[ K ′ : K ] (cid:16) Z ′ · c ( ρ ∗ L ) ℓ +1 · c ( ρ ∗ π ∗ N ) dim B − (cid:17) We thus can assume in the sequel that Z is defined over K . We can also define b h f ( Z ) = lim n →∞ d n ( ℓ +1) h X,L (( f n ) ∗ ( Z ))Remark that the above implies have b h f ( f ∗ ( Z )) = d ℓ +1 · b h f ( Z ). In particular, if Z ispreperiodic, i.e. if there exists n > m ≥ f n ( Z ) = f m ( Z ), we have b h f ( Z ) = 0.We can also compute the canonical height of Z as the mass of its bifurcation current.Let us now prove Theorem B. Proof of Theorem B.
Note that [ Z ] is a horizontal ( k − ℓ, k − ℓ )-current on X Λ , so that I n := (cid:16) ( f n ) ∗ ( Z ) · c ( L ) ℓ +1 · π ∗ c ( M ) dim B − (cid:17) is I n = Z X Λ ( f n ) ∗ [ Z ] ∧ b ω ℓ +1 ∧ ( π ∗ ω B ) dim B − . Note that, as ω dim B +1 B = 0, we have k ( f n ) ∗ [ Z ] ∧ ( π ∗ ω B ) dim B − k =( ℓ + 1) Z X Λ ( f n ) ∗ [ Z ] ∧ ( π ∗ ω B ) dim B ∧ b ω ℓ + Z X Λ ( f n ) ∗ [ Z ] ∧ ( π ∗ ω B ) dim B − ∧ b ω ℓ +1 , so that we have k ( f n ) ∗ [ Z ] ∧ ( π ∗ ω B ) dim B − k = I n + ( ℓ + 1) Z X Λ ( f n ) ∗ [ Z ] ∧ b ω ℓ ∧ ( π ∗ ω B ) dim B . As shown in the proof of Proposition 13, we have Z X Λ ( f n ) ∗ [ Z ] ∧ b ω ℓ ∧ ( π ∗ ω B ) dim B = O ( d ℓn ) , which gives 1 d n ( ℓ +1) h X,L (( f n ) ∗ ( Z )) = 1 d n ( ℓ +1) k ( f n ) ∗ [ Z ] ∧ ( π ∗ ω B ) dim B − k + O ( d − n ) . We use Proposition 13 to conclude. (cid:3)
HE GEOMETRIC DYNAMICAL NORTHCOTT AND BOGOMOLOV PROPERTIES 19
Remark.
This is a a more precise statement than [CGHX, Theorem B]. Let us recalltheir result: when A is an abelian variety, L is an ample and symmetric linebundle on A and f = [ n ] is the multiplication by an integer, then b h f is nothing but the N´eron-Tateheight function of the Abelian variety A , relative to L . Theorem B from [CGHX] can berephrased as follows: if Z has height 0, then the integral vanishes: Z A Λ b T ℓ +1[ n ] ∧ [ Z ] ∧ ( π ∗ ν ) dim B − = 0 , for any smooth closed positive (1 , ν which represents c ( N ), but it is not shownthat the converse holds true.Even though we don’t need it in the sequel, we can prove that a stronger statementholds true in full generality, compare with [CGHX, Corollary 3.3]. Corollary 15.
Under the hypothesis of Theorem B, b h f ( Z ) = 0 if and only if, for anyclosed positive (1 , -current ν on B with continuous potentials we have Z X Λ b T ℓ +1 f ∧ [ Z ] ∧ ( π ∗ ν ) dim B − = 0 . Proof.
The proof is actually easy. Theorem B implies b h f ( Z ) = 0 if and only if Z X Λ b T ℓ +1 f ∧ [ Z ] ∧ ( π ∗ ω B ) dim B − = 0 . (3)It is thus sufficient to prove that if equation (3) holds, then Z π − ( U ) b T ℓ +1 f ∧ [ Z ] ∧ ( π ∗ ν ) dim B − = 0(4)for any closed positive (1 , ν on B with continuous potentials and any an opensubset U of Λ which is relatively compact in Λ. Again, by continuity, it is sufficient toconsider the case where ν is smooth since, by Richberg’s theorem, any continuous pshfunction can be locally approximated by smooth psh functions and we have the continuityof the wedge product in (4). Pick such a smooth current ν . As ω B is strictly positive on U , there exists C > Cω B ≥ ν ≥ ≤ Z π − ( U ) b T ℓ +1 f ∧ [ Z ] ∧ ( π ∗ ν ) dim B − ≤ Z X Λ b T ℓ +1 f ∧ [ Z ] ∧ ( π ∗ ω B ) dim B − = 0 . Taking the supremum over all U ⋐ Λ ends the proof. (cid:3) Stability of algebraic dynamical pairs
Several characterizations of stability.Definition 7. A p -dynamical pair (( X , f , L ) , [ Z ]) is algebraic if ( † ) ( X , f , L ) is an algebraic family of polarized endomorphisms of relative dimension k . ( † ) Z is an irreducible algebraic subvariety of X of codimension p ≤ k which is flatover the regular part Λ of ( X , f , L ) .We say that the p -dynamical pair (( X , f , L ) , [ Z ]) is stable if ( X Λ , f , L , [ Z ]) is stable. Example. (1) Let B be a normal projective variety, and Λ be a Zariski open set of B . If f : ( z, λ ) ∈ P k × Λ ( f λ ( z ) , λ ) ∈ P k × Λis an algebraic family of degree d endomorphisms of P k , then ( P k × B , f , O P k (1))is an algebraic family of polarized endomorphisms. If in addition a is a markedpoint, i.e. a morphism a : Λ → P k , the pair ( P k × B , f , O P k (1) , [Γ a ]), where Γ a isthe graph of a , is an algebraic k -dynamical pair.Notice that up to taking a finite branched cover of B , any k -dynamical pair ofthe form ( P k × B , f , O P k (1) , [ Z ]) can be decomposed as a finite collection of pairsgiven by graphs. Nevertheless, it is convenient in what follows to allow Z not tobe a graph.(2) Let ( X , f , L ) be an algebraic family of polarized endomorphisms. SetPer f ( n, m ) := { z ∈ X : f n ( z ) = f m ( z ) } for any n > m ≥
0. Lemma 7 in particular implies that the set Per f ( n, m ) definesa subvariety of X which has pure dimension dim B and which is flat over Λ. As aconsequence, the current [Per f ( n, m )] is a closed positive horizontal ( k, k )-currenton X Λ . As an immediate application of Theorem B, we have the following, seealso Corollary 3. Corollary 16.
For any irreducible component C of Per f ( n, m ) , the k -dynamicalpair (( X , f , L ) , [ C ]) is stable. Recall that, when Z is a subvariety of X of dimension ℓ + dim B , the degree of Z relatively to the ample linebundle M = L ⊗ π ∗ ( N ) (where N is a very ample linebundleon B , b ω is cohomologous to c ( L ) and ω B is cohomologous to c ( N )) is given bydeg M ( Z ) = (cid:16) Z · M ℓ +dim B (cid:17) = Z Z Λ ( b ω + π ∗ ω B ) ℓ +dim B = k [ Z ] k . We are now in position to prove the following, which is a more general version of item(1) of Theorem A.
Theorem 17.
Let (( X , f , L ) , [ Z ]) be an algebraic dynamical pair, which is a model of ( X, f, L, Z ) over the field K of rational functions of a normal complex projective variety B . Let ℓ := dim Z and let Z n be the Zariski closure of f n ( Z Λ ) in X .Then the following are equivalent: (1) (( X , f , L ) , [ Z ]) is stable, (2) T f , [ Z ] = 0 as a closed positive (1 , -current on Λ , (3) there exists C > such that for all n ≥ , C − d nℓ deg( f n | Z ) ≤ deg M ( Z n ) ≤ C d nℓ deg( f n | Z ) , (4) b h f ( Z ) = 0 .Proof. The equivalence between points 1 and 2 is the content of Proposition 10 and theequivalence between 2 and 4 is an immediate consequence of Theorem B. We thus justneed to prove the equivalence between 2 and 3. We now recall that, as algebraic cycles,we have ( f nλ ) ∗ Z λ = deg( f nλ | Z λ ) · f nλ ( Z λ ) = deg( f n | Z ) · f nλ ( Z λ ) . HE GEOMETRIC DYNAMICAL NORTHCOTT AND BOGOMOLOV PROPERTIES 21
We infer that, as Z → Λ is flat, as currents on X Λ , we also have( f n ) ∗ [ Z ] = deg( f n | Z ) · [ f n ( Z )] . Set α n := d nℓ / deg( f n | Z ). Recall also thatdeg M ( Z n ) = 1deg( f n | Z ) Z X Λ ( f n ) ∗ [ Z ] ∧ ( b ω + π ∗ ω B ) dim B + ℓ = 1deg( f n | Z ) dim B X j =0 (cid:18) dim B + ℓj + ℓ (cid:19) Z X Λ ( f n ) ∗ [ Z ] ∧ b ω j + ℓ ∧ ( π ∗ ω B ) dim B − j ≥ f n | Z ) Z X Λ ( f n ) ∗ [ Z ] ∧ b ω ℓ +1 ∧ ( π ∗ ω B ) dim B − . If T f , [ Z ] is non-zero, we use the second point of Proposition 13. The above impliesdeg M ( f n ( Z )) ≥ d n α n Z X Λ b T ℓ +1 f ∧ [ Z ] ∧ ( π ∗ ω B ) dim B − + O ( α n ) , so there is no C > M ( f n ( Z )) ≤ Cα n . Hence 3 implies 2.We finally assume T f , [ Z ] = 0 and we want to prove deg M ( Z n ) = O ( α n ). As b T ℓ +1 f ∧ [ Z ] =0, applying point 1 of Proposition 13 to S = b T ℓ f ∧ [ Z ] ∧ ( π ∗ ω B ) τ (for any τ ), and animmediate induction, for all 0 ≤ s ≤ ℓ + 1, we have k ( f n ) ∗ ( b ω ℓ +1 − s ) ∧ b T s f ∧ [ Z ] ∧ ( π ∗ ω B ) τ k = O ( d n ( ℓ − s ) ) . Again point 1 of Proposition 13 and an easy induction, we get for all 1 ≤ j ≤ dim B , k [ Z ] ∧ ( f n ) ∗ ( b ω ℓ + j − s ) ∧ b T s f ∧ ( π ∗ ω B ) dim B − j k = O ( d n ( ℓ − s ) ) . By point 2, for any 1 ≤ j ≤ dim B , since b T ℓ +1 f ∧ [ Z ] = 0, the quantity Z X Λ ( f n ) ∗ [ Z ] ∧ b ω j + ℓ ∧ ( π ∗ ω B ) dim B − j is thus bounded above by C X q ≤ ℓ d nq k b T q f ∧ ( f n ) ∗ ( b ω j + ℓ − − q ) ∧ [ Z ] ∧ ( π ∗ ω B ) dim B − j k≤ C X q ≤ ℓ d nq · d n ( ℓ − q − ≤ C d nℓ , for some constants C , C , C > n and j . This summarizes as0 ≤ Z X Λ ( f n ) ∗ [ Z ] ∧ b ω j + ℓ ∧ ( π ∗ ω B ) dim B − j ≤ C d nℓ , for some constant C > n . Finally, for j = 0, we have Z X Λ ( f n ) ∗ [ Z ] ∧ b ω ℓ ∧ ( π ∗ ω B ) dim B = Z X Λ [ Z ] ∧ ( f n ) ∗ ( b ω ℓ ) ∧ ( π ∗ ω B ) dim B ≤ C d nℓ k [ Z k , for some C >
0, since π ◦ f n = π . To see that this ends the proof, it is sufficient to remarkthat we always have Z X Λ b T ℓ f ∧ [ Z ] ∧ ( π ∗ ω B ) dim B > . Indeed, the slice along X λ of b T ℓ f ∧ [ Z ] is T ℓf λ ∧ [ Z λ ] which has mass deg( Z λ ) by B´ezout,and since it has continuous potentials, the global mass of R X Λ b T ℓ f ∧ [ Z ] ∧ ( π ∗ ω B ) dim B hasto be positive. (cid:3) Remark. (1) When Z is a point, we have ℓ = dim Z = 0, and obviously deg( f | Z ) = 1,so that item 3 means deg M ( Z n ) = O (1), as claimed in the introduction.(2) When dim Z >
0, the degree deg( f n | Z ) could as well grow as d ℓn or remain con-stant equal to 1. However, we expect to have deg M ( Z n ) = O (1), when the pair(( X , f , L ) , [ Z ]) is stable unless we have some isotriviality. For example, considera family of polarized endomorphism of P k given by polynomial maps of C k , thenthe hyperplane at infinity H ∞ defines an irreducible algebraic subvariety of X ofcodimension 1, it is periodic hence its height is zero and deg( f n | H ∞ ) = d n ( k − .4.2. A criterion for unstability: similarity.Definition 8.
Let ( Y, g, L ) be a complex polarized endomorphism with dim( Y ) = k . Wesay that a periodic point z of g is ( J, p )-repelling if it is J -repelling and for any germ Z of analytic set of dimension p and any small enough neighborhood U of z , we have that R U T pg ∧ [ Z ] > . Remark.
By definition, a J -repelling point z is ( J, k )-repelling and by non-normality of( g n ) on any analytic curve Z passing through z , it is also ( J, J -repelling point is ( J, p )-repelling for all p .The following is an adaptation of Tan Lei’s similarity between the Julia set and theMandelbrot set at a Misiurewicz parameter [T] and follows the idea of the proof of [AGMV,Theorem B]. Lemma 18.
Let (( X , f , L ) , [ Z ]) be an algebraic p -dynamical pair with regular part Λ .Pick λ ∈ Λ . Assume there exists a ( J, p ) -repelling periodic point z ∈ ( X λ ) reg ∩ Z for f λ . Let z : λ ∈ (Λ , λ ) z ( λ ) ∈ ( X , z ) be the local analytic continuation of z as a J -repelling periodic point of f λ . Then (1) either λ ∈ supp( T f , [ Z ] ) , in particular (( X , f , L ) , [ Z ]) is not stable, (2) or there is a neighborhood U of λ in Λ such that the graph Γ z satisfies Γ z ⊂ Z ∩ π − ( U ) .Proof. Up to replacing f by an iterate, we can assume that z is fixed. We assume Γ z Z and we want to show T f , [ Z ] = 0. First, as T f , [ Z ] is a closed positive current on Λ withcontinuous potential, we can reduce to the case when dim B = 1. Indeed, for any curve C , supp (cid:0) T f , [ Z ] ∧ [ C ] (cid:1) ⊂ supp( T f , [ Z ] ).We thus assume that dim B = 1, and that there exists an open neighborhood U of λ in B such that the intersection Z ∩ Γ z ∩ π − ( U ) is finite. We want to prove (( X , f , L ) , [ Z ]) isnot stable. We assume by contradiction it is stable. As z ( λ ) is repelling for all λ ∈ B ( λ , ε ),up to reducing ε , we can assume there exist K > δ > d X λ ( f λ ( x ) , f λ ( z ( λ ))) ≥ Kd X λ ( x, z ( λ )) , for all λ ∈ B ( λ , ε ) and all z ∈ B X λ ( z ( λ ) , δ ) ⊂ X λ . In particular, there exists a neighbor-hood Ω of z in X such that f : Ω → f (Ω) ⊃ Ω is proper. In addition, if λ ∈ π (Ω), we HE GEOMETRIC DYNAMICAL NORTHCOTT AND BOGOMOLOV PROPERTIES 23 have Ω ∩ X λ ⋐ f (Ω ∩ X λ ). Then d n ( k +1 − p ) Z Ω b T k +1 − p f ∧ [ Z ] = Z Ω ( f n ) ∗ b T k +1 − p f ∧ [ Z ] ≥ Z Ω b T k +1 − p f ∧ ( f n ) ∗ [ Z ] , and the above implies lim inf n →∞ Ω ( f n ) ∗ [ Z ] ≥ α [ Z ] in the weak sense of currents, forsome α > Z of Ω of codimension p .Now, since z is ( J, p )-repelling, we have R Ω b T k +1 − p f ∧ α [ Z ] >
0. This contradicts thefact that (( X , f , L ) , [ Z ]) is stable as R Ω ( f n ) ∗ b T k +1 − p f ∧ [ Z ] > n large enough. (cid:3) The Geometric Dynamical Northcott property.
Here we prove the main resultof the paper:
Theorem 19.
Let ( X , f , L ) be a non-isotrivial algebraic family of polarized endomor-phisms, with regular part Λ . Fix an integer D ≥ .There exist a subvariety Y D of X which is flat over Λ and N ≥ such that (1) f ( Y D ) = Y D and for any periodic irreducible component V of Y D with f n ( V ) = V ,the family of polarized endomorphisms ( V , f n | V , L | V ) is isotrivial, (2) for any irreducible subvariety C ⊂ X of dimension dim B with deg M ( f n ( C )) ≤ D for all n ≥ which is flat over Λ , then Y D contains f N ( C ) , Remark. (1) In the case where any irreducible component of Y D has dimensiondim B , the set Y D itself is in fact a (possibly reducible) preperiodic subvarietyunder iteration of f , i.e. there exist n > m ≥ f n ( x ) = f m ( x ), for all x ∈ Y D .(2) This theorem is both more precise than Theorem A, since its deals with subvarieties of X which are not necessarily the images of rational sections, and less precise thanTheorem A since the subvariety Y D depends a prior on D . In fact we will see inthe next paragraph that we can make Y D independent of D .(3) Of course, Y D can be empty for small D but Corollary 16 guarantees it is notempty for D large enough. Proof.
Define Z D as the Zariski closure of the set of irreducible subvarieties of X whichhave dimension dim B , are flat over Λ, and such thatdeg M ( f n ( V )) ≤ D, for all n ≥ . Remark that, by construction, Z D is a projective variety, that a general V ∈ Z D isirreducible and flat over Λ, and that for any V ∈ Z D , we have f ( V ) ∈ Z D . In particular,there is an irreducible component of Z D which is periodic under iteration of f . Let Z D be a periodic irreducible component of Z D of highest dimension. As we have proved inLemma 12 that ( X , f , L ) is isotrivial if and only if ( X , f n , L ) is for some n ≥
1, we mayassume f ( Z D ) = Z D .We now define b Z D := { ( V , x ) ∈ Z D × X : x ∈ V } . Assume, by contradiction, that b Z D has dimension ≥ dim X = k + dim B , so that thecanonical projection Π : ( V , x ) x is dominant, whence Π is surjective. If we set b Z Λ D := Π − ( X Λ ), we have Lemma 20.
The map Π | b Z Λ D : b Z Λ D → X Λ is an isomorphism. We take the lemma for granted. Let p : b Z Λ D → Z D be the projection given by p ( V , z ) = V . By Lemma 20 above, we have dim Z D = k and the mapΨ := p ◦ (cid:16) Π | b Z Λ D (cid:17) − : X Λ −→ Z D is regular and its fibers are of the form V Λ for some V ∈ Z D . Up to normalizing Z D , wemay assume it is normal. The action induced by f defines an endomorphism : Z D → Z D which makes the following diagram commute: X Λ f / / Ψ (cid:15) (cid:15) X ΛΨ (cid:15) (cid:15) Z D g / / Z D Note that, up to making a base change by C → B for any C ∈ Z D , we can assume thereis C ∈ Z D with ( C · X λ ) = 1, where X λ is a general fiber π − { λ } = X λ of π : X → B .As all fibers of π (resp. all varieties in Z D ) are cohomologous and as the intersection canbe computed in cohomology, we deduce that( V · X λ ) = 1for all λ ∈ Λ and all V ∈ Z D . Let now ψ λ := Ψ | X λ : X λ → Z D . As for any point z ∈ X λ ,( z, λ ) is contained in one and only one variety V ∈ Z , ψ λ is a morphism. Finally, sincethe topological degree of ψ λ is exactly ( V · X λ ) for a general variety V ∈ Z D , we get that ψ λ is an isomorphism which conjugates f λ to g .We have proved that for any λ ∈ Λ, the morphism f λ is conjugated by an isomorphism ψ λ : X λ → Z D to the endomorphism g . In particular, for any λ, λ ′ ∈ Λ, the map ψ − λ ′ ◦ ψ λ : X λ → X λ ′ is an isomorphism which conjugates f λ to f λ ′ . More precisely, forany λ ∈ Λ and any small ball B ⊂ Λ centered at λ , we have in fact built an analytic map φ : X λ × B → π − ( B ) such that the above isomorphism X λ → X λ is in fact φ λ := φ ( · , λ ).In particular, φ ∗ λ L λ depends continuously on λ , whence its class in the Picard groupPic( X λ ) also. As the Picard group of a complex projective variety is discrete and B isconnected, it must be constant. We thus have proved ( X, f, L ) is isotrivial. This is acontradiction.As a consequence, the image Y of Π b Z D : b Z D → X is a strict subvariety of X whichis invariant by f and flat over Λ. Moreover, we have f ( Y ) = Y and, if ( Y , f | Y , L | Y )is non-isotrivial, up to taking the normalization n : b Y → Y of Y , we can assume Y isnormal and ( Y , f | Y , L | Y ) is polarized, since n is a finite morphism. We thus end up witha contradiction exactly as above. In particular, ( Y , f | Y , L | Y ) is isotrivial.Finally, if W , . . . , W ℓ are all periodic irreducible components of Z D , we let Y D be theunion of the images of b W i under the natural projections Π b W i : b W i → X defined as above.By construction of the varieties W i , there exists an integer N ≥ C of X of dimension dim B which is flat over Λ and with sup n deg M ( f n ( C )) ≤ D satisfies f N ( C ) ⊂ Y D . (cid:3) We now prove Lemma 20.
Proof of Lemma 20.
Pick λ ∈ Λ and let z ∈ ( X λ ) reg be such that z is a J -repellingperiodic point for f λ and let p ≥ HE GEOMETRIC DYNAMICAL NORTHCOTT AND BOGOMOLOV PROPERTIES 25 there exist a neighborhood U ⊂ Λ of λ and a local section z : U → X of π : X → B suchthat f pλ ( z ( λ )) = z ( λ ) , λ ∈ U, and z ( λ ) ∈ ( X λ ) reg is J -repelling for f λ for all λ ∈ U . By assumption, there exists V ∈ b Z D such that ( z , λ ) ∈ V . In this case, Lemma 18 states that V ∩ π − ( U ) = z ( U ). Thisimplies there exists a Zariski dense subset S ⊂ X Λ such that Π − { x } is a singleton forall x ∈ S . Whence there exist dense Zariski open sets U ⊂ b Z Λ D and W ⊂ X Λ such thatΠ | U : U −→ W is a biholomorphism. Up to replacing Λ with Λ \ F where F is a subvariety,we can assume W = π − (Λ) \ ( X ′ ∪ X ′′ ), where X ′ and X ′′ are strict subvariety of X , flatover Λ, which are the respective Zariski closures of { x ∈ X Λ ; 1 < Card(Π − { x } ) < + ∞} and { x ∈ X Λ ; Π − { x } is infinite } .We now want to prove X ′′ is empty. First, we note that f − ( X ′′ Λ ) = X ′′ Λ = f ( X ′′ Λ ). Thisfollows easily from the fact that f is a morphism on X Λ , which lifts as a morphism on Z D :if x ∈ X Λ \ X ′′ Λ , then any curve V ∈ Z D passing through x has a unique image under f and finitely many preimages, whence f − ( X Λ \ X ′′ Λ ) = X Λ \ X ′′ Λ = f ( X Λ \ X ′′ Λ ) , and thus f − ( X ′′ Λ ) = X ′′ Λ = f ( X ′′ Λ ). We thus may apply Lemma 18 as above. The set X ′′ Λ is at worst a subvariety of dimension dim B which is flat over Λ and totally invariant by f . But it is supposed to contain infinitely many distinct such subvarieties of Z D whichpass by points in X ′′ Λ . This is impossible, whence X ′′ = ∅ .We thus have proved so far that the map Π | b Z Λ D : b Z Λ D → X Λ is a finite birationalmorphism. Since X Λ is normal, this implies Π | b Z Λ D is an isomorphism, as required. (cid:3) Proof of the second part of Theorem A.
We now can deduce the second partTheorem A from Theorem 19. Let us rephrase the second part of Theorem A.
Theorem 21.
Let ( X , f , L ) be a non-isotrivial family of polarized endomorphisms, where π : X → B is a family of complex normal varieties. There exist a strict subvariety Y of X which is flat over B and integers N ≥ and D ≥ such that (1) f ( Y ) = Y and for any periodic irreducible component V of Y with f n ( V ) = V , thefamily of polarized endomorphisms ( V , f n | V , L | V ) is isotrivial, (2) Pick any rational section a : B X of π such that (( X , f , L ) , a ) is stable. Then (a) if a N is defined by a N ( λ ) := f Nλ ( a ( λ )) , then a N is a section of π | Y : Y → B , (b) if C a is the Zariski closure of a (Λ) , we have deg M ( C a ) ≤ D .Proof. By Theorem 19, for any integer D , there exist a subvariety Y D of X which is flatover Λ and an integer N D ≥ f ( Y D ) = Y D and for any periodic irreducible component V of Y D with f n ( V ) = V ,the family of polarized endomorphisms ( V , f n | V , L | V , ) is isotrivial,(2) Y D contains f N D ( C ), for any irreducible subvariety C ⊂ X of dimension dim B which is flat over Λ and with deg M ( f n ( C )) ≤ D for all n ≥
1. In particular, forany rational section a : B X , if C a is the Zariski closure of a (Λ) in X , and ifdeg M ( f n ( C a )) ≤ D , then f N ( C a ) ⊂ Y D .Moreover, if D ≤ D ′ , then Y D ⊂ Y D ′ . All there is left to prove is the existence of D ≥ M ( C a ) > D , then (( X , f , L ) , a ) is not stable. Let X be the generic fiber of π , L := L | X and f := f | X , so that ( X , f , L ) is a modelfor ( X, f, L ) with regular part Λ. Let b h f := lim n →∞ d n h X,L ◦ f be the canonical height function of ( X, f, L ) as above. Any rational section correspondsto some a ∈ X K , where K = C ( B ) and h X,L ( a ) = (cid:16) C a · c ( L ) · c ( N ) dim B − (cid:17) = deg M ( C a ) − (cid:16) C a · c ( N ) dim B (cid:17) = deg M ( C a ) − , since C a is the Zariski closure of the image of a rational section of π . Indeed, the size ofthe field extension K ( a ) of K is exactly the intersection number ( C a · c ( N ) dim B ). Recallfrom Section 3.3 that h X,L = b h f + O (1), so there exists a constant C > b h f ≥ h X,L − C , whence b h f ( a ) ≥ deg M ( C a ) − ( C + 1) . If deg M ( C a ) > C +1, we thus find b h f ( a ) >
0. By Theorem 17, we deduce that (( X , f , L ) , a )is not stable, ending the proof. (cid:3) We now easily deduce Corollary 2 from Theorem 21.
Proof of Corollary 2.
Let (
X, f, L ) be a polarized endomorphism over K and let ( X , f , L )be a model of ( X, f, L ) over B . Applying Theorem 21 gives the conclusion, since X isisomorphic to the generic fiber of π : X → B , L = L | X and f = f | X . (cid:3) Example (The elementary Desboves family, continued I) . We consider again the familyof degree 4 endomorphisms of P defined by f λ ([ x : y : z ]) = [ − x ( x + 2 z ) : y ( z − x + λ ( x + y + z ) : z (2 x + z )] , λ ∈ C ∗ , and we keep the notations of the example of Section 3.1. The critical set of f λ is the unionof three lines L j passing through ρ = [0 : 1 : 0] and [1 : 0 : α j ] with α = 1, for j = 0 , , C ′ λ := { [ x : y : z ] ∈ P : − x + z + λ (4 y + x + z ) = 0 } . Note that the lines L j correspond to preimages of critical points of the Latt`es map g := f λ | Y by the fibration p : P Y which semi-conjugates f λ to g and whose fibers are thelines of the pencil P .We now make a base change for our family : Let K ⊃ C ( λ ) be the splitting field ofthe equation w = (1 + λ ) / (1 − λ ). Then K = C ( B ) for some smooth complex projectivecurve B and there exists a finite branched cover τ : B → P of degree [ K : C ( λ )] andchanging base by τ allows to define the three intersection points of Y = { y = 0 } with C ′ λ to define marked points a , a , a : B → P where a i ( λ ) = [ x i ( λ ) : 0 : z i ( λ )] with(1 + λ ) z i ( λ ) + (1 − λ ) x i ( λ ) = 0. We have defined a family ( P × B , f B , O P (1)) withregular part Λ := τ − ( C ∗ ). Let Y ⊂ P × B be the subvariety such that f B ( Y ) = Y givenby Theorem 21. As seen in Section 3.1, Y × B ⊂ Y and C × B ⊂ Y , but X × B Y and Z × B Y . The subvariety Y may have other irreducible components,which would have to have period at least 2 under iteration of f B . HE GEOMETRIC DYNAMICAL NORTHCOTT AND BOGOMOLOV PROPERTIES 27
Any constant marked point α ∈ Y ∪ C is stable (but can have infinite orbit). Note alsothat a i ( λ ) ∈ Y λ for any λ ∈ C ∗ . However a i is not stable. Indeed, one has b h f ( a i ) = Z P × Λ [Γ a i ] ∧ b T f B = Z B ˜ a i ∗ ( µ g ) = deg( ˜ a i ) > , where ˜ a i ( λ ) = [ x i ( λ ) : z i ( λ )] for all λ ∈ C ∗ and µ g is the Green measure of g : Y → Y which gives no mass to proper analytic sets. This example emphasizes that the conclusionthat Γ a i ⊂ Y is not sufficient to have stability.4.5. Applications.
We briefly give applications of Theorems A and B to the arithmeticof dynamical systems defined over function fields K = k ( B ) where k has characteristic 0and let B be a normal projective k -variety.4.5.1. The case of Abelian varieties. If A is an abelian variety defined over K , L a sym-metric ample line bundle on A and if [ n ] the morphism of multiplication by the integer n on A , the N´eron-Tate height function b h A : A ( ¯ K ) → R of A is defined by b h A = lim n →∞ n h A,L ◦ [ n ] . In particular, it satisfies b h A ◦ [ n ] = n b h A for any n ≥
2. Applying Corollary 2 to the polar-ized endomorphism ( A, [ n ] , L ) directly gives the next result due to Lang and N´eron [LN]: Theorem 22 (Lang-N´eron) . Let A be a non-isotrivial abelian variety defined over afunction field of characteristic zero K , L a symmetric ample line bundle on A and let τ : A → A be the K / k -Trace of A . Then, there exists an integer m such that for any z ∈ A ( K ) with ˆ h A ( z ) = 0 , we have [ m ] z ∈ τ ( A ( K )) .Proof. As before, we can assume k = C without loss of generality. We apply Corollary 2to the polarized endomorphism ( A, [2] , L ). We have a possibly reducible subvariety B such that [2] B = B such that, if B , . . . , B M are the periodic irreducible components of B , i.e. such that there exist integers n j > n j ] B j = B j , for all 1 ≤ j ≤ M , thenall the subvarieties B j are thus translate by a torsion point p j of an abelian subvariety A j of A , which is isotrivial.Note that, since A j is isotrivial, there exist an abelian variety C j defined over C and amorphism τ j : C j → A such that A j = τ j ( C j ). In particular, A j ⊂ τ ( A ), which gives thesought statement. (cid:3) This has applications to the case of Latt`es maps. Recall that an endomorphism g : P N → P N is a Latt`es map if there exist an Abelian variety A of dimension N , an isogeny ℓ : A → A and a finite branched cover Θ : A → P N such that Θ ◦ ℓ = g ◦ Θ. Combining [AL,Theorem 1.1] and [C, Theorem 2.3 and the paragraph before], we can lift a non-isotrivialLatt`es map to an abelian variety whose K / k -Trace is reduced to 0, so that Theorem 21and Theorem 22 imply Corollary 23 (Northcott’s property for Latt`es maps) . Let f : P N → P N be a non-isotrivial Latt`es map of degree d ≥ over the function field K of a projective variety ovea field of characteristic zero . Then, for any z ∈ P N ( K ) , b h f ( z ) = 0 if and only if z ispreperiodic.Proof. As previously, we may assume k = C . Let π : A → B be a non-isotrivial familyof abelian varieties of relative dimension N ≥ L a symmetric relatively ample line-bundle on A . Let A be the generic fiber of π and L | A be a symmetric ample line bundle. Let [ n ] : A A be the rational map such that ( A , [ n ] , L ) is a model for ( A, [ n ] , L ) withregular part Λ.The family ( A , [ n ] , L ) induces a non-isotrivial family of Latt`es maps if there exists afinite group G acting on A such that, for all λ ∈ Λ, • the action of G fixes the origin of A λ and • the quotient A λ /G is isomorphic to P N , • the rational map f λ : P N → P N induced by [ n ] is an endomorphism (of degree d = n ).Following [AL, Theorem 1.1], this implies A λ is isogenous to the self power E N of acomplex elliptic curve E and moreover, • either G is isomorphic to C N ⋊ S N for some cyclic group C of automorphisms of E fixing the origin and the action of C N is coordinate-wise and that of S N permutescoordinates, • or G is isomorphic to S N +1 and acts by permutation on A ≃ { ( x , . . . , x N +1 ) ∈ E N +1 ; x + · · · + x N +1 = 0 } . In any cases, this implies A is isogenous to the fiber power E N B of a non-isotrivial familyof elliptic curves π E : E → B . We thus only need to consider the case when A = E N B .Let K := C ( B ) be the field of rational functions on B and let E be the generic fiberof π E . According to [C, Theorem 2.3 and the paragraph before], since E is non-isotrivial,the K / C -Trace of E is reduced to 0, whence the K / C -Trace of A is also trivial. We thusmay apply directly Theorem 21 and Theorem 22 to conclude.Indeed, if a is a rational map a : B P N , since B is a smooth curve and P N isprojective, it extends as a morphism a : B → P N and it lifts as a rational multisection of π : E N → B . Up to making a base change B ′ → B , we can assume it is a rational section,whence a morphism, b : B → E N . Theorem 21 gives the existence of Y ⊂ A which isinvariant by [ n ] such that, by Theorem 22, any irreducible component of Y is mappedunder iteration of [ n ] to A , the Zariski closure in E N of the K / C -Trace A of E N . Bythe discussion above, this implies Y is a curve of E N which is flat over Λ. In particular, b ( B ) is preperiodic under iteration of [ n ], which is equivalent to saying that the graph Γ a of a is preperiodic under iteration of f , which ends the proof of Corollary 23. (cid:3) A conjecture by Kawaguchi and Silverman.
Finally, we remark here that Theorem Aimplies a conjecture of Kawaguchi and Silverman [KS2, Conjecture 6] in the case of po-larized endomorphisms over the field K of rational functions of a projective variety over afield of characteristic zero.Let X be a normal projective variety defined over a global K of characteristic zero (i.e. K is either a number field or a function field as above) and f : X X a dominantrational map. Definition 9.
For ≤ j ≤ dim X , the j - dynamical degree λ j ( f ) of f is λ j ( f ) := lim n → + ∞ (cid:16) ( f n ) ∗ H j · H dim X − j (cid:17) /n , where H is any big and nef Cartier divisor on X . Let X f ( ¯ K ) be the set of points whose full forward orbit is well-defined. In this case,one can also define the arithmetic degree of a point P ∈ X f ( ¯ K ) as α f ( P ) := lim n → + ∞ max ( h X ( f n ( P )) , /n , HE GEOMETRIC DYNAMICAL NORTHCOTT AND BOGOMOLOV PROPERTIES 29 where h X is any given Weil height function h X : X ( ¯ K ) → R , when the limit exists.Kawaguchi and Silverman’s conjecture [KS2, Conjecture 6] says that:(1) the limit α f ( P ) exists and is an algebraic integer for all P ∈ X f ( ¯ K ),(2) the set { α f ( Q ) : Q ∈ X f ( ¯ K ) } is finite,(3) if P ∈ X f ( ¯ K ) has Zariski dense orbit, then α f ( P ) = λ ( f ).This conjecture is known, over number fields, in several cases including the case of polarizedendomorphisms [KS2, Theorem 5], see also e.g. [KS1, LS, MSS, S, Mat].As a consequence of Theorem 19, we deduce this conjecture holds true for polarizedendomorphisms over function fields of characteristic zero. Theorem 24.
Let k be a field of characteristic zero and B be a normal projective k -variety.Let ( X, f, L ) be a non-isotrivial polarized endomorphism over K := k ( B ) of degree d . Fixa point P ∈ X ( ¯ K ) . Then (1) the limit α f ( P ) exists and is an integer, (2) the set { α f ( Q ) : Q ∈ X ( ¯ K ) } coincides with { , d } , (3) if P has Zariski dense orbit, we have α f ( P ) = λ ( f ) = d .Proof. Let us first remark that, since (
X, f, L ) is polarized, we have f ∗ L ≃ L ⊗ d , so that (cid:16) ( f n ) ∗ L · L dim X − (cid:17) = (cid:16) L ⊗ d n · L dim X − (cid:17) = d n (cid:16) L dim X (cid:17) , which gives λ ( f ) = lim n → + ∞ d (cid:0) L dim X (cid:1) /n = d .We now take x ∈ X ( ¯ K ). Let b h f : X ( ¯ K ) → R + be the canonical height function of f .Assume b h f ( x ) >
0. Then h X,L ( f n ( x )) = b h f ( f n ( x )) + O (1) = d n b h f ( x ) + O (1) , so that α f ( x ) = d , as sought. Note also that if b h f ( x ) = 0, this implies h X,L ( f n ( x )) = O (1) , so that α f ( x ) = 1. We are thus left with proving that, if x has a Zariski dense orbit, then b h f ( x ) >
0. Assume x has Zariski dense orbit, but b h f ( x ) = 0. Let K ′ be the field of rationalfunctions of some normal projective variety B ′ such that x is defined over K ′ and let( X , f , L ) be a non-isotrivial model of ( X, f, L ) such that π : X → B ′ is a family of normalvarieties, with regular part Λ. To x , we can associate a rational section x : B ′ X anda subvariety C ⊂ X (which is the Zariski closure of x (Λ) in X and which is flat over Λ).According to Theorem 17, we havesup n deg M ( f n ( C )) = D < + ∞ . We now can apply Theorem 19, and we deduce the existence of Y D such that Z := S n ≥ f n ( Y D ) is a strict subvariety of X and f n ( C ) ⊂ Z for all n ≥ N . If Z is the genericfiber of Z → B ′ , we have f n ( x ) ∈ Z for any n ≥
1, which is a contradiction. (cid:3)
We refer the reader to [DGHLS] for generalization of the above conjecture where TheoremB provides partial answers.5.
The Geometric Dynamical Bogomolov Conjecture
In the whole section, we let k be a field of characteristic 0, we also let B be a normalprojective k -variety and we let K := k ( B ) be the field of rational functions of B . Motivation of the Conjecture.
As mentioned in the introduction, the GeometricBogomolov conjecture has been proved recently in [CGHX]. Let us first state their result.
Theorem 25 ([CGHX]) . Let A be a non-isotrivial abelian variety, L be a symmetricample linebundle on A and b h A be the N´eron-Tate height of A relative to L . Let Z be anirreducible subvariety of A ¯ K . Assume for all ε > , the set Z ε := { x ∈ Z ( ¯ K ) : b h A ( x ) < ε } is Zariski dense in Z . Then, there exist a torsion point a ∈ A ( ¯ K ) , an abelian subvariety C ⊂ A and a subvariety W ⊂ A of the ¯ K / k -trace ( A , τ ) of A such that Z = a + C + τ ( W ⊗ k ¯ K ) . Let us now give a dynamical formulation of the conclusion of their result. Fix n ≥ n ] : A → A and ˜[ n ] : A → A be the respective multiplication by n morphisms.Let M ≥ b := [ n ] M ( a ) is periodic under iteration of [ n ], i.e. there exists k ≥ n ] k b = b , and let V := b + C + τ ( A ⊗ k ¯ K ) . The variety V is fixed by [ n ] k , i.e. [ n ] k V = V and, if V := τ ( A ⊗ k ¯ K ), there is afibration p : V → τ ( A ⊗ k ¯ K ) which is invariant by [ n ] k , i.e. such that the followingdiagram commutes: V [ n ] k / / p (cid:15) (cid:15) V p (cid:15) (cid:15) V n ] k | V / / V and such that ( V , [ n ] k | V , L | V ) is isotrivial. We thus may rephrase the conclusion asfollows: There exist subvarieties V, V ⊂ A invariant by [ n ] k and an integer N ≥ p : V → V such that(1) p ◦ ([ n ] k | V ) = ([ n ] k | V ) ◦ p and ( V , [ n ] | V , L | V ) is isotrivial,(2) [ n ] N ( Z ) = p − ( W ) where W is an isotrivial subvariety of V .In particular, Conjecture 1 in the Introduction is a generalization of the above dynamicalformulation of the Geometric Bogomolov Conjecture to the case where ( X, f, L ) is a non-isotrivial polarized endomorphism defined over K , X is normal and Z ⊂ X ¯ K is irreduciblesubvariety.Recall that, by [Fab, G1], b h f is induced by a M B -metric in the sense of Gubler [G2], and,that in this case, if Z ⊂ X ¯ K is an irreducible subvariety, we have the Zhang inequalitieswhich imply the following by [G2, Corollary 4.4]: • the height of Z is zero, i.e. b h f ( Z ) = 0, • the essential minimum of the metrization vanishes: e := sup Y inf x ∈ Z ( ¯ K ) \ Y b h f ( x ) = 0 , where Y ranges over all hypersurfaces of Z , • for any ε > Z ε is Zariski-dense in Z .In particular, it is sufficient to characterize subvarieties Z with b h f ( Z ) = 0. HE GEOMETRIC DYNAMICAL NORTHCOTT AND BOGOMOLOV PROPERTIES 31
Remark.
Note also that, using the same argument as in the proof of Corollary 23, wesee that the main Theorem of [CGHX] directly implies a strengthened version of theConjecture for Latt`es maps: if f : P k → P k is a Latt`es map defined over K and Z ⊂ P k isan irreducible subvariety defined over K , then b h f ( Z ) = 0 if and only if Z is preperiodic. b h f ( x ) = 0 if and only if b h f i ( x i ) = 0. Since we assumed f i non-isotrivial for all i , this isequivalent to the fact that x is preperiodic.5.2. Stable fibers of an invariant fibration.
We want here to explore basic propertiesof subvarieties with height 0. The first thing we want to do is to relate, when f preservesa fibration, the height of the fiber over y to the height of y .Let us fix the context before we give a more precise statement: we let K := C ( B ) bethe field of rational function of a complex normal projective variety B . When X and Y are projective varieties over K and p : X Y is a dominant rational map defined over K , for any irreducible subvariety W ⊂ Y ¯ K , we let X W be the “fiber” p − ( W ) of p over W . Lemma 26.
Let ( X, f, L ) be a polarized endomorphism over K of degree d > . Assumethere exist a polarized endomorphism ( Y, g, E ) over K of degree d with dim Y < dim X ,and a dominant rational map p : X Y defined over K with p ◦ f = g ◦ p . For anysubvariety W ⊂ Y ¯ K , we have b h f ( X W ) = b h g ( W ) . In particular, b h f ( X W ) = 0 if and only if b h g ( W ) = 0 .Proof. Let ( X , f , L ) and ( Y , g , E ) be respective models for ( X, f, L ) and (
Y, g, E ) withcommon regular part Λ, and let p : X Y be a model for p : X Y . For anysubvariety W ⊂ Y ¯ K , we let W be the Zariski closure of W in Y . Recall that we have set X W := P − ( W ). We let X W := p − ( W ) and ℓ := dim X − dim Y > s := dim W .We remark that b T s +1 g = p ∗ (cid:16) b T ℓ + s +1 f (cid:17) , so that the projection formula and Theorem B give b h g ( W ) = b h f ( X W ) , where we used that π X ◦ p = π Y . (cid:3) Consider a trivial family ( g ( z ) , λ ) on P × Λ and a marked point a : Λ → P . Then, it iseasy to see that the graph of a is stable if and only if a is constant (if not, a ( λ ) is a repellingperiodic point of g at some λ ). More generally, when p : X P and g : P → P isisotrivial, there exists an automorphism φ : P → P defined over ¯ K such that φ − ◦ g ◦ φ isdefined over C and we say that X z is an isotrivial fiber of p if X z = p − { z } where φ − ( z )is defined over C .As a particular case of Lemma 26, we have Corollary 27.
Let ( X, f, L ) be a non-isotrivial polarized endomorphism over K of degree d > . Assume there exist a polarized endomorphism ( P , g, O P (1)) over K of degree d ,and a dominant rational map p : X P defined over K with p ◦ f = g ◦ p . For any y ∈ P ( K ) , let Z y be the fiber p − { y } of p . (1) If ( P , g, O P (1)) is non-isotrivial, then b h f ( Z y ) = 0 if and only if Z y is preperiodicunder iteration of f , (2) if g is isotrivial, then b h f ( Z y ) = 0 if and only if Z y is an isotrivial fiber of p . Proof.
Assume first g is non-isotrivial. According to Lemma 26, we have b h g ( Z y ) = 0 ifand only if b h g ( y ) = 0. Since ( P , g, E ) is non-isotrivial, b h g ( y ) = 0 if and only if y is g -preperiodic. As f ( Z y ) = Z g ( y ) , the variety Z y has to be preperiodic.Assume now g is isotrivial. Then there exist a finite extension K ′ of K and an affineautomorphism φ : P → P , defined over K ′ such that g := φ − ◦ g ◦ φ is defined over C .Let ρ : B ′ → B be a finite branched cover with K ′ = C ( B ′ ) and let ( P × B ′ , g , O P (1))be a model for ( P , g, O P (1)) over B ′ . Let σ : B ′ P be the rational map induced by φ − ( y ) and Y be the Zariski closure of y in P × B ′ . If Φ : P × B ′ P × B ′ is a modelof φ and Λ is a common regular part for all above models, then b h g ( y ) = Z P × Λ [ Y ] ∧ b T g ∧ ( π ∗ B ′ ω B ′ ) dim B ′ − = Z P × Λ [ Y ] ∧ Φ ∗ (cid:0) π ∗ P ( µ g ) (cid:1) ∧ ( π ∗ B ′ ω B ′ ) dim B ′ − = Z P × Λ [Γ σ ] ∧ π ∗ P ( µ g ) ∧ ( π ∗ B ′ ω B ′ ) dim B ′ − = Z Λ σ ∗ ( µ g ) ∧ ω dim B ′ − B ′ = deg N ( σ ) . In particular, b h g ( y ) = 0 if and only if σ is constant, i.e. φ − ( y ) ∈ C . This concludes theproof. (cid:3) Some cases of the Conjecture on P We here show the Geometric Dynamical Bogomolov Conjecture holds for skew-productpolynomial maps of P and, under some restrictions on the growth of the degree of f n | Z ,for a general endomorphism of P . We first establish properness of the composition mapon the space of monic centered polynomials, as we will use this property to prove theconjecture for polynomial skew-products.In the whole section, B is a normal complex projective variety and K := C ( B ) is its fieldof rational functions. We also let N be a very ample linebundle on B , and ω B a K¨ahlerform on B which represents c ( N ), so that M := O P (1) ⊗ π ∗ ( N ) is ample on P × B , asbefore.6.1. The case with an invariant pencil of lines.
Let K be the field of rational func-tions of a normal complex projective variety B . Let ( P , f, O P (1)) be a polarized endo-morphism of degree d . We assume there exists a pencil of lines P on P which is invariantby f and which is trivial. This means there is a a dominant rational map p : P P such that all fibers of p are lines of P and an endomorphism ( P , g, O P (1)) of degree d ,which is defined over C , such that p ◦ f = g ◦ p .We want here to prove the following. Theorem 28.
Let ( P , f, O P (1)) be a non-isotrivial degree d endomorphism defined overthe field K of rational functions of a normal complex projective variety B . Assume f leaves invariant a trivial pencil of lines. Then for any irreducible curve Z ⊂ P definedover K with b h f ( Z ) = 0 , (1) either Z is preperiodic under iteration of f , (2) or Z is a line of the isotrivial fiber of the invariant pencil.Proof. Let p : P P be the invariant fibration and ( P , g, O P (1)) be such that p ◦ f = g ◦ p . Note that, if Z is a fiber of the projection p , then the result follows from Corollary 27.We thus assume that the projection p : P P restricts to Z as a dominant rationalfunction p : Z → P . HE GEOMETRIC DYNAMICAL NORTHCOTT AND BOGOMOLOV PROPERTIES 33
Fix a model ( P × B , f , O P (1)) of ( P , f, O P (1)) with regular part Λ. The family( P × B , g × id B , O P (1)) is a model of ( P , g, O P (1)) and there is a map p : P × B → P × B such that p ◦ f = ( g × id) ◦ p on P × Λ. Let Z be the Zariski closure of Z in P × B . Up toreducing Λ we can assume the projection p : Z Λ → P × Λ is a finite branched cover. Weclaim S n ≥ f n ( Z ) is a proper subvariety of P × Λ. Recall that b T f = b T f ∧ p ∗ ( b T g ) where b T g is the Green current for the map g × id so it can be written as b T g = π ∗ ( µ g ) where π : P × Λ → P is the projection on P and µ g is the Green measure of g . By hypothesisand by Fubini Theorem, we have0 = Z P × Λ b T f ∧ [ Z ] ∧ ( π ∗ ω B ) dim B − = Z x ∈ P Z p − ( { x }× Λ) b T f ∧ [ Z ] ∧ ( π ∗ ω B ) dim B − ! d µ g ( x )= Z x ∈ P (cid:18)Z P × Λ b T f ∧ [ Z x ] ∧ ( π ∗ ω B ) dim B − (cid:19) d µ g ( x ) , where Z x := p − ( { x } × Λ) ∩ Z . This implies Z P × Λ b T f ∧ [ Z x ] ∧ ( π ∗ ω B ) dim B − = 0for µ g -a.e. x ∈ P . We now may apply Theorems 17 and 21 to the Z x ’s: we have D x := sup n deg M ( f n ( Z x )) < + ∞ , for µ g -a.e. x , and D x = D is independent of x . In particular, there exist a subvariety Y ⊂ P × B which is periodic under iteration of f with dim Y = dim B + 1, and an integer N ≥ f N ( Z x ) ⊂ Y for µ g -almost every x . As µ g has continuous potential, thisimplies the set { ( z, λ ) ∈ f N ( Z ) : ( z, λ ) ∈ Y } is Zariski dense in f N ( Z ). Since dim Z = dim B + 1 = dim Y , we deduce that f N ( Z ) andthus also Z is preperiodic. (cid:3) Remark that this case covers the case of polynomial skew-products, as well as the caseod the elementary Desboves family.Recall that an endomorphism f : P → P is a polynomial skew-product over a field K of characteristic zero if f ( z, w ) = ( p ( z ) , q ( z, w )) , ( z, w ) ∈ A , where p ∈ K [ z ] is a degree d polynomial and q ∈ K [ z, w ] has degree d and deg w ( q ) = d and that, automatically, such a polynomial mapping extends as an endomorphism of P of degree d .We now can prove easily that the conjecture is true for polynomial skew-products. Theorem 29.
Let ( P , f, O P (1)) be a non-isotrivial degree d polynomial skew-productdefined over the field K of rational functions of a normal complex projective variety B andwrite f = ( p, q ) . Then (1) If p is isotrivial, for any irreducible curve Z ⊂ P defined over K with b h f ( Z ) = 0 , (a) either Z is preperiodic under iteration of f , (b) or Z is an isotrivial fiber of the projection onto the first coordinate. (2) The critical set
Crit( f ) of f is not small, i.e. b h f (Crit( f )) > . Let Poly mc d be the set of degree d monic and centered polynomials, i.e. polynomials ofthe form p ( z ) = z d + a z d − + · · · + a d , with a , . . . , a d in a fixed field of characteristic 0. Assume the field contains ( d − d − p is a degree d monic centered polynomial defined over the field K = C ( B ), then p is isotrivial if and only if p is actually defined over C .For the proof of Theorem 29, we need the following folklore lemma whose proof is leftto the reader: Lemma 30.
For any n ≥ and any d , . . . , d n ≥ , the composition map Φ d ,...,d n : ( f , . . . , f n ) ∈ Poly mc d × · · · × Poly mc d n f ◦ · · · ◦ f n ∈ Poly mc d ··· d n is well-defined and proper over C . We are now in position to prove Theorem 29:
Proof of Theorem 29.
To start the proof, we note that up to taking a finite branched cover B ′ → B , we can conjugate f over K ′ = C ( B ′ ) by an affine map of the form φ ( z, w ) =( α z + β , α z + γw + β ), with α i , β i , γ ∈ K ′ , to a polynomial skew product (˜ p, ˜ q ) where˜ p is monic centered and ˜ q ( z, w ) = w d + O z ( w d − ). We thus assume f is of this form. As p is isotrivial and monic centered, it is trivial. We thus can directly apply Theorem 28 toget the first point.The second point can be easily deduced from the first one as follows: assume b h f (Crit( f )) =0 and f is non-isotrivial. A simple computation shows that we have the decompositioninto two divisors Crit( f ) = π − (Crit( p )) ∪ Crit h ( f ) , where π | Crit h ( f ) : Crit h ( f ) → P is finite. By Fubini and b T f ∧ π ∗ ( b T p ) = b T f b h f (Crit( f )) = b h p (Crit( p )) + b h f (Crit h ( f )) . In particular, b h p (Crit( p )) = 0, so that by [Mc] the polynomial p is isotrivial, whence trivial p = p λ for all λ , since p is monic centered.We now apply the first point to the curve Crit h ( f ): it has to be preperiodic. We nowuse the properness of the composition maps. For any z ∈ C , let q z,n : P × Λ → P × Λ bethe family of degree d n monic centered polynomials defined by f nλ ( z, w ) = ( p n ( z ) , q z,n,λ ( w )) , ( z, w ) ∈ C . The polynomial q z,n,λ is defined by the composition q z,n,λ ( w ) = q λ ( p n − ( z ) , q z,n − ,λ ( w ))where q z, ,λ ( w ) = q λ ( z, w ).Let z ∈ C be a periodic point of p of period n ≥
1. Then q z ,n is a postcritically finitefamily of degree d n polynomials since Crit h ( f n ) ∩ π − ( { z } × Λ) = Crit( q z ,n ). As before,the family f n | { z }× Λ = q z ,n is stable, whence trivial. By properness of the compositionmap, q z , is trivial. So, the set A ′ := { z ∈ C : ∀ ( w, λ ) ∈ C × Λ , f λ ( z, w ) = f λ ( z, w ) } contains z , whence it is Zariski dense in C and Zariski closed, so that A ′ = C . HE GEOMETRIC DYNAMICAL NORTHCOTT AND BOGOMOLOV PROPERTIES 35
We thus have proved that f is trivial, a contradiction. In particular, ( P , f, O P (1)) isisotrivial. (cid:3) Example (The elementary Desboves family, continued II) . We now come back to theexample of the elementary Desboves family. Recall the Desboves family is given by f λ ([ x : y : z ]) = [ − x ( x + 2 z ) : y ( z − x + λ ( x + y + z ) : z (2 x + z )] , λ ∈ C ∗ . The above Theorem 28 directly implies that the Geometric Bogomolov Conjecture is truefor the elementary Desboves family.6.2.
Fast growth of the topological degree implies the Conjecture on P . Here,we prove the conjecture holds when deg M ( Z n ) is bounded. Let B be a complex projectivevariety. By Theorem 17, this is equivalent to the fact that deg( f n | Z ) ≥ Cd n for some C > Theorem 31.
Let ( P , f, O P (1)) be a non-isotrivial polarized endomorphism of degree d ≥ defined over K = C ( B ) . Let Z be an irreducible curve defined over K with b h f ( Z ) =0 . Assume in addition there exists C > such that deg( f n | Z ) ≥ C · d n for all n ≥ . Then (1) either Z is preperiodic under iteration of f , (2) or there exist an integer k ≥ , an isotrivial polarized endomorphism ( P , g, O P (1)) ,a dominant rational map p : X P defined over K with p ◦ f k = g ◦ p such that Z is an isotrivial fiber of p .Proof. Let ( P × B , f , O P (1) , Z ) be a model for ( P , f, O P (1) , Z ) with regular part Λ and N be a very ample linebundle on B and let M := O P (1) ⊗ N . For any n ≥
1, let Z n be the Zariski closure of f n ( Z Λ ). Our assumption implies deg M ( Z n ) = O (1). If Z Λ ispreperiodic under iteration of f , there is nothing to prove. We thus assume the collection { Z n , n ≥ } is infinite.Let A be the Zariski closure of { Z n , n ≥ } in the Chow variety Ch dim B +1 ,D ( P × B ),where D := sup n deg M ( Z n ) < + ∞ . By construction, the variety A is invariant bythe map f which sends Z to the Zariski closure of f ( Z Λ ) so there exists an irreduciblecomponent A of A of largest dimension which is invariant by f n for some n ≥
1. Up toreplacing f with an iterate, we can assume f ( A ) = A . By construction, the endomorphism( A , f , L ) is a polarized endomorphism of degree d of the complex projective variety A ,by e.g. [Z], where L is the restriction to A of the ample linebundle on the Chow varietyCh dim B +1 ,D ( P × B ) induced by the natural embedding of this Chow variety into some P M ( C ). In particular, f has topological degree d dim A .By construction, a general C ∈ A is irreducible and the projection π | C : C → B is flat with fibers of dimension 1. For any two distinct elements C , C ′ ∈ A , the set X C , C ′ := C ∩ C ′ is finite and π | X C , C ′ : X C , C ′ → B is flat with finite fibers. Moreover, byB´ezout’s Theorem, we havedeg M ( X C , C ′ ) ≤ deg M ( C ) · deg M ( C ′ ) ≤ D . Note that this holds for all pairs ( C , C ′ ) ∈ A × A \ ∆ (where ∆ is the diagonal in A ), sothat for all ( C , C ′ ) ∈ A × A \ (cid:0) ∆ ∪ ( f , f ) − (∆) (cid:1) , we have[ f ( X C , C ′ )] ≤ [ X f ( C ) , f ( C ′ ) ] . Letting ∆ f := S n ≥ ( f , f ) − n (∆), for any ( C , C ′ ) ∈ A × A \ ∆ f , we havedeg M ( f n ( X C , C ′ )) ≤ D . Let A ∧ be the Zariski closure of the family { X C , C ′ ; ( C , C ′ ) ∈ A × A \ ∆ f } in the Chow variety Ch dim B ,D ( P × B ). By Theorem 19, any periodic component of A ∧ has dimension at most 1, since we assumed ( P , f, O P (1)) is non-isotrivial.As in the proof of Theorem 19, if d A ∧ Λ := { ( X , z, λ ) ∈ A ∧ × P × Λ ; ( z, λ ) ∈ X } and if Y is the image of d A ∧ Λ under the projection onto P × Λ, then we have dim Y = dim B + e with e ∈ { , } and f k ( Y ) = Y . Up to replacing f with f k , we assume k = 1 in the restof the proof. We claim e = 0. Indeed, if e = 1, pick any irreducible C ∈ A and let Y C := C ∩ Y . This is a subvariety of P × B and the map π | Y C : Y C → B is finite, i.e. Y C is a curve and, for any C ′ in a Zariski dense open set U of A , theintersection between C and C ′ is contained in Y C . This is a contradiction. We thus haveproved there exists a subvariety Y ⊂ P × B such that • the set Y is f -invariant, i.e. f ( Y ) = Y , • for any two C , C ′ ∈ A distinct, C ∩ C ′ ⊂ Y .In particular, dim A = 1 and, for any ( z, λ ) ∈ P × Λ \ Y , there is one and only onecurve C ∈ A such that ( z, λ ) ∈ C . As dim A = 1, up to taking a finite quotient of thenormalization of A , we can assume A ≃ P ( C ). We thus can define a rational map p : P × B P × B such that p ◦ f = g ◦ p , where g := ( f × id B ) as follows (recall that f sends Z ∈ A to theZariski closure of f ( Z Λ )): Let b A Λ = { ( C , λ, z ) ∈ A × P × Λ ; ( z, λ ) ∈ C } . By the above,the canonical projection Π : b A Λ → P × Λ is a birational morphism. If τ : b A Λ → A is theprojection onto the first factor, we let p ( z, λ ) := ( τ ◦ Π − ( z, λ ) , λ ) whenever it is defined.Moreover, the map p restricts to the generic fiber P K of the projection π : P × B → B as a rational map p defined over K , Z is a fiber of p and the restriction : P → P of g tothe generic fiber of P × B → B is a trivial endomorphism over K , since the map g is aproduct of the form g ( z, λ ) = ( f ( z ) , λ ). We conclude the proof using Corollary 27. (cid:3) Remark.
An immediate consequence of the proof is that, given a positive integer
D > Z with max n deg M ( Z n ) = D , where Z n is the Zariski closure of f n ( Z ) in P × B .6.3. Application in complex dynamics.
Fix integers k ≥ d ≥
2. Let Λ be acomplex manifold and f : P k × Λ → P k × Λ be an analytic family of complex endomorphismsof P k of degree d . The function L : λ ∈ Λ L ( f λ ) := Z P k log | det Df λ | d µ f λ ∈ R which to a parameter λ ∈ Λ associates the sum of the Lyapunov exponents of f λ withrespect to its unique maximal entropy measure is a plurisubharmonic and continuousfunction, see e.g. [P]. Moreover, for any λ ∈ Λ, we know that L ( f λ ) ≥ k log d , see [BD2].In fact, it has been proved in[BB] that, if Crit( f ) is the critical set of f Crit( f ) = { ( z, λ ) ∈ P k × Λ : det( D z f λ ) = 0 } , HE GEOMETRIC DYNAMICAL NORTHCOTT AND BOGOMOLOV PROPERTIES 37 then dd c L = T f , [Crit( f )] as closed positive currents on Λ. Definition 10.
The family f is J - stable if the function L is pluriharmonic on Λ , orequivalently if the pair ( P k × Λ , f , O P k (1) , [Crit( f )]) is stable. When k = 1, by [D1], this is equivalent to the existence of a holomorphic motion ofthe support of µ f λ and, when k >
1, it is equivalent by [BBD] to the existence of weakholomorphic motion of the support of µ f λ . If we assume in addition that Λ is a smoothquasi-projective complex curve and that f is a morphism, we say the family f is algebraic .In dimension 1, a striking result of McMullen [Mc] states that if f : P × Λ → P × Λ is astable family, then(1) either f is isotrivial, i.e. for any λ, λ ′ ∈ Λ, there exists an automorphism φ ∈ Aut( P ) such that f λ ◦ φ = φ ◦ f λ ′ ,(2) or f is a family of Latt`es maps, i.e. it is induced by the multiplication by an integeron a non-isotrivial elliptic surface, see [Mc] for more details.In particular, if f is a family of polynomials of degree d , the above implies that f is stableif and only if it is isotrivial.One may naturally ask whether a similar statement holds for algebraic families of endo-morphisms of the projective space P k of dimension k > P of degree d extends as a family f : P × Λ → P × Λwhich is stable, since L ( f λ ) = log d in this case). Theorems 29 immediately implies thefirst point of Theorem C. Moreover, Theorem 31 directly gives the second point of TheoremC. Namely, it gives a sufficient condition for stable algebraic families f : P × Λ → P × Λof endomorphisms of P to be postcritically finite, i.e. so that the setPC( f ) := [ n ≥ f n (Crit( f ))is an algebraic subvariety of P × Λ. Example (The elementary Desboves family, continued III) . Recall one last time that theDesboves family is given by f λ ([ x : y : z ]) = [ − x ( x + 2 z ) : y ( z − x + λ ( x + y + z ) : z (2 x + z )] , λ ∈ C ∗ . Finally, let us show how we can prove easily that this family is not J -stable. This has beenproved by Bianchi-Taflin [BT] to provide a counter-example to the equivalence between thecontinuous dependence of the Julia set on the parameter and the J -stability of the familyin dimension at least 2. They proved the stronger statement that supp( T f , Crit( f ) ) = C ∗ ,also giving the first example of a family with bifurcation locus of non-empty interior.We want here to give a quick proof of the fact that T f , Crit( f ) = 0 as a positive measureon C ∗ . Recall that Crit( f λ ) is the union of three lines L j passing through ρ = [0 : 1 : 0]and [1 : 0 : α j ] with α = 1, for j = 0 , ,
2, each of them counted with multiplicity 2 (the L j ’s are lines of the invariant pencil P ) together with the curve C ′ λ := { [ x : y : z ] ∈ P : − x + z + λ (4 y + x + z ) = 0 } . Let C ′ := { ( a, λ ) ∈ P × C : a ∈ C ′ λ } . As before, we have Z P × C ∗ b T f ∧ [Crit( f )] = Z P × C ∗ b T f ∧ [ C ′ ] + 2 X j =0 Z P × C ∗ b T f ∧ [ L j × C ∗ ]= Z P × C ∗ b T f ∧ [ C ′ ] , where we used Corollary 27. To prove the family is not J -stable it is thus necessary toprove T f , [ C ′ ] = 0. By Theorem 28, the hypothesis T f , [ C ′ ] = 0 implies C ′ is preperiodic underiteration of f . In particular, since the Fermat curve C is fixed by f λ for any λ ∈ C ∗ , thisimplies the intersection Z λ := C ′ λ ∩ C is preperiodic under iteration of f λ for all λ ∈ C ∗ . Asthe finite set Z λ depends non-trivially on λ and f λ | C is independent of λ , this is impossible. Acknowledgment.
The first author would like to heartily thank S´ebastien Boucksomfor useful comments and discussions about families of varieties and Damian Brotbek fordiscussions which led to the formulation of Theorem A in the context of polarized endo-morphisms of normal projective varieties. We would like to thank Matthieu Astorg forinteresting discussions at an early stage of this work. We also would like to thank AntoineChambert-Loir and Charles Favre. Finally, we would like to thank the anonymous ref-eree for his/her very interesting comments and suggestions which helped to improve thepresentation of this work.
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