Arithmetic degrees and dynamical degrees of endomorphisms on surfaces
aa r X i v : . [ m a t h . AG ] J a n ARITHMETIC DEGREES AND DYNAMICALDEGREES OF ENDOMORPHISMS ON SURFACES
YOHSUKE MATSUZAWA, KAORU SANO, AND TAKAHIRO SHIBATA
Abstract.
For a dominant rational self-map on a smooth projec-tive variety defined over a number field, Kawaguchi and Silvermanconjectured that the (first) dynamical degree is equal to the arith-metic degree at a rational point whose forward orbit is well-definedand Zariski dense. We prove this conjecture for surjective endo-morphisms on smooth projective surfaces. For surjective endomor-phisms on any smooth projective varieties, we show the existenceof rational points whose arithmetic degrees are equal to the dynam-ical degree. Moreover, we prove that there exists a Zariski denseset of rational points having disjoint orbits if the endomorphism isan automorphism.
Contents
1. Introduction 2Notation 4Outline of this paper 52. Dynamical degree and Arithmetic degree 53. Some reductions for Conjecture 1.1 63.1. Reductions 63.2. Birational invariance of the arithmetic degree 73.3. Applications of the birational invariance 94. Endomorphisms on surfaces 105. Some properties of P -bundles over curves 116. P -bundles over curves 156.1. P -bundles over P P -bundles over genus one curves 156.3. P -bundles over curves of genus ≥ κ ( X ) = 1 219. Existence of a rational point P satisfying α f ( P ) = δ f Introduction
Let k be a number field, X a smooth projective variety over k , and f : X X a dominant rational self-map on X over k . Let I f ⊂ X bethe indeterminacy locus of f . Let X f ( k ) be the set of k -rational points P on X such that f n ( P ) / ∈ I f for every n ≥
0. For P ∈ X f ( k ) , its forward f -orbit is defined as O f ( P ) := { f n ( P ) : n ≥ } . Let H be an ample divisor on X defined over k . The ( first ) dynamicaldegree of f is defined by δ f := lim n →∞ (( f n ) ∗ H · H dim X − ) /n . The first dynamical degree of a dominant rational self-map on a smoothcomplex projective variety was first defined by Dinh and Sibony in[7, 8]. In [32], Truong gave an algebraic definition of dynamical degrees.The arithmetic degree , introduced by Silverman in [30], of f at a k -rational point P ∈ X f ( k ) is defined by α f ( P ) := lim n →∞ h + H ( f n ( P )) /n if the limit on the right hand side exists. Here, h H : X ( k ) −→ [0 , ∞ )is the (absolute logarithmic) Weil height function associated with H ,and we put h + H := max { h H , } .Then we have two types of quantity concerned with the iteration ofthe action of f . It is natural to consider the relation between dynam-ical degrees and arithmetic degrees. In this direction, Kawaguchi andSilverman formulated the following conjecture. Conjecture 1.1 (The Kawaguchi–Silverman conjecture (see [21, Con-jecture 6])) . For every k -rational point P ∈ X f ( k ) , the arithmetic de-gree α f ( P ) exists. Moreover, if the forward f -orbit O f ( P ) is Zariskidense in X , the arithmetic degree α f ( P ) is equal to the dynamical de-gree δ f , i.e., we have α f ( P ) = δ f . Remark 1.2.
Let X be a complex smooth projective variety with κ ( X ) >
0, Φ : X W the Iitaka fibration of X , and f : X X adominant rational self-map on X . Nakayama and Zhang proved thatthere exists an automorphism g : W −→ W of finite order such thatΦ ◦ f = g ◦ Φ (see [28, Theorem A]). This implies that any dominantrational self-map on a smooth projective variety of positive Kodairadimension does not have a Zariski dense orbit. So the latter half ofConjecture 1.1 is meaningful only for smooth projective varieties ofnon-positive Kodaira dimension. However, we do not use their resultin this paper.When f is a dominant endomorphism (i.e. f is defined everywhere),the existence of the limit defining the arithmetic degree was proved in EGREES OF ENDOMORPHISMS ON SURFACES 3 [19]. But in general, the convergence is not known. It seems difficultat the moment to prove Conjecture 1.1 in full generality.In this paper, we prove Conjecture 1.1 for any endomorphisms onany smooth projective surfaces:
Theorem 1.3.
Let k be a number field, X a smooth projective surfaceover k , and f : X −→ X a surjective endomorphism on X . ThenConjecture 1.1 holds for f . As by-products of our arguments, we also obtain the following twocases for which Conjecture 1.1 holds:
Theorem 1.4 (Theorem 3.6) . Let k be a number field, X a smoothprojective irrational surface over k , and f : X X a birational au-tomorphism on X . Then Conjecture 1.1 holds for f . Theorem 1.5 (Theorem 3.7) . Let k be a number field, X a smoothprojective toric variety over k , and f : X −→ X a toric surjectiveendomorphism on X . Then Conjecture 1.1 holds for f . As we will see in the proof of Theorem 1.3, there does not alwaysexist a Zariski dense orbit for a given self-map. For instance, a self-mapcannot have a Zariski dense orbit if it is a self-map over a variety ofpositive Kodaira dimension. So it is also important to consider whethera self-map has a k -rational point whose orbit has full arithmetic com-plexity, that is, whose arithmetic degree coincides with the dynamicaldegree. We prove that such a point always exists for any surjectiveendomorphism on any smooth projective variety. Theorem 1.6.
Let k be a number field, X a smooth projective varietyover k , and f : X −→ X a surjective endomorphism on X . Then thereexists a k -rational point P ∈ X ( k ) such that α f ( P ) = δ f . If f is an automorphism, we can construct a “large” collection ofpoints whose orbits have full arithmetic complexity. Theorem 1.7.
Let k be a number field, X a smooth projective varietyover k , and f : X −→ X an automorphism. Then there exists a subset S ⊂ X ( k ) which satisfies all of the following conditions. (1) For every P ∈ S , α f ( P ) = δ f . (2) For
P, Q ∈ S with P = Q , O f ( P ) ∩ O f ( Q ) = ∅ . (3) S is Zariski dense in X . Remark 1.8.
Kawaguchi, Silverman, and the second author provedConjecture 1.1 in the following cases (for details, see [19], [20], [29],[30], [31]).(1) ([20, Theorem 2 (a)]) f is an endomorphism and the N´eron-Severi group of X has rank one.(2) ([20, Theorem 2 (b)]) f is the extension to P N of a regular affineautomorphism on A N . YOHSUKE MATSUZAWA, KAORU SANO, AND TAKAHIRO SHIBATA (3) ([18, Theorem A], [20, Theorem 2 (c)]) X is a smooth projectivesurface and f is an automorphism on X .(4) ([30, Proposition 19]) f is the extension to P N of a monomialendomorphism on G Nm and P ∈ G Nm ( k ).(5) ([19, Corollary 31], [31, Theorem 2]) X is an abelian variety.Note that any rational map between abelian varieties is auto-matically a morphism.(6) ([29, Theorem 1.3]) f is an endomorphism and X is the product Q ni =1 X i of smooth projective varieties, with the assumptionthat each variety X i satisfies one of the following conditions: – the first Betti number of ( X i ) C is zero and the N´eron–Severigroup of X i has rank one, – X i is an abelian variety, – X i is an Enriques surface, or – X i is a K f is an endomorphism and X is the product X × X of positive dimensional varieties such that one of X or X is of general type. (In fact, there do not exist Zariski denseforward f -orbits on such X × X .) Notation. • Throughout this paper, we fix a number field k . • A variety always means an integral separated scheme of finitetype over k in this paper. • A divisor on a variety X means a divisor on X defined over k . • An endomorphism on a variety X means a morphism from X toitself defined over k . A non-trivial endomorphism is a surjectiveendomorphism which is not an automorphism. • A curve (resp. surface ) simply means a smooth projective vari-ety of dimension 1 (resp. dimension 2) unless otherwise stated. • For any curve C , the genus of C is denoted by g ( C ). • When we say that P is a point of X or write as P ∈ X , itmeans that P is a k -rational point of X . • The N´eron–Severi group of a smooth projective variety X isdenoted by NS( X ). It is well-known that NS( X ) is a finitelygenerated abelian group. We put NS( X ) R := NS( X ) ⊗ Z R . • The symbols ≡ , ∼ , ∼ Q and ∼ R mean algebraic equivalence, lin-ear equivalence, Q -linear equivalence, and R -linear equivalence,respectively. • Let X be a smooth projective variety and f : X X a domi-nant rational self-map. A point P ∈ X f ( k ) is called preperiodic if the forward f -orbit O f ( P ) of P is a finite set. This is equiv-alent to the condition that f n ( P ) = f m ( P ) for some n, m ≥ n = m . EGREES OF ENDOMORPHISMS ON SURFACES 5 • Let f , g and h be real-valued functions on a domain S . Theequality f = g + O ( h ) means that there is a positive constant C such that | f ( x ) − g ( x ) | ≤ C | h ( x ) | for every x ∈ S . The equality f = g + O (1) means that there is a positive constant C ′ suchthat | f ( x ) − g ( x ) | ≤ C ′ for every x ∈ S . Outline of this paper.
In Section 2, we recall the definitions andsome properties of dynamical and arithmetic degrees. In Section 3,at first we recall some lemmata about reduction for Conjecture 1.1,which were proved in [29] and [31]. Then, we prove the birationalinvariance of arithmetic degree, and prove Theorem 1.4 and Theorem1.5. In Section 4, we reduce Theorem 1.3 to three cases, i.e. the case of P -bundles, hyperelliptic surfaces, and surfaces of Kodaira dimensionone. In Section 5 we recall fundamental properties of P -bundles overcurves. In Section 6, Section 7, and Section 8, we prove Theorem 1.3in each case explained in Section 4. Finally, in Section 9, we proveTheorem 1.6 and Theorem 1.7.2. Dynamical degree and Arithmetic degree
Let H be an ample divisor on a smooth projective variety X . The( first ) dynamical degree of a dominant rational self-map f : X X isdefined by δ f := lim n →∞ (( f n ) ∗ H · H dim X − ) /n . The limit defining δ f exists, and δ f does not depend on the choice of H (see [8, Corollary 7], [12, Proposition 1.2]). Note that if f is anendomorphism, we have ( f n ) ∗ = ( f ∗ ) n as a linear self-map on NS( X ).But if f is merely a rational self-map, then ( f n ) ∗ = ( f ∗ ) n in general. Remark 2.1 ([8, Proposition 1.2 (iii)], [21, Remark 7]) . Let ρ (( f n ) ∗ ) bethe spectral radius of the linear self-map ( f n ) ∗ : NS( X ) R −→ NS( X ) R . The dynamical degree δ f is equal to the limit lim n →∞ ( ρ (( f n ) ∗ )) /n . Thus we have δ f n = δ nf for every n ≥ X f ( k ) be the set of points P on X such that f is defined at f n ( P )for every n ≥ . The arithmetic degree of f at a point P ∈ X f ( k ) isdefined as follows. Let h H : X ( k ) −→ [0 , ∞ )be the (absolute logarithmic) Weil height function associated with H (see [14, Theorem B3.2]). We put h + H ( P ) := max { h H ( P ) , } . YOHSUKE MATSUZAWA, KAORU SANO, AND TAKAHIRO SHIBATA
We call α f ( P ) := lim sup n →∞ h + H ( f n ( P )) /n and α f ( P ) := lim inf n →∞ h + H ( f n ( P )) /n the upper arithmetic degree and the lower arithmetic degree of f at P ,respectively. It is known that α f ( P ) and α f ( P ) do not depend on thechoice of H (see [21, Proposition 12]). If α f ( P ) = α f ( P ), the limit α f ( P ) := lim n →∞ h + H ( f n ( P )) /n is called the arithmetic degree of f at P . Remark 2.2.
Let D be a divisor on X , H an ample divisor on X , and f a dominant rational self-map on X . Take P ∈ X f ( k ). Then we caneasily check that α f ( P ) ≥ lim sup n →∞ h + D ( f n ( P )) /n , and α f ( P ) ≥ lim inf n →∞ h + D ( f n ( P )) /n . So when these limits exist, we have α f ( P ) ≥ lim n →∞ h + D ( f n ( P )) /n . Remark 2.3.
When f is an endomorphism, the existence of the limitdefining the arithmetic degree α f ( P ) was proved by Kawaguchi andSilverman in [19, Theorem 3]. But it is not known in general. Remark 2.4.
The inequality α f ( P ) ≤ δ f was proved by Kawaguchiand Silverman, and the third author (see [21, Theorem 4],[25, Theorem1.4]). Hence, in order to prove Conjecture 1.1, it is enough to provethe opposite inequality α f ( P ) ≥ δ f .3. Some reductions for Conjecture 1.1
Reductions.
We recall some lemmata which are useful to reducethe proof of some cases of Conjecture 1.1 to easier cases.
Lemma 3.1.
Let X be a smooth projective variety and f : X −→ X asurjective endomorphism. Then Conjecture 1.1 holds for f if and onlyif Conjecture 1.1 holds for f t for some t ≥ .Proof. See [29, Lemma 3.3]. (cid:3)
Lemma 3.2 ([31, Lemma 6]) . Let ψ : X −→ Y be a finite surjectivemorphism between smooth projective varieties. Let f X : X −→ X and f Y : Y −→ Y be surjective endomorphisms on X and Y , respectively.Assume that ψ ◦ f X = f Y ◦ ψ . Then Conjecture 1.1 holds for f X if andonly if Conjecture 1.1 holds for f Y . EGREES OF ENDOMORPHISMS ON SURFACES 7
Proof.
Since ψ is a finite surjective morphism, we have dim X = dim Y .For a point P ∈ X ( k ), the forward f X -orbit O f X ( P ) is Zariski densein X if and only if the forward f Y -orbit O f Y ( ψ ( P )) is Zariski dense in Y . Let H be an ample divisor on Y . Then ψ ∗ H is an ample divisor on X . Hence, we can calculate the dynamical degree and the arithmeticdegree of f X as follows: δ f X = lim n →∞ (( f nX ) ∗ ψ ∗ H · ( ψ ∗ H ) dim X − ) /n = lim n →∞ ( ψ ∗ ( f nY ) ∗ H · ( ψ ∗ H ) dim Y − ) /n = lim n →∞ (deg( ψ )(( f nY ) ∗ H · H dim Y − )) /n = δ f Y .α f X ( P ) = lim n →∞ h + ψ ∗ H ( f nX ( P )) /n = lim n →∞ h + H ( ψ ◦ f nX ( P )) /n = lim n →∞ h + H ( f nY ◦ ψ ( P )) /n = α f Y ( ψ ( P )) . Our assertion follows from these calculations. (cid:3)
Birational invariance of the arithmetic degree.
We showthat arithmetic degree is invariant under birational conjugacy.
Lemma 3.3.
Let µ : X Y be a birational map of smooth projectivevarieties. Take Weil height functions h X , h Y associated with ampledivisors H X , H Y on X, Y , respectively. Then there are constants M ∈ R > and M ′ ∈ R such that h X ( P ) ≥ M h Y ( µ ( P )) + M ′ for any P ∈ X ( k ) \ I µ ( k ) .Proof. Take a smooth projective variety Z and a birational morphism p : Z −→ X such that p is isomorphic over X \ I µ and q = µ ◦ p : Z −→ Y is a morphism. Set E = p ∗ p ∗ q ∗ H Y − q ∗ H Y . Then E is a p -exceptionaldivisor on Z such that − E is p -nef. By the negativity lemma (cf. [22,Lemma 3.39]), E is an effective and p -exceptional divisor on Z . Takea sufficiently large integer N such that N H X − p ∗ q ∗ H Y is very ample. YOHSUKE MATSUZAWA, KAORU SANO, AND TAKAHIRO SHIBATA
Then, for P ∈ X \ I µ , we have h X ( P ) = 1 N ( h NH X − p ∗ q ∗ H Y ( P ) + h p ∗ q ∗ H Y ( P )) + O (1) ≥ N h p ∗ q ∗ H Y ( P ) + O (1)= 1 N h p ∗ p ∗ q ∗ H Y ( p − ( P )) + O (1)= 1 N h q ∗ H Y ( p − ( P )) + h E ( p − ( P )) + O (1)= 1 N h Y ( µ ( P )) + h E ( p − ( P )) + O (1) . We know that h E ≥ O (1) on Z \ Supp E (cf. [14, Theorem B.3.2(e)]).Since Supp E ⊂ p − ( I µ ), h E ( p − ( P )) ≥ O (1) for P ∈ X \ I µ . Eventu-ally, we obtain that h X ( P ) ≥ (1 /N ) h Y ( µ ( P ))+ O (1) for P ∈ X \ I µ . (cid:3) Theorem 3.4.
Let f : X X and g : Y Y be dominant rationalself-maps on smooth projective varieties and µ : X Y a birationalmap such that g ◦ µ = µ ◦ f . (i) Let U ⊂ X be a Zariski open subset such that µ | U : U −→ µ ( U ) is an isomorphism. Then α f ( P ) = α g ( µ ( P )) and α f ( P ) = α g ( µ ( P )) for P ∈ X f ( k ) ∩ µ − ( Y g ( k )) such that O f ( P ) ⊂ U ( k ) . (ii) Take P ∈ X f ( k ) ∩ µ − ( Y g ( k )) . Assume that O f ( P ) is Zariskidense in X and both α f ( P ) and α g ( µ ( P )) exist. Then α f ( P ) = α g ( µ ( P )) .Proof. (i) Using Lemma 3.3 for both µ and µ − , there are constants M , L ∈ R > and M , L ∈ R such that( ∗ ) M h Y ( µ ( P )) + M ≤ h X ( P ) ≤ L h Y ( µ ( P )) + L for P ∈ U ( k ). The claimed equalities follow from ( ∗ ).(ii) Since O f ( P ) is Zariski dense in X , we can take a subsequence { f n k ( P ) } k of { f n ( P ) } n contained in U . Using ( ∗ ) again, it follows that α f ( P ) = lim k →∞ h + X ( f n k ( P )) /n k = lim k →∞ h + Y ( g n k ( µ ( P ))) /n k = α g ( µ ( P )) . (cid:3) Remark 3.5.
In [30], Silverman dealt with a height function on G nm induced by an open immersion G nm ֒ → P n and proved Conjecture 1.1for monomial maps on G nm . It seems that it had not be checked inthe literature that the arithmetic degrees of endomorphisms on quasi-projective varieties do not depend on the choice of open immersionsto projective varieties. Now by Theorem 3.4, the arithmetic degree ofa rational self-map on a quasi-projective variety at a point does notdepend on the choice of an open immersion of the quasi-projective va-riety to a projective variety. Furthermore, by the birational invariance EGREES OF ENDOMORPHISMS ON SURFACES 9 of dynamical degrees, we can state Conjecture 1.1 for rational self-mapson quasi-projective varieties, such as semi-abelian varieties.3.3.
Applications of the birational invariance.
In this subsection,we prove Theorem 1.4 and Theorem 1.5 as applications of Theorem 3.4.
Theorem 3.6 (Theorem 1.4) . Let X be an irrational surface and f : X X a birational automorphism on X . Then Conjecture 1.1holds for f .Proof. Take a point P ∈ X f ( k ). If O f ( P ) is finite, the limit α f ( P )exists and is equal to 1. Next, assume that the closure O f ( P ) of O f ( P )has dimension 1. Let Z be the normalization of O f ( P ) and ν : Z −→ X the induced morphism. Then an endomorphism g : Z −→ Z satisfying ν ◦ g = f ◦ ν is induced. Take a point P ′ ∈ Z such that ν ( P ′ ) = P .Then α g ( P ′ ) = α f ( P ) since ν is finite. It follows from [19, Theorem2] that α g ( P ′ ) exists (note that [19, Theorem 2] holds for possiblynon-surjective endomorphisms on possibly reducible normal varieties).Therefore α f ( P ) exists.Assume that O f ( P ) is Zariski dense. If δ f = 1, then 1 ≤ α f ( P ) ≤ α f ( P ) ≤ δ f = 1 by Remark 2.4, so α f ( P ) exists and α f ( P ) = δ f = 1.So we may assume that δ f >
1. Since X is irrational and δ f > κ ( X ) must be non-negative (cf. [6, Theorem 0.4, Proposition 7.1 andTheorem 7.2]). Take a birational morphism µ : X −→ Y to the minimalmodel Y of X and let g : Y Y be the birational automorphism on Y defined as g = µ ◦ f ◦ µ − . Then g is in fact an automorphismsince, if g has indeterminacy, Y must have a K Y -negative curve. Itis obvious that O g ( µ ( P )) is also Zariski dense in Y . Since µ (Exc( µ ))is a finite set, there is a positive integer n such that µ ( f n ( P )) = g n ( µ ( P )) µ (Exc( µ )) for n ≥ n . So we have f n ( P ) Exc( µ ) for n ≥ n . Replacing P by f n ( P ), we may assume that O f ( P ) ⊂ X \ Exc( µ ).Applying Theorem 3.4 (i) to P , it follows that α f ( P ) = α g ( µ ( P )). Weknow that α g ( µ ( P )) exists since g is a morphism. So α f ( P ) also exists.The equality α g ( µ ( P )) = δ g holds as a consequence of Conjecture 1.1for automorphisms on surfaces (cf. Remark 1.8 (3)). Since dynamicaldegree is invariant under birational conjugacy, it follows that δ g = δ f .So we obtain the equality α f ( P ) = δ f . (cid:3) Theorem 3.7 (Theorem 1.5) . Let X be a smooth projective toric va-riety and f : X −→ X a toric surjective endomorphism on X . ThenConjecture 1.1 holds for f .Proof. Let G dm ⊂ X be the torus embedded as an open dense subsetin X . Then f | G dm : G dm −→ G dm is a homomorphism of algebraic groupsby assumtion. Let G dm ⊂ P d be the natural embedding of G dm to theprojective space P d and g : P d P d be the induced rational self-map.Then g is a monomial map. Take P ∈ X ( k ) such that O f ( P ) is Zariski dense. Note that α f ( P )exists since f is a morphism. Since O f ( P ) is Zariski dense and f ( G dm ) ⊂ G dm , there is a positive integer n such that f n ( P ) ∈ G dm for n ≥ n . Byreplacing P by f n ( P ), we may assume that O f ( P ) ⊂ G dm . ApplyingTheorem 3.4 (i) to P , it follows that α f ( P ) = α g ( P ).The equality α g ( P ) = δ g holds as a consequence of Conjecture 1.1for monomial maps (cf. Remark 1.8 (4)). Since dynamical degree isinvariant under birational conjugacy, it follows that δ g = δ f . So weobtain the equality α f ( P ) = δ f . (cid:3) Endomorphisms on surfaces
We start to prove Theorem 1.3. Since Conjecture 1.1 for automor-phisms on surfaces is already proved by Kawaguchi (see Remark 1.8(3)), it is sufficient to prove Theorem 1.3 for non-trivial endomor-phisms, that is, surjective endomorphisms which are not automor-phisms.Let f : X −→ X be a non-trivial endomorphism on a surface. Firstwe divide the proof of Theorem 1.3 according to the Kodaira dimensionof X .(I) κ ( X ) = −∞ ; we need the following result due to Nakayama. Lemma 4.1 (cf. [27, Proposition 10]) . Let f : X −→ X be a non-trivial endomorphism on a surface X with κ ( X ) = −∞ . Then there isa positive integer m such that f m ( E ) = E for any irreducible curve E on X with negative self-intersection.Proof. See [27, Proposition 10]. (cid:3)
Let µ : X −→ X ′ be the contraction of a ( − E on X . ByLemma 4.1, there is a positive integer m such that f m ( E ) = E . Then f m induces an endomorphism f ′ : X ′ −→ X ′ such that µ ◦ f m = f ′ ◦ µ .Using Lemma 3.1 and Theorem 3.4, the assertion of Theorem 1.3 for f follows from that for f ′ . Continuing this process, we may assume that X is relatively minimal.When X is irrational and relatively minimal, X is a P -bundle overa curve C with g ( C ) ≥ X is rational and relatively minimal, X is isomorphic to P or the Hirzebruch surface F n = P ( O P ⊕ O P ( − n )) for some n ≥ n = 1. Note that Conjecture 1.1 holds for surjective endomorphismson projective spaces (see Remark 1.8 (1)).(II) κ ( X ) = 0; for surfaces with non-negative Kodaira dimension, weuse the following result due to Fujimoto. Lemma 4.2 (cf. [9, Lemma 2.3 and Proposition 3.1]) . Let f : X −→ X be a non-trivial endomorphism on a surface X with κ ( X ) ≥ . Then X is minimal and f is ´etale.Proof. See [9, Lemma 2.3 and Proposition 3.1] (cid:3)
EGREES OF ENDOMORPHISMS ON SURFACES 11 So X is either an abelian surface, a hyperelliptic surface, a K3 sur-face, or an Enriques surface. Since f is ´etale, we have χ ( X, O X ) =deg( f ) χ ( X, O X ). Now deg( f ) ≥ χ ( X, O X ) = 0(cf. [9, Corollary 2.4]). Hence X must be either an abelian surface ora hyperelliptic surface because K3 surfaces and Enriques surfaces havenon-zero Euler characteristics. Note that Conjecture 1.1 is valid forendomorphisms on abelian varieties (see Remark 1.8 (5)).(III) κ ( X ) = 1; this case will be treated in Section 8.(IV) κ ( X ) = 2; the following fact is well-known. Lemma 4.3.
Let X be a smooth projective variety of general type.Then any surjective endomorphisms on X are automorphisms. Fur-thermore, the group of automorphisms Aut( X ) on X has finite order.Proof. See [9, Proposition 2.6], [17, Theorem 11.12], or [24, Corollary2]. (cid:3)
So there is no non-trivial endomorphism on X . As a summary, theremaining cases for the proof of Theorem 1.3 are the following: • Non-trivial endomorphisms on P -bundles over a curve. • Non-trivial endomorphisms on hyperelliptic surfaces. • Non-trivial endomorphisms on surfaces of Kodaira dimension 1.
Remark 4.4.
Fujimoto and Nakayama gave a complete classificationof surfaces which admit non-trivial endomorphisms (cf. [11, Theorem1.1], [9, Proposition 3.3], [27, Theorem 3], and [10, Appendix to Section4]). 5.
Some properties of P -bundles over curves In this section, we recall and prove some properties of P -bundles(see [13, Chapter V.2], [15], [16] for detail). In this section, let X be a P -bundle over a curve C . Let π : X −→ C be the projection. Proposition 5.1.
We can represent X as X ∼ = P ( E ) , where E is alocally free sheaf of rank 2 on C such that H ( E ) = 0 but H ( E ⊗L ) = 0 for all invertible sheaves L on C with deg L < . The integer e := − deg E does not depend on the choice of such E . Furthermore, thereis a section σ : C −→ X with image C such that O X ( C ) ∼ = O X (1) . Proof.
See [13, Proposition 2.8]. (cid:3)
Lemma 5.2.
The Picard group and the N´eron–Severi group of X havethe structure as follows. Pic( X ) ∼ = Z ⊕ π ∗ Pic( C ) , NS( X ) ∼ = Z ⊕ π ∗ NS( C ) ∼ = Z ⊕ Z . Furthermore, the image C of the section σ : C −→ X in Proposition5.1 generates the first direct factor of Pic( X ) and NS( X ) . Proof.
See [13, V, Proposition 2.3]. (cid:3)
Lemma 5.3.
Let F ∈ NS( X ) be a fiber π − ( p ) = π ∗ p over a point p ∈ C ( k ) , and e the integer defined in Proposition 5.1. Then the inter-section numbers of generators of NS( X ) are the following. F · F = 0 ,F · C = 1 ,C · C = − e. Proof.
It is easy to see that the equalities F · F = 0 and F · C = 1hold. For the last equality, see [13, V, Proposition 2.9]. (cid:3) We say that f preserves fibers if there is an endomorphism f C on C such that π ◦ f = f C ◦ π . In our situation, since there is a section σ : C −→ X , f preserves fibers if and only if, for any point p ∈ C ,there is a point q ∈ C such that f ( π − ( p )) ⊂ π − ( q ).The following lemma appears in [1, p. 18] in more general form. Butwe need it only in the case of P -bundles on a curve, and the proof ingeneral case is similar to our case. So we deal only with the case of P -bundle on a curve. Lemma 5.4.
For any surjective endomorphism f on X , the iterate f preserves fibers.Proof. By the projection formula, the fibers of π : X −→ C can becharacterized as connected curves having intersection number zero withany fibers F p = π ∗ p , p ∈ C . Hence, to check that the iterate f sendsfibers to fibers, it suffices to show that ( f ) ∗ ( π ∗ NS( C ) R ) = π ∗ NS( C ) R .Since π ∗ NS( C ) R is a hyperplane in NS( X ) R such that any divisor class D from this hyperplane satisfies D · D = 0, its pullback f ∗ π ∗ NS( C ) R is a hyperplane with the same property. There are at most two suchhyperplanes, because the form of self-intersection NS( X ) R −→ R is aquadratic form associated to the coefficients of C and F . Hence, f ∗ fixes or interchanges them and so ( f ) ∗ fixes them. (cid:3) Lemma 5.5.
A surjective endomorphism f preserves fibers if and onlyif there exists a non-zero integer a such that f ∗ F ≡ aF . Here, F is thenumerical class of a fiber.Proof. Assume f ∗ F ≡ aF . For any point p ∈ C , we set F p := π − ( p ) = π ∗ p . If f does not preserve fibers, there is a point p ∈ C such that f ( F p ) · F >
0. Now we can calculate the intersection number as follows:0 = F · aF = F · ( f ∗ F ) = F p · ( f ∗ F )= ( f ∗ F p ) · F = deg( f | F p ) · ( f ( F p ) · F ) > . This is a contradiction. Hence f preserves fibers. EGREES OF ENDOMORPHISMS ON SURFACES 13
Next, assume that f preserves fibers. Write f ∗ F = aF + bC . Thenwe can also calculate the intersection number as follows: b = F · ( aF + bC ) = F · f ∗ F = ( f ∗ F ) · F = deg( f | F ) · ( F · F ) = 0 . Further, by the injectivity of f ∗ , we have a = 0. The proof is complete. (cid:3) Lemma 5.6. If E splits, i.e., if there is an invertible sheaf L on C such that E ∼ = O C ⊕ L , the invariant e of X = P ( E ) is non-negative.Proof. See [13, V, Example 2.11.3]. (cid:3)
Lemma 5.7.
Assume that e ≥ . Then for a divisor D = aF + bC ∈ NS( X ) , the following properties are equivalent. • D is ample. • a > be and b > . Proof.
See [13, V, Proposition 2.20]. (cid:3)
We can prove a result stronger than Lemma 5.4 as follows.
Lemma 5.8.
Assume that e > . Then any surjective endomorphism f : X −→ X preserves fibers.Proof. By Lemma 5.5, it is enough to prove f ∗ F ≡ aF for some integer a >
0. We can write f ∗ F ≡ aF + bC for some integers a, b ≥ aF + bC = ( a − be ) F + b ( eF + C )and f preserves the nef cone and the ample cone, either of the equalities a − be = 0 or b = 0 holds.We have 0 = deg( f )( F · F ) = ( f ∗ f ∗ F ) · F = ( f ∗ F ) · ( f ∗ F ) = ( aF + bC ) · ( aF + bC )= 2 ab − b e = b (2 a − be ) . So either of the equalities b = 0 or 2 a − be = 0 holds.If we have b = 0, we have a − be = 0 and 2 a − be = 0. So we get a = 0. But since e = 0, we obtain b = 0. This is a contradiction.Consequently, we get b = 0 and f ∗ F ≡ aF . (cid:3) Lemma 5.9.
Fix a fiber F = F p for a point p ∈ C ( k ) . Let f be asurjective endomorphism on X preserving fibers, f C the endomorphismon C satisfying π ◦ f = f C ◦ π , f F := f | F : F −→ f ( F ) the restrictionof f to the fiber F . Set f ∗ F ≡ aF and f ∗ C ≡ cF + dC . Then wehave a = deg( f C ) , d = deg( f F ) , deg( f ) = ad , and δ f = max { a, d } . Proof.
Our assertions follow from the following equalities of divisorclasses in NS( X ) and of intersection numbers: aF = f ∗ F = f ∗ π ∗ p = π ∗ f ∗ C p = π ∗ (deg( f C ) p )= deg( f C ) π ∗ p = deg( f C ) F, deg( f ) F = f ∗ f ∗ F = f ∗ f ∗ π ∗ p = f ∗ π ∗ f ∗ C p = f ∗ π ∗ (deg( f C ) p )= deg( f C ) f ∗ F = deg( f C ) deg( f F ) f ( F )= deg( f C ) deg( f F ) F deg( f ) = deg( f ) C · F = ( f ∗ f ∗ C ) · F = ( f ∗ C ) · ( f ∗ F ) = ( cF + dC ) · aF = ad. The last assertion δ f = max { a, d } follows from the functoriality of f ∗ and the equality δ f = lim n →∞ ρ (( f n ) ∗ ) /n (cf. Remark 2.1). (cid:3) Lemma 5.10.
Let Notation be as in Lemma 5.9. Assume that e ≥ .Then both F and C are eigenvectors of f ∗ : NS( X ) R −→ NS( X ) R .Further, if e is positive, then we have deg( f C ) = deg( f F ) .Proof. Set f ∗ F = aF and f ∗ C = cF + dC in NS( X ). Then we have − ead = − e deg f = ( f ∗ f ∗ C ) · C = ( f ∗ C ) = ( cF + dC ) = 2 cd − ed . Hence, we get c = e ( d − a ) /
2. We have the following equalities inNS( X ): f ∗ ( eF + C ) = aeF + ( cF + dC ) = ( ae + c ) F + dC . By the fact that f ∗ D is ample if and only if D is ample, it follows that eF + C is an eigenvector of f ∗ . Thus, we have de = ae + c = ae + e ( d − a ) / e ( d + a ) / . Therefore, the equality e ( d − a ) = 0 holds. So c = e ( d − a ) / e >
0. Then it follows that d − a = 0. Sowe have deg( f C ) = a = d = deg( f F ). (cid:3) The following lemma is used in Subsection 6.2.
Lemma 5.11.
Let L be a non-trivial invertible sheaf of degree on acurve C with g ( C ) ≥ , E = O C ⊕ L , and X = P ( E ) . Let C , C besections corresponding to the projections E −→ L and
E −→ O C . If σ : C −→ X is a section such that ( σ ( C )) = 0 , then σ ( C ) is equal to C or C .Proof. Note that e = 0 in this case and thus ( C ) = 0. Moreover, O X ( C ) ∼ = O X (1) and O X ( C ) ∼ = O X (1) ⊗ π ∗ L − . Let F be the numer-ical class of a fiber. Set σ ( C ) ≡ aC + bF . Then a = ( σ ( C ) · F ) = 1 EGREES OF ENDOMORPHISMS ON SURFACES 15 and 2 ab = ( σ ( C ) ) = 0. Thus σ ( C ) ≡ C . Therefore, O X ( σ ( C )) ∼ = O X ( C ) ⊗ π ∗ N for some invertible sheaf N of degree 0 on C . Then0 = H ( X, O X ( σ ( C ))) = H ( C, π ∗ O X ( C ) ⊗N )= H ( C, ( L⊕O C ) ⊗ N )and this implies N ∼ = O C or N ∼ = L − . Hence O X ( σ ( C )) is isomorphicto O X ( C ) or O X ( C ) ⊗ π ∗ L − = O X ( C ). Since L is non-trivial, wehave H ( O X ( C )) = H ( O X ( C )) = k and we get σ ( C ) = C or C . (cid:3) P -bundles over curves In this section, we prove Conjecture 1.1 for non-trivial endomor-phisms on P -bundles over curves. We divide the proof according tothe genus of the base curve.6.1. P -bundles over P .Theorem 6.1. Let π : X −→ P be a P -bundle over P and f : X → X be a non-trivial endomorphism. Then Conjecture 1.1 holds for f .Proof. Take a locally free sheaf E of rank 2 on P such that X ∼ = P ( E ) and deg E = − e (cf. Proposition 5.1). Then E splits (see [13,V. Corollary 2.14]). When X is isomorphic to P × P , i.e. the caseof e = 0, the assertion holds by [29, Theorem 1.3]. When X is notisomorphic to P × P , i.e. the case of e >
0, the endomorphism f preserves fibers and induces an endomorphism f P on the base curve P . By Lemma 5.10, we have δ f = δ f P . Fix a point p ∈ P and set F = π ∗ p . Let P ∈ X ( k ) be a point whose forward f -orbit is Zariskidense in X . Then the forward f P -orbit of π ( P ) is also Zariski densein P . Now the assertion follows from the following computation. α f ( P ) ≥ lim n →∞ h F ( f n ( P )) /n = lim n →∞ h π ∗ p ( f n ( P )) /n = lim n →∞ h p ( π ◦ f n ( P )) /n = lim n →∞ h p ( f n P ◦ π ( P )) /n = δ f P = δ f . (cid:3) P -bundles over genus one curves. In this subsection, we proveConjecture 1.1 for any endomorphisms on a P -bundle on a curve C ofgenus one.The following result is due to Amerik. Note that Amerik in factproved it for P -bundles over varieties of arbitrary dimension (cf. [1]). Lemma 6.2 (Amerik) . Let X = P ( E ) be a P -bundle over a curve C .If X has a fiber-preserving surjective endomorphism whose restrictionto a general fiber has degree greater than 1, then E splits into a directsum of two line bundles after a finite base change. Furthermore, if E is semistable, then E splits into a direct sum of two line bundles afteran ´etale base change. Proof.
See [1, Theorem 2 and Proposition 2.4]. (cid:3)
The following lemma is used when we take the base change by an´etale cover of genus one curve.
Lemma 6.3.
Let E be a curve of genus one with an endomorphism f : E −→ E . If g : E ′ −→ E is a finite ´etale covering of E , there existsa finite ´etale covering h : E ′′ −→ E ′ and an endomorphism f ′ : E ′′ −→ E ′′ such that f ◦ g ◦ h = g ◦ h ◦ f ′ . Furthermore, we can take h assatisfying E ′′ = E .Proof. At first, since E ′ is an ´etale covering of genus one curve E , E ′ is also a genus one curve. By fixing a rational point p ∈ E ′ ( k ) and g ( p ) ∈ E ( k ), these curves E and E ′ are regarded as elliptic curves,and g can be regarded as an isogeny between elliptic curves. Let h :=ˆ g : E −→ E ′ be the dual isogeny of g . The morphism f is decomposedas f = τ c ◦ ψ for a homomorphism ψ and a translation map τ c by c ∈ E ( k ). Fix a rational point c ′ ∈ E ( k ) such that [deg( g )]( c ′ ) = c andconsider the translation map τ c ′ , where [deg( g )] is the multiplicationby deg( g ). We set f ′ = τ c ′ ◦ ψ . Then we have the following equalities. f ◦ g ◦ h = τ c ◦ ψ ◦ g ◦ ˆ g = τ c ◦ ψ ◦ [deg( g )] = τ c ◦ [deg( g )] ◦ ψ = [deg( g )] ◦ τ c ′ ◦ ψ = g ◦ h ◦ f ′ . This is what we want. (cid:3)
Proposition 6.4.
Let E be a locally free sheaf of rank on a genusone curve C and X = P ( E ) . Suppose Conjecture 1.1 holds for any non-trivial endomorphism on X with E = O C ⊕ L where L is a line bundleof degree zero on C . Then Conjecture 1.1 holds for any non-trivialendomorphism on X = P ( E ) for any E .Proof. By Lemma 5.4 and Lemma 3.1, we may assume that f preservesfibers. We can prove Conjecture1.1 in the case of deg( f | F ) = 1 by thesame way as in the case of g ( C ) = 0 since deg( f | F ) = 1 ≤ deg( f C ).Since we are considering the case of g ( C ) = 1, if E is indecomposable,then E is semistable (see [26, 10.2 (c), 10.49] or [13, V. Exercise 2.8(c)]). By Lemma 6.2, if deg( f | F ) > E is indecomposable, there isa finite ´etale covering g : E −→ C satisfying that E × C X ∼ = P ( O E ⊕ L )for an invertible sheaf L over E . Furthermore, by Lemma 6.3, wecan take E equal to C and there is an endomorphism f ′ C : C −→ C satisfying f C ◦ g = g ◦ f ′ C . Then by the universality of cartesian product X × C,g C , an endomorphism f ′ : X × C,g C −→ X × C,g C is induced. ByLemma 3.2, it is enough to prove Conjecture 1.1 for the endomorphism f ′ . Thus, we may assume that E is decomposable, i.e., X ∼ = P ( O C ⊕ L ).Then the invariant e is non-negative by Lemma 5.6. When e is positive,by the same way as the proof of Theorem 1.3 in the case of g ( C ) = 0, EGREES OF ENDOMORPHISMS ON SURFACES 17 the proof is complete. When e = 0, we have deg L = 0 and the assertionholds by the assumption. (cid:3) In the rest of this subsection, we keep the following notation. Let C be a genus one curve and L an invertible sheaf on C with degree 0. Let X = P ( O C ⊕ L ) = Proj(Sym( O C ⊕ L )) and π : X −→ C the projection.When L is trivial, we have X ∼ = C × P , and by [29, Theorem1.3],Conjecture 1.1 is true for X . Thus we may assume L is non-trivial.In this case, we have two sections of π : X −→ C corresponding to theprojections O C ⊕ L −→ L and O C ⊕ L −→ O C . Let C and C denotethe images of these sections. Then we have O X ( C ) = O X (1) and O X ( C ) = O X (1) ⊗ π ∗ L − . Since L is non-trivial, we have C = C .But since deg L = 0, C and C are numerically equivalent. Thus( C · C ) = ( C ) = 0 and therefore C ∩ C = ∅ .Let f be a non-trivial endomorphism on X such that there is asurjective endomorphism f C : C −→ C with π ◦ f = f C ◦ π . Lemma 6.5.
When L is a torsion element of Pic C , Conjecture 1.1holds for f .Proof. We fix an algebraic group structure on C . Since L is torsion,there exists a positive integer n > n ] ∗ L ∼ = O C . Then thebase change of π : X −→ C by [ n ] : C −→ C is the trivial P -bundle P × C −→ C . Applying Lemma 6.3 to g = [ n ], we get a finite morphism h : C −→ C such that the base change of π : X −→ C by h : C −→ C is P × C −→ C and there exists a finite morphism f ′ C : C −→ C with f C ◦ h = h ◦ f ′ C . Then f induces a non-trivial endomorphism f ′ : P × C −→ P × C . By [29, Theorem1.3], Conjecture 1.1 holds for f ′ . By Lemma 3.2, Conjecture 1.1 holds also for f . (cid:3) Now, let F be the numerical class of a fiber of π . By Lemma 5.10,we have f ∗ F ≡ aF,f ∗ C ≡ bC for some integers a, b ≥
1. Note that a = deg f C , b = deg f | F and ab = deg f (cf. Lemma 5.9). Lemma 6.6. (1)
One of the equalities f ( C ) = C , f ( C ) = C and f ( C ) ∩ C = f ( C ) ∩ C = ∅ holds. The same is true for f ( C ) . (2) If f ( C ) ∩ C i = ∅ for i = 0 , , then the base change of π : X −→ C by f C : C −→ C is isomorphic to P × C . In particular, f ∗ C L ∼ = O C and L is a torsion element of Pic C . The sameconclusion holds under the assumption that f ( C ) ∩ C i = ∅ for i = 0 , . Proof. (1) Since f ∗ C i ≡ bC i , C ≡ C and ( C ) = 0, we have ( f ∗ C i · C j ) = 0 for every i and j . Thus the assertion follows.(2) Assume f ( C ) ∩ C i = ∅ for i = 0 ,
1. Consider the followingCartesian diagram. Y g / / π ′ (cid:15) (cid:15) X π (cid:15) (cid:15) C f C / / C Then Y is a P -bundle over C associated with the vector bundle O C ⊕ f ∗ C L . The pull-backs C i = g − ( C i ) , i = 0 , π ′ . By theprojection formula, we have ( C ′ i ) = 0. Let σ : C −→ X be the sectionwith σ ( C ) = C . Since π ◦ f ◦ σ = f C , we get a section s : C −→ Y of π ′ . C s (cid:7) (cid:7) ✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍ σ (cid:15) (cid:15) id (cid:22) (cid:22) X f (cid:15) (cid:15) Y π ′ (cid:15) (cid:15) g / / X π (cid:15) (cid:15) C f C / / C Note that g ( s ( C )) = f ( C ) = C , C . Thus s ( C ) , C ′ , C ′ are distinctsections of π ′ . Moreover, by the projection formula, we have ( s ( C ) · C ′ ) = 0. Thus we have three sections which are numerically equivalentto each other. Then Lemma 5.11 implies f ∗ C L ∼ = O C and Y ∼ = P × C .Since f ∗ C : Pic C −→ Pic C is an isogeny, the kernel of f ∗ C is finite andthus L is a torsion element of Pic C . (cid:3) Lemma 6.7. (1)
Suppose that • L is non-torsion in
Pic C , • f ( C ) = C or C , and • f ( C ) = C or C .Then f ( C ) = C and f ( C ) = C , or f ( C ) = C and f ( C ) = C . (2) If the equalities f ( C ) = C and f ( C ) = C hold, then f ∗ C i ∼ Q bC i for i = 0 and .Proof. (1) Assume that f ( C ) = C and f ( C ) = C . Then f ∗ C = aC and f ∗ C = aC as cycles. Since f ∗ C : Pic C −→ Pic C is surjective,there exists a degree zero divisor M on C such that f ∗ C O C ( M ) ∼ = L .Then C ∼ C − π ∗ f ∗ C M . Hence aC = f ∗ C ∼ ( f ∗ C − f ∗ π ∗ f ∗ C M ) = ( aC − f ∗ π ∗ f ∗ C M ) EGREES OF ENDOMORPHISMS ON SURFACES 19 and 0 ∼ f ∗ π ∗ f ∗ C M ∼ f ∗ f ∗ π ∗ M ∼ (deg f ) π ∗ M. Thus π ∗ M is torsion and so is M . This implies that L is torsion, whichcontradicts the assumption.The same argument shows that the case when f ( C ) = C and f ( C ) = C does not occur.(2) In this case, we have f ∗ C ∼ aC . We can write f ∗ C ∼ bC + π ∗ D for some degree zero divisor D on C . Thus(deg f ) C ∼ f ∗ f ∗ C ∼ abC + f ∗ π ∗ D = (deg f ) C + f ∗ π ∗ D and f ∗ π ∗ D ∼
0. Since f ∗ C : Pic C −→ Pic C is surjective, there existsa degree zero divisor D ′ on C such that f ∗ C D ′ ∼ D . Then0 ∼ f ∗ π ∗ D ∼ f ∗ π ∗ f ∗ C D ′ ∼ f ∗ f ∗ π ∗ D ′ ∼ (deg f ) π ∗ D ′ . Hence π ∗ D ′ ∼ Q D ′ ∼ Q
0. Therefore D ∼ Q f ∗ C ∼ Q bC .Similarly, we have f ∗ C ∼ Q bC . (cid:3) Lemma 6.8.
Suppose a < b . If f ∗ C i ∼ Q bC i for i = 0 , , the linebundle L is a torsion element of Pic C .Proof. Let L be a divisor on C such that O C ( L ) ∼ = L . Note that C ∼ C − π ∗ L . Thus f ∗ π ∗ L ∼ f ∗ ( C − C ) ∼ Q bC − bC ∼ bπ ∗ L and f ∗ C L ∼ Q bL hold.Thus, from the following lemma, L is a torsion element. (cid:3) Lemma 6.9.
Let a, b be integers such that ≤ a < b . Let C be acurve of genus one defined over an algebraically closed field k . Let f C : C −→ C be an endomorphism of deg f C = a . If L is a divisor on C of degree satisfying f ∗ C L ∼ Q bL, the divisor L is a torsion element of Pic ( C ) Proof.
By the definition of Q -linear equivalence, we have f ∗ C rL ∼ brL for some positive integer r . Since the curve C is of genus one, the groupPic ( C ) is an elliptic curve. Assume the (group) endomorphism f ∗ C − [ b ] : Pic ( C ) −→ Pic ( C )is the 0 map. Then we have the equalities a = deg f C = deg f ∗ C =deg[ b ] = b . But this contradicts to the inequality 1 ≤ a < b . Hencethe map f ∗ C − [ b ] is an isogeny, and Ker( f ∗ C − [ b ]) ⊂ Pic ( C ) is a finitegroup scheme. In particular, the order of rL ∈ Ker( f ∗ C − [ b ])( k ) is finite.Thus, L is a torsion element. (cid:3) Remark 6.10.
We can actually prove the following. Let X be asmooth projective variety over Q and f : X −→ X be a surjectivemorphism over Q with first dynamical degree δ . If an R -divisor D on X satisfies f ∗ D ∼ R λD for some λ > δ , then one has D ∼ R Sketch of the proof.
Consider the canonical heightˆ h D ( P ) = lim n →∞ h D ( f n ( P )) /λ n where h D is a height associated with D (cf. [5]). If ˆ h D ( P ) = 0 forsome P , then we can prove α f ( P ) ≥ λ . This contradicts to the fact δ ≥ α f ( P ) and the assumption λ > δ . Thus one has ˆ h D = 0 andtherefore h D = ˆ h D + O (1) = O (1). By a theorem of Serre, we get D ∼ R (cid:3) Proposition 6.11.
Let L be an invertible sheaf of degree zero on agenus one curve C and X = P ( O C ⊕ L ) . For any non-trivial endomor-phism f : X −→ X , Conjecture 1.1 holds.Proof. By Lemma 6.5 and Proposition 6.9 we may assume a ≥ b . Inthis case, δ f = a and Conjecture 1.1 can be proved as in the proof ofProposition 6.1. (cid:3) Proof of Theorem 1.3 for P -bundles over genus one curves. As we ar-gued at the first of Section 4, we may assume that the endomorphism f : X −→ X is not an automorphism. Then the assertion follows fromProposition 6.4 and Proposition 6.11. (cid:3) Remark 6.12.
In the above setting, the line bundle L is actually aneigenvector for f ∗ C up to linear equivalence. More precisely, for a P -bundle π : X = P ( O C ⊕ L ) −→ C over a curve C with deg L = 0and an endomorphism f : X −→ X that induces an endomorphism f C : C −→ C , there exists an integer t such that f ∗ C L ∼ = L t . Indeed,let C and C be the sections defined above. Since ( f ∗ ( C ) · C ) = 0,we can write O X ( f − ( C )) ∼ = O X ( mC ) ⊗ π ∗ N for some integer m anddegree zero line bundle N on C . Since0 = H ( O X ( f − ( C ))) = H ( O X ( mC ) ⊗ π ∗ N )= H (Sym m ( O C ⊕ L ) ⊗N ) = m M i =0 H ( L i ⊗N ) , we have N ∼ = L r for some − m ≤ r ≤
0. Thus f ∗ O X ( C ) ∼ = O X ( mC ) ⊗ π ∗ L r .The key is the calculation of global sections using projection formula.Since O X ( C ) ∼ = O X ( C ) ⊗ π ∗ L − , we have π ∗ O X ( mC ) ∼ = π ∗ O X ( mC ) ⊗L − m .Moreover, since C and C are numerically equivalent, we can similarlyget f ∗ O X ( C ) ∼ = O X ( mC ) ⊗ π ∗ L s for some integer s . Thus, f ∗ π ∗ L ∼ = EGREES OF ENDOMORPHISMS ON SURFACES 21 π ∗ L r − s . Therefore, π ∗ f ∗ C L ∼ = π ∗ L r − s . Since π ∗ : Pic C −→ Pic X isinjective, we get f ∗ C L ∼ = L r − s .6.3. P -bundles over curves of genus ≥ . By the following propo-sition, Conjecture 1.1 trivially holds in this case.
Proposition 6.13.
Let C be a curve with g ( C ) ≥ and π : X −→ C be a P -bundle over C . Let f : X −→ X be a surjective endomorphism.Then there exists an integer t > such that f t is a morphism over C ,that is, f t satisfies π ◦ f t = π . In particular, f admits no Zariski denseorbit.Proof. By Lemma 5.4, we may assume that f induces a surjective endo-morphism f C : C −→ C with π ◦ f = f C ◦ π . Since C is of general type, f C is an automorphism of finite order and the assertion follows. (cid:3) Remark 6.14.
The fact that f does not admit any Zariski dense orbitsalso follows from the Mordell conjecture (Faltings’s theorem). Indeed,assume there exists a Zariski dense orbit O f ( P ) on X . Then π ( O f ( P ))is also Zariski dense in C . We may assume that X, C, f, π, P are definedover a number field K . Since g ( C ) ≥
2, by the Mordell conjecture, theset of K -rational points C ( K ) is finite and therefore π ( O f ( P )) is alsofinite. This is a contradiction.7. Hyperelliptic surfaces
Theorem 7.1.
Let X be a hyperelliptic surface and f : X −→ X anon-trivial endomorphism on X . Then Conjecture 1.1 holds for f .Proof. Let π : X −→ E be the Albanese map of X . By the universalityof π , there is a morphism g : E −→ E satisfying π ◦ f = g ◦ π . It iswell-known that E is a genus one curve, π is a surjective morphismwith connected fibers, and there is an ´etale cover φ : E ′ −→ E suchthat X ′ = X × E E ′ ∼ = F × E ′ , where F is a genus one curve (cf. [2,Chapter 10]). In particular, X ′ is an abelian surface. By Lemma 6.3,taking a further ´etale base change, we may assume that there is anendomorphism h : E ′ −→ E ′ such that φ ◦ h = g ◦ φ . Let π ′ : X ′ −→ E ′ and ψ : X ′ −→ X be the induced morphisms. Then, by the universalityof fiber products, there is a morphism f ′ : X ′ −→ X ′ satisfying π ′ ◦ f ′ = π ′ ◦ h and ψ ◦ f ′ = f ◦ ψ . Applying Lemma 3.2, it is enough to proveConjecture 1.1 for the endomorphism f ′ . Since X ′ is an abelian variety,it holds by [19, Corollary 31] and [31, Theorem 2]. (cid:3) Surfaces with κ ( X ) = 1Let f : X −→ X be a non-trivial endomorphism on a surface X with κ ( X ) = 1. In this section we shall prove that f does not admit anyZariski dense forward f -orbit. Although this result is a special case of[28, Theorem A] (see Remark 1.2), we will give a simpler proof of it. By Lemma 4.2, X is minimal and f is ´etale. Since deg( f ) ≥
2, wehave χ ( X, O X ) = 0.Let φ = φ | mK X | : X −→ P N = P H ( X, mK X ) be the Iitaka fibrationof X and set C = φ ( X ). Since f is ´etale, it induces an automorphism g : P N −→ P N such that φ ◦ f = g ◦ φ (cf. [11, Lemma 3.1]). Therestriction of g to C gives an automorphism f C : C −→ C such that φ ◦ f = f C ◦ φ . Take the normalization ν : C −→ C of C . Then φ factors as X π −→ C ν −→ C and π is an elliptic fibration. Moreover, f C lifts to an automorphism f C : C −→ C such that π ◦ f = f C ◦ π .So we obtain an elliptic fibration π : X −→ C and an automorphism f C on C such that π ◦ f = f C ◦ π In this situation, the following holds.
Theorem 8.1.
Let X be a surface with κ ( X ) = 1 , π : X −→ C an elliptic fibration, f : X −→ X a non-trivial endomorphism, and f C : C −→ C an automorphism such that π ◦ f = f C ◦ π . Then f tC = id C for a positive integer t .Proof. Let { P , . . . , P r } be the points over which the fibers of π aremultiple fibers (possibly r = 0, i.e. π does not have any multiple fibers).We denote by m i denotes the multiplicity of the fiber π ∗ P i for every i .Then we have the canonical bundle formula: K X = π ∗ ( K C + L ) + r X i =1 m i − m i π ∗ P i , where L is a divisor on C such that deg( L ) = χ ( X, O X ). Thendeg( L ) = 0 because f is ´etale and deg( f ) ≥ κ ( X ) = 1, the divisor K C + L + P ri =1 m i − m i P i must have positivedegrees.So we have( ∗ ) 2( g ( C ) −
1) + r X i =1 m i − m i > . For any i , set Q i = f − C ( P i ). Then π ∗ Q i = π ∗ f ∗ C P i = f ∗ π ∗ P i is amultiple fiber. So ( f C ) | { P ,...,P r } is a permutation of { P , . . . , P r } since f C is an automorphism.We divide the proof into three cases according to the genus g ( C ) of C :(1) g ( C ) ≥
2; then the automorphism group of C is finite. So f tC =id C for a positive integer t .(2) g ( C ) = 1; by ( ∗ ), it follows that r ≥
1. For a suitable t , all P i arefixed points of f tC . We put the algebraic group structure on C such that P is the identity element. Then f tC is a group automorphism on C . So f tsC = id C for a suitable s since the group of group automorphisms on C is finite.(3) g ( C ) = 0; again by ( ∗ ), it follows that r ≥
3. For a suitable t ,all P i are fixed points of f tC . Then f tC fixes at least three points, whichimplies that f tC is in fact the identity map. (cid:3) EGREES OF ENDOMORPHISMS ON SURFACES 23
Immediately we obtain the following corollary.
Corollary 8.2.
Let f : X −→ X be a non-trivial endomorphism on asurface X with κ ( X ) = 1. Then there does not exist any Zariski dense f -orbit.Therefore Conjecture 1.1 trivially holds for non-trivial endomor-phisms on surfaces of Kodaira dimension 1.9. Existence of a rational point P satisfying α f ( P ) = δ f In this section, we prove Theorem 1.6 and Theorem 1.7. Theorem1.6 follows from the following lemma. A subset Σ ⊂ V ( k ) is called a set of bounded height if for an (every) ample divisor A on V , the heightfunction h A associated with A is a bounded function on Σ. Lemma 9.1.
Let X be a smooth projective variety and f : X −→ X a surjective endomorphism with δ f > . Let D be a nef R -divisorsuch that f ∗ D ≡ δ f D . Let V ⊂ X be a closed subvariety of positivedimension such that ( D dim V · V ) > . Then there exists a non-emptyopen subset U ⊂ V and a set Σ ⊂ U ( k ) of bounded height such that forevery P ∈ U ( k ) \ Σ we have α f ( P ) = δ f . Remark 9.2.
By Perron-Frobenius-type result of [4, Theorem], thereis a nef R -divisor D f ∗ D ≡ δ f D since f ∗ preserves the nef cone. Proof.
Fix a height function h D associated with D . For every P ∈ X ( k ), the following limit exists (cf. [21, Theorem 5]).ˆ h ( P ) = lim n →∞ h D ( f n ( P )) δ nf The function ˆ h has the following properties (cf. [21, Theorem 5]).(i) ˆ h = h D + O ( √ h H ) where H is any ample divisor on X and h H ≥ H .(ii) If ˆ h ( P ) >
0, then α f ( P ) = δ f .Since ( D dim V · V ) >
0, we have ( D | V dim V ) > D | V is big. Thuswe can write D | V ∼ R A + E with an ample R -divisor A and an effective R -divisor E on V . Therefore we haveˆ h | V ( k ) = h A + h E + O ( p h A )where h A , h E are height functions associated with A, E and h A is takento be h A ≥
1. In particular, there exists a positive real number
B > h A + h E − ˆ h | V ( k ) ≤ B √ h A . Then we have the following inclusions. { P ∈ V ( k ) | ˆ h ( P ) ≤ } ⊂ { P ∈ V ( k ) | h A ( P ) + h E ( P ) ≤ B p h A ( P ) }⊂ Supp E ∪ { P ∈ V ( k ) | h A ( P ) ≤ B p h A ( P ) } = Supp E ∪ { P ∈ V ( k ) | h A ( P ) ≤ B } . Hence we can take U = V \ Supp E and Σ = { P ∈ U ( k ) | ˆ h ( P ) ≤ } . (cid:3) Corollary 9.3.
Let X be a smooth projective variety of dimension N and f : X −→ X a surjective endomorphism. Let C be a irre-ducible curve which is a complete intersection of ample effective divi-sors H , . . . , H N − on X . Then for infinitely many points P on C , wehave α f ( P ) = δ f . Proof.
We may assume δ f >
1. Let D be as in Lemma 9.1. Then( D · C ) = ( D · H · · · H N − ) > C ( k ) is nota set of bounded height, the assertion follows from Lemma 9.1. (cid:3) To prove Theorem 1.7, we need the following theorem which is acorollary of the dynamical Mordell–Lang conjecture for ´etale finite mor-phisms.
Theorem 9.4 (Bell–Ghioca–Tucker [3, Corollary 1.4]) . Let f : X −→ X be an ´etale finite morphism of smooth projective variety X . Let P ∈ X ( k ) . If the orbit O f ( P ) is Zariski dense in X , then any properclosed subvariety of X intersects O f ( P ) in at most finitely many points.Proof of Theorem 1.7. We may assume dim X ≥
2. Since we are work-ing over k , we can write the set of all proper subvarieties of X as { V i ( X | i = 0 , , , . . . } . By Corollary 9.3, we can take a point P ∈ X \ V such that α f ( P ) = δ f .Assume we can construct P , . . . , P n satisfying the following conditions.(1) α f ( P i ) = δ f for i = 0 , . . . , n .(2) O f ( P i ) ∩ O f ( P j ) = ∅ for i = j .(3) P i / ∈ V i for i = 0 , . . . , n .Now, take a complete intersection curve C ⊂ X satisfying the followingconditions. • For i = 0 , . . . , n , C
6⊂ O f ( P i ) if O f ( P i ) = X . • For i = 0 , . . . , n , C
6⊂ O f − ( P i ) if O f − ( P i ) = X . • C V n +1 .By Theorem 9.4, if O f ± ( P i ) is Zariski dense in X , then O f ± ( P i ) ∩ C isa finite set. By Corollary 9.3, there exists a point P n +1 ∈ C \ [ ≤ i ≤ n O f ( P i ) ∪ [ ≤ i ≤ n O f − ( P i ) ∪ V n +1 ! EGREES OF ENDOMORPHISMS ON SURFACES 25 such that α f ( P n +1 ) = δ f . Then P , . . . , P n +1 satisfy the same condi-tions. Therefore we get a subset S = { P i | i = 0 , , , . . . } of X whichsatisfies the desired conditions. (cid:3) Acknowledgements
The authors would like to thank Professors Tetsushi Ito, OsamuFujino, and Tomohide Terasoma for helpful advice. They would alsolike to thank Takeru Fukuoka and Hiroyasu Miyazaki for answeringtheir questions.
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Graduate school of Mathematical Sciences, the University of Tokyo,Komaba, Tokyo, 153-8914, JapanDepartment of Mathematics, Faculty of Science, Kyoto University,Kyoto 606-8502, JapanDepartment of Mathematics, Faculty of Science, Kyoto University,Kyoto 606-8502, Japan
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