Higher arithmetic degrees of dominant rational self-maps
Nguyen-Bac Dang Dragos Ghioca Fei Hu John Lesieutre Matthew Satriano









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#1http://www.ams.org/mathscinet-getitem?mr=#1MR#1











Institute for Mathematical Sciences, Stony Brook University, Stony Brook, NY 11794-3660, USA
nguyen-bac.dang@stonybrook.edu


Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC V6T 1Z2, Canada
dghioca@math.ubc.ca


Pacific Institute for the Mathematical Sciences, 2207 Main Mall, Vancouver, BC V6T 1Z4, Canada
hf@u.nus.edu


Department of Mathematics, The Pennsylvania State University, 204 McAllister Building, University Park, PA 16802, USA
jdl@psu.edu


Department of Pure Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada
msatrian@uwaterloo.ca



Suppose that f  X  X is a dominant rational self-map of a smooth projective variety defined over 𝐐.  Kawaguchi and Silverman conjectured that if P ∈ X(𝐐) is a point with well-defined forward orbit, then the growth rate of the height along the orbit exists, and coincides with the first dynamical degree λ_1(f) of f if the orbit of P is Zariski dense in X.

In this note, we extend the Kawaguchi–Silverman conjecture to the setting of orbits of higher-dimensional subvarieties of X.  We begin by defining a set of arithmetic degrees of f, independent of the choice of cycle, and we then develop the theory of arithmetic degrees in parallel to existing results for dynamical degrees.  We formulate several conjectures governing these higher arithmetic degrees, relating them to dynamical degrees.


[2010]
37P15, 
14G40, 
11G50, 
32H50. 





	D.G. and M.S. were partially supported by Discovery Grants from NSERC.
	F.H. was partially supported by a UBC-PIMS Postdoctoral Fellowship.
	J.L. was partially supported by NSF grants.


Higher arithmetic degrees of dominant rational self-maps
    Matthew Satriano
    December 30, 2023
========================================================










§ INTRODUCTION



Suppose that f  X  X is a rational self-map of a d-dimensional smooth projective variety defined over .  Let X_f() denote the set of rational points P for which the full forward orbit is well-defined.  Fix a Weil height function h_X  X() → and set h_X^+ max{ h_X, 1 }.
Following Kawaguchi–Silverman <cit.>, the arithmetic degree of a point P ∈ X_f() is defined to be the limit

    α_f(P) = lim_n →∞ h_X^+(f^n(P))^1/n,

supposing that it exists.  This is a measure of the “arithmetic complexity” of the orbit of P.

Another basic invariant of f is its set of dynamical degrees.  For 0 ⩽ k ⩽ d, we define

    λ_k(f) = lim_n →∞ ((f^n)^∗ H^k · H^d-k)^1/n,

where H is a fixed ample divisor on X.  The limit is known to exist (cf. <cit.>). Also, note that it is independent of the choice of the ample divisor H.  There is expected to be a close relationship between the arithmetic and dynamical degrees: a conjecture of Kawaguchi and Silverman <cit.> states that if P has Zariski dense f-orbit, then α_f(P) = λ_1(f).

Although the theory of dynamical degrees is comparably well-developed, many basic questions remain open about the arithmetic degree α_f(P) of a point P (see <cit.> for various results).  The  dynamical degrees of f satisfy certain inequalities, and there is a well-defined notion of relative dynamical degrees of a rational map preserving a fibration.
We refer to <cit.>, <cit.>, and references therein for a comprehensive exposition on dynamical degrees, topological and algebraic entropies.

Our aim in this note is to develop a theory of higher arithmetic degrees in parallel to the existing theory for dynamical degrees,  which was suggested by Kawaguchi-Silverman in <cit.>.
  We introduce the so-called k-th arithmetic degree of a self-map as a rough measure of the height growth of (k-1)-dimensional algebraic subvarieties of X, and show that a number of the basic properties of dynamical degrees extend to the arithmetic setting.  Although the arguments are along familiar lines, considerable technical difficulties arise in handling arithmetic intersection numbers. Our arithmetic degrees are independent of the choice of the subvarieties (unlike the α_f(P) of Kawaguchi–Silverman, which depends on P).
On the other hand, we also formulate analogous invariants depending on cycles and a higher-dimensional analogue of the Kawaguchi–Silverman conjecture.





 §.§ Examples



Before giving the precise definition, we consider some examples in the first new case: the growth rate of heights of hypersurfaces under self-maps of projective spaces.  In this case, the computations are quite concrete.  Suppose that f ^d ^d is a dominant rational map.  Fix coordinates on ^d.  Given a hypersurface V ⊂^d defined over ℚ, write out its defining equation in coordinates using coprime integer coefficients.
Let c_V be the largest of these coefficients (in absolute value) and define the height h(V) = (V) + log(c_V).

We will define the arithmetic degree of V by the limit α_d(f;V) = lim_n →∞ h(f^n(V))^1/n; the more general and precise definition appears later as Definition <ref>.

The conjecture of Kawaguchi and Silverman is that if P is a point with dense orbit, we have α_1(f;P) = λ_1(f).  Considering some simple two-dimensional examples shows the most obvious analogue of this conjecture is too optimistic: it may not be true that if V has dense orbit then 
α_k(f;V) = λ_k(f).


Let f ^2 ^2 be a rational self-map of ^2 defined in an affine chart by f(x,y) = (2y,xy).  The iterates are given by f^n(x,y) = ( x^F_n-1 (2y)^F_n, x^F_n (2y)^F_n+1/2 ),
where {F_n} = 0,1,1,2,3,5, … is the Fibonacci sequence.
The map f is birational, and its inverse is given by the formula f^-1(x,y) = (2y/x,x/2); the iterates of the inverse are given by 

    f^-n(x,y) = ( x^(-1)^n F_n+1/(2y)^(-1)^n F_n, (2y)^(-1)^n F_n-1/2x^(-1)^n F_n).

Clearly, the dynamical degrees of f are given by

    λ_0(f) = 1,   λ_1(f) = 1+√(5)/2,   λ_2(f) = 1.


Consider now the height growth of some subvarieties of ^2.  If P is a point with dense orbit, it is easy to see that α_1(f;P) = 1+√(5)/2.

If V is a general curve on ^2, its image under f^n can be computed by pulling back its defining equations via the formula for f^-n.  From the definition of the height of a hypersurface in ^d, we then see that α_2(f;V) = 1 + √(5)/2 as well.

In fact, all arithmetic degrees for this map can be computed as

    α_0(f) = 1,   α_1(f) = 1+√(5)/2,   α_2(f) = 1+√(5)/2,   α_3(f) = 1.




Consider next the map f ^2 ^2 which is given in coordinates by the formula f(x,y) = (x^2,y^2).  Then f^n(x,y) = (x^2^n,y^2^n).  This map has dynamical degrees

    λ_0(f) = 1,   λ_1(f) = 2,   λ_2(f) = 4.


If P is a point with dense orbit, then α_1(f;P) = 2.  Consider now the curve V ⊂^2 defined by x+y = 1.  Since the curve is parametrized by the map γ(t) = (t,1-t), the strict transform of V under f^n is parametrized by (t^2^n,(1-t)^2^n).
The arithmetic degrees this time are given by

    α_0(f) = 1,   α_1(f) = 2,   α_2(f) = 4,   α_3(f) = 8.




Observe that although in both examples we have α_1(f) = λ_1(f), it is not true in general that α_2(f) = λ_2(f).  These examples, together with a heuristic from the function field case (see also <cit.>), naturally lend themselves to the conjecture that α_k(f) = max{λ_k(f),λ_k-1(f) }.

Suppose that (C) is the function field of a smooth projective curve C defined over .  Let π→ C be a model of X, i.e., a smooth family for which the generic fiber is isomorphic to X.  Fix an ample divisor H_ on .
Given an algebraic k-cycle V ⊂ X, take  to denote the closure of V in .  The height of V is then defined to be the intersection number · H_^k+1, computed on . Now, f  X  X extends to a rational map f_.  

Suppose further that the closure  is a positive cycle (in a reasonable sense). Then we have

    α_k+1(f;V)    = lim_n →∞ h_X(f^n(V))^1/n = lim_n →∞ (f_^n() · H_^k+1)^1/n
       = lim_n →∞ ((f_^n)^∗ H_^k+1·)^1/n = λ_k+1(f_).

However, λ_k+1(f_) is related to the dynamical degrees of f itself by the product formula of Dinh–Nguyen <cit.>.  Since f_ preserves a fibration over the curve C, we have

    α_k+1(f;V) = λ_k+1(f_) = max{λ_k+1(f), λ_k(f) }.







 §.§ Definition of higher arithmetic degrees


We turn at last to the general definitions.  Let X be a smooth projective variety of dimension d defined  over . 
We choose a regular model π→ of X, i.e., a regular projective scheme over  such that π is flat and the generic fiber _ is isomorphic to X.
In particular,  is an arithmetic variety of relative dimension d over .
We refer to <cit.> for a quick tour to the Arakelov intersection theory and various arithmetic positivity properties.
Fix an arithmetically ample line bundle = (, ·) on .
The corresponding ample line bundle _ on X will be simply denoted by H.
For an arithmetic k-cycle Z = (Z, g) ∈Z_D^d+1-k() on , the Faltings height h_(Z) of Z with respect to  is defined by the Arakelov intersection

    h_ (Z) ( Z ·c_1()^k),

where c_1() ∈_D^1() is the first arithmetic Chern class of , and

    _D^d+1()

is the arithmetic degree map.
When k≥ 1, let _(Z) denote the degree of Z_ with respect to the ample line bundle H on X.
Namely

    _(Z)  (Z_· H^k-1).


Let  be a dominant rational self-map of , and f the corresponding dominant rational self-map of X.
Denote by π_1, π_2 Γ_→ the projection from the normalization of the graph of  in ×_ onto the first and second component respectively. 

Recall that given an integer 0⩽ k⩽ d, the k-th (algebraic) degree of f with respect to H is defined as follows:

    _k(f)  ( π_2^* H^k·π_1^* H^d-k ).

Then the k-th dynamical degree of f is then interpreted as the growth rate of the k-th degree of f with respect to any ample divisor, i.e.,

    λ_k(f) = lim_n→∞_k(f^n)^1/n.

The existence of the limit is a non-trivial result due to <cit.>.
We can also consider analogous versions of these two quantities measuring the degree growth of an algebraic subvariety V of dimension k. Namely, we define the k-th degree of f along V as

    _k(f;V)  ( π_2^* H^k·π_1^* V)

and the k-th dynamical degree of f along V by

    λ_k(f;V) lim sup_n →∞_k(f^n;V)^1/n.

Note that we are taking lim sup since in general it is not known whether the limit actually exists.
Similar phenomena also happen in the definition of arithmetic degrees below.



For any integer 0⩽ k ⩽ d+1, let _k(f) denote the k-th degree of f with respect to , i.e.,

    _k(f) ( c_1(π_2^*)^k ·c_1(π_1^*)^d+1-k).

Then the k-th arithmetic degree of f is defined as follows:

    α_k(f) lim sup_n→∞_k(f^n)^1/n.


Similarly, let  denote the closure of the above V in .
We define

    _k+1(f; V) ( c_1(π_2^* )^k+1·π_1^*),

and

    α_k+1(f; V) lim sup_n→∞_k+1(f^n; V)^1/n.



When f is a morphism, it is not necessary to take a resolution when defining _k and _k.
In this case, we have

    _k(f; V) = π_2^∗ H^k ·π_1^∗ V = H^k · f_*V = _k(f_*V),  and


    _k+1(f; V) = ( c_1(π_2^* )^k+1·π_1^*) = ( c_1()^k+1·_*) = h_(f_*V).

Thus λ_k(f; V) (respectively, α_k+1(f; V)) measure the degree (respectively, the height growth of the iterates) of an algebraic subvariety.




Here f_∗ V denotes the cycle-theoretic pushforward of V, not just its strict transform; if f is ramified along V, this class V multiplied by the degree of the ramification.  Subtleties can arise when some iterate(s) of f is ramified along V, and when infinitely many different iterates are ramified along V the ramification is reflected in the arithmetic degree of the cycle: f^n_∗ V is equal to the class of the strict transform f^n(V) with a factor of the ramification degree.  Consider for contrast the following examples.


Define f ^2 ^2 by setting f(x,y) = (x^2,y), and let V be the curve defined by y = 0.
Then V is invariant under f and f|_V is the squaring map on ^1.  As a result, f_*^n(V) = 2^n V, and we have α_2(f;V) = 2.

On the other hand, let W be the curve defined by x=y; this curve is equivalent to V as a cycle and its iterates are also disjoint from the indeterminacy point [0:1:0].  Then f^n(W) is defined by the equation x = y^2^n, so that h(f^n_∗(W)) = 2^n.  Thus we find that α_2(f;W) = 2 as well.


It is also possible that f is ramified along a curve only a finite number of times, in which the extra factor corresponding to ramification disappears in the limit of h(f^n_∗ (V))^1/n.

It is also conceivable that f could ramify along V infinitely many times, but with such infrequency that the ramification is not reflected in the arithmetic degree.  This is impossible, assuming the following special case of a generalization of the dynamical Mordell–Lang conjecture (see <cit.> and also <cit.>).


Suppose that f  X  X is a rational map and that V ⊂ X is a subvariety with well-defined forward orbit.  Is

    R(f,V) = {n : f^n(V) ⊆Ram(f) }

the union of a finite set and finitely many arithmetic progressions?




 §.§ Conjectures and results


As we have mentioned before, it is known that the limit (<ref>) defining λ_k(f) indeed exists; in the other three cases (<ref>)-(<ref>), this is unknown and it is necessary to take the lim sup in the definition.  We will show later that the limit defining α_1(f) actually exists and is equal to λ_1(f).
We expect that the limit exists for the other α_k(f) as well.  It is a more subtle question whether the limits exist for the invariants which depend on a particular cycle and not just the map f.  In the case of α_1(f;P), the existence of the limit is one part of Kawaguchi–Silverman conjecture. Even in the case of λ_k(f;V) (a question with no arithmetic content), this seems to be an open question.


Let f  X  X be a dominant rational map, and let V be a subvariety of dimension k.  Then the limits defining λ_k(f;V), α_k(f), and α_k+1(f;V) exist and are independent of the choices of H and  as well as the model .


We also expect that α_k+1(f) provides an upper bound on the growth rate of the heights of algebraic subvarieties of dimension k.



Let f  X  X be a dominant rational map, and let V be a subvariety of dimension k.  Then α_k+1(f;V) ⩽α_k+1(f).  If V has Zariski dense orbit, then α_k+1(f;V) = α_k+1(f).




Although we are not able to prove that the lim sup is actually a limit independent of any choice when k⩾ 2,
recent work due to Ikoma <cit.> based on the arithmetic Hodge index theorem of Yuan–Zhang <cit.> shows that the arithmetic intersection shares many similarities with the algebraic intersection.
This allows us to extend a number of basic properties of dynamical degrees to the arithmetic setting.
For example, it is shown that the Khovanski–Teissier inequalities hold in arithmetic intersection theory, and the log concavity of the arithmetic degrees follows.  We list some basic properties of the arithmetic degrees in the next theorem.



Using notation as above, the following assertions hold.


  * The sequence k ↦α_k(f) is log-concave.

  * If f is birational, then α_k(f) = α_d+1-k(f^-1).






We also formulate an analogue of the Kawaguchi–Silverman conjecture, which predicts that the height growth of cycles is controlled by dynamical degrees.  In this case, the natural prediction derives from the case of function fields (see Section <ref>), in which it reduces to the familiar Dinh–Nguyen product formula for dynamical degrees (cf. <cit.>).



Let f be a dominant rational self-map of X.  Then for any integer 1 ⩽ k ⩽ d, one has

    α_k(f) = max{λ_k(f), λ_k-1(f) }.



We prove this conjecture in the case that X = 2 and f is birational (see Theorem <ref>) and in the case that f  X → X is a polarized endomorphism (see Theorem <ref>).



Our next result provides an inequality of Conjecture <ref>.



Let f be a dominant rational self-map of X.  Then for any integer 1 ⩽ k ⩽ d, one has

    α_k(f) ⩾max{λ_k(f), λ_k-1(f) }.


 
Observe that combining Conjectures <ref> and <ref> provides a precise prediction about the growth rate of heights of iterates f^n(V) for V with dense orbit:

    α_k+1(f;V) = α_k+1(f) = max{λ_k+1(f),λ_k(f) }.

We have seen in Section <ref> that either of these two numbers can appear, even in the case that V is a curve on a surface.  These conjectures should be understood as the higher-dimensional formulation of the Kawaguchi–Silverman conjecture.

The proof of Theorem <ref> relies on an arithmetic Bertini argument (see Lemma <ref>) which allows us to relate the arithmetic degree with a height of a given particular cycle. Then one applies an inequality due to Faltings <cit.> to bound below the height of the cycles by its algebraic degree. 


In the particular case where k=1, we show that the sequence _1(f^n) is submultiplicative.




Let X, Y, Z be smooth projective varieties defined over  of dimension d.
Choose the regular models ,  and  of X,Y and Z respectively.
Fix three arithmetically ample line bundles _, _, and _ on ,  and  respectively.  
Then there exists a constant C>0 such that for any dominant rational maps f  X  Y and g  Y  Z, one has

    _1( g∘ f) ⩽ C _1(f) _1(g),

where the intersection numbers are taken with respect to the chosen arithmetically ample line bundles on each model.


The above result relies heavily on Yuan's work <cit.> on the properties of the arithmetic volume (see also <cit.>). From Yuan's arithmetic volume estimates, one deduces an effective way to obtain an arithmetically pseudo-effective line bundle. 
The above theorem yields the following consequence.



The first arithmetic degree α_1(f) of f is equal to the limit:

    α_1(f) = lim_n →∞_1(f^n)^1/n,

it is a finite number and independent of the choices of the regular model  of X, the birational model of X, or the arithmetically ample line bundle on . 


More precisely, Theorem <ref> shows that the growth rate of the sequence _1(f^n) is also a birational invariant.
In this particular case, the relationship between the arithmetic degree and the dynamical degree is  more constrained.




Let f be a dominant rational self-map of X. Then one has

    α_1(f)=λ_1(f).



This last result together with Theorem <ref> implies that every arithmetic degree α_k(f) is finite, precisely:

    α_k(f) ⩽λ_1(f)^k.


Let us explain how one can understand the above theorem.
Since Theorem <ref> proves that λ_1(f) ⩽α_1(f), the proof of this last result amounts in proving the converse inequality α_1(f) ⩽λ_1(f). 
Using an arithmetic Bertini argument (see Section <ref>) and a generalization of Siu's analytic estimates in higher codimension due to Xiao <cit.>, we prove a more general statement (see Theorem <ref>), namely that for all integer n, there exists an algebraic cycle Z of dimension k-1 on X such that

    _k(f^n) ⩽ h_(f^n Z) + C _k(f^n)

for a positive constant C>0 which is independent of p and Z.
In the case where k=1, the degree of f^n with respect to  is controlled by the height of a point and the algebraic degree of f^n.
We finally conclude using the fact that the height of the images of a rational point by f^n is controlled by the first dynamical degree of f (see <cit.>).



 §.§ Acknowledgements


The first, fourth and fifth authors thank Yohsuke Matsuzawa and Joe Silverman for helpful discussions. The first author also thanks Charles Favre, Thomas Gauthier and Mattias Jonsson for their suggestions and remarks. Part of this work was done during the Simons Symposium: Algebraic, Complex and Arithmetic Dynamics in May 2019 at Schloss Elmau. We are grateful to the Simons foundation for its generous support.



§ ON ARITHMETIC DEGREES OF CODIMENSION K





 §.§ Proof of Theorem <ref>


Fix a model  of X and an arithmetically ample line bundle  on . We take f  X  X a dominant rational self-map of X.
By abuse of notation, we also denote by f the dominant rational map on the regular model .
Let π_1n,π_2n be the projections from the normalization of the graph of f^n in ×_ to the first and second component respectively.
By <cit.> applied the arithmetic π_1n^*, π_2n^*, we have:

    _k(f^n)^2 = ( c_1(π_2n^*)^k ·c_1(π_1n^*)^d+1-k)^2 ⩾


    ( c_1(π_2n^*)^k+1·c_1(π_1n^*)^d-k) ( c_1(π_2n^*)^k-1·c_1(π_1n^*)^d+2-k).

We thus obtain that

    _k(f^n)^2 ⩾_k+1(f^n) _k-1(f^n).

Taking the n-th root as n tends to infinity yields that

    α_k(f)^2 ⩾α_k-1(f) α_k+1(f)

and the sequence k ↦α_k(f) is locally log-concave at each point.
It is thus globally log-concave, as required.

The second assertion immediately follows from the definition of α_k(f), with the roles of π_1 and π_2 reversed.





 §.§ A technical lemma




In this section, we prove a technical lemma which relates the two different k-th degrees of a rational self-map f as well as the height of certain points on the f-orbit. To do so, let us introduce a few notations.

We fix a dominant rational map f  X  X defined over .
Denote by π_1, π_2 the projections from a generically smooth birational model Γ_f of the graph of f in × onto the first and the second factor respectively. 
We fix an arithmetically ample line bundle  on  and an integer k ⩽ d.
Take a sequence of sections s_1,ϵ, … , s_d, ϵ of an arithmetically ample line bundle such that (s_i,ϵ) = _ϵ is arbitrarily close to π_1^* and whose norm is uniformly bounded by 1. We denote by ω_1,ϵ the smooth (1,1)-forms associated to the class (s_i,ϵ).
For an integer 1 ⩽ i ⩽ d+1-k, denote by Z_i,ϵ the smooth subvarieties given by:

    Z_i, ϵ = (s_i,ϵ) ·…·(s_d+1-k, ϵ).

Observe that the support of the cycles Z_i,ϵ forms an increasing sequence of closed subvarieties of Γ_f where Z_d+1-k,ϵ= (s_d+1-k,ϵ).
For the next proof, we will also set Z_d+2-k,ϵΓ_f.  
   
  One has:

    _k(f) = lim_ϵ→ 0(c_1(π_2^* )^k · Z_1,ϵ) -  ∑_j=2^d+2-k∫_Z_j,ϵ()log(||s_j-1,ϵ||) π_2^* ω_X^k ∧ω_1,ϵ^j-2.



Let us write _k(f) as follows:

    _k(f) = (c_1(π_2^* )^k ·(s_1,ϵ) ·…·(s_d+1-k,ϵ) ).

By <cit.> applied to the arithmetic divisor (s_d+1-k), we have:

    _k(f)    = lim_ϵ→ 0(c_1(π_2^* )^k ·(s_1,ϵ) ·…·(s_d-k,ϵ) ·(s_d+1 - k ,ϵ) ) 
          - ∫_Γ_f()log(|| s_d+1-k,ϵ||) π_2^* ω_X^k ∧ω_1,ϵ^d-k
        = lim_ϵ→ 0(c_1(π_2^* )^k ·(s_1,ϵ) ·…·(s_d-k,ϵ) · Z_d+1-k,ϵ) 
          - ∫_Γ_f()log(|| s_d+1-k,ϵ||) π_2^* ω_X^k ∧ω_1,ϵ^d-k.

We apply inductively <cit.> as follows:

    ( c_1(π_2^* )^k ·c_1(_ϵ)^d+1-k· [X] ) =


    ( c_1(π_2^* )^k ·c_1(_ϵ)^d-k· Z_d+1-k,ϵ) - ∫_X()log(|| s_d+1-k,ϵ||) π_2^* ω_X^k ∧ω_1,ϵ^d-k,


    ( c_1(π_2^* )^k ·c_1(_ϵ)^d-k· Z_d+1-k,ϵ) =


    ( c_1(π_2^* )^k ·c_1(_ϵ)^d-k-1· Z_d-k,ϵ) - ∫_Z_d+1-k,ϵ()log(|| s_d-k,ϵ||) π_2^* ω_X^k ∧ω_1,ϵ^d-k-1,


    ( c_1(π_2^* )^k ·c_1(_ϵ)^d-k-1· Z_d-k,ϵ) =


    ( c_1(π_2^* )^k ·c_1(_ϵ)^d-k-2· Z_d-k-1,ϵ) - ∫_Z_d-k,ϵ()log(|| s_d-k-1,ϵ||) π_2^* ω_X^k ∧ω_1,ϵ^d-k-2,


    ⋮


    ( c_1(π_2^* )^k ·c_1(_ϵ)^2· Z_3,ϵ) =


    ( c_1(π_2^* )^k ·c_1(_ϵ) · Z_2,ϵ) -∫_Z_3,ϵ()log(|| s_2,ϵ||) π_2^* ω_X^k ∧ω_1,ϵ,


    ( c_1(π_2^* )^k ·c_1(_ϵ) · Z_2,ϵ) = ( c_1(π_2^* )^k · Z_1,ϵ) -∫_Z_2,ϵ()log(|| s_1,ϵ||) π_2^* ω_X^k.


From the above sequence of equalities, we deduce that

    _k(f) = lim_ϵ→ 0(c_1(π_2^* )^k · Z_1,ϵ) - ∑_j=2^d+2-k∫_Z_j,ϵ()log(||s_j-1,ϵ||) π_2^* ω_X^k ∧ω_1,ϵ^j-2 .

This finishes the proof of the lemma.





 §.§ Proof of Theorem <ref>



We shall use the following inequality due to Faltings <cit.>:

There exists a constant C>0 depending only on the arithmetically ample line bundle  on  (depending on the supremum of the norm of a section of the corresponding line bundle at infinite places) such that:

    h_ (Z) ⩾ C ( H^k · Z),

for any effective  cycle Z ∈ Z^d-k(). 
By Lemma <ref>, for any integer p, we have:

    _k(f^p) = lim_ϵ→ 0 h_ (f^p π_1_*(Z_1,ϵ)) - ∑_j=2^d+2-k∫_Z_j,ϵ()log(||s_j-1,ϵ||) π_2^* ω_X^k ∧ω_1,ϵ^j-2,

where Z_1,ϵ and the s_i,ϵ are defined in Section <ref>. 
Observe that this cycle depends on each iterate p since it a cycle on the graph of f^p.
Since the supremum of the norm of the sections s_j,ϵ is bounded by 1, all the terms in the integral are non-negative.
We thus have

    _k(f^p) ⩾ h_ (f^p π_1_* (Z_1,ϵ)).

Using the Faltings inequality, we thus deduce _k(f^p) ⩾ C (H^k-1· Z_1,ϵ). 
Taking the p-th root as p tends to infinity finally gives:

    α_k(f) ⩾λ_k-1(f).

For the other inequality, we observe that the roles played by the line bundles π_1^* and π_2^* are symmetric and the technical lemma also proves that _k(f^p) ⩾ C h_ (f^-p Z) for an appropriate complete intersection cycle of dimension k and we conclude that α_k(f) ⩾λ_k(f). 
We have thus proven that α_k(f) ⩾max{λ_k(f), λ_k-1(f) }, as required.


We have conjectured that in fact the inequality of Theorem <ref> is always an equality (see Conjecture <ref>).
There are two cases in which we can prove this conjecture.



If f  X  X is a birational self-map of a surface, then

    α_k(f) = max{λ_k(f),λ_k-1(f) }

for all 1 ⩽ k ⩽ d.  



It is always true that α_0(f) = 1, while α_1(f) = λ_1(f) for any map according to Theorem <ref>.  We have

    α_2(f) = α_1(f^-1) = λ_1(f^-1) = λ_1(f),

and since λ_2(f) = 1 as f is birational, we have α_2(f) = max{λ_1(f),λ_2(f) } by definition.  


Recall that a surjective self-morphism f  X → X is polarized, if there exists an ample line bundle L on X such that f^*L ≃ L^q for some integer q>1.



If f  X → X is a polarized endomorphism with respect to the ample line bundle H=_ on X, then

    α_k(f) = max{λ_k(f),λ_k-1(f) }

for all 1 ⩽ k ⩽ d. 



We fix an isomorphism ϕ^q ≃ f^* on the arithmetic model  of X,
where = (, || · ||) is a fixed arithmetically ample line bundle on .
By <cit.>, there exists a unique metric ·_0 on  satisfying:

    || · ||_0 = (ϕ^* f^* || · ||_0)^1/q.

This metric is obtained from the initial metric || · || on  using Tate's limiting argument. As the initial  is arithmetically ample, so is the new arithmetic line bundle _0  (, || ·||_0).

Let us compute the k-th degree of f^n with respect to the choice of the arithmetically ample line bundle _0:

    _k(f^n) = ( c_1((f^n)^* _0)^k ·c_1(_0)^d+1-k) = q^kn( c_1(_0)^d+1).

This proves that α_k(f) = q^k.
On the other hand, λ_k(f) = q^k for any k.
We hence conclude that

    α_k(f) = max{λ_k(f), λ_k-1(f)},

as required.




 §.§ Upper bound for the arithmetic degree

We now give an upper bound for the k-th degree in the following result.
To do so, we will need the following generalization of Siu's analytic estimates in higher codimension due to Xiao <cit.>.

 Take ω_1, ω_2 two big nef (1,1) classes on a smooth compact Kähler manifold X of dimension d. 
Then for any integer k ⩽ n,  the (k,k)-form

    dk∫_X ω_1^k ∧ω_2^d-k∫_X ω_2^dω_2^k - ω_1^k

is represented by a closed positive current.


We now state our next result, which gives an upper bound of the k-th degree of f^p. 



Let f  X  X be a dominant rational map on a projective variety defined over . 
There exists a constant C>0 such that for any integers k⩽ d+1, for any integer p, there exists a cycle Z of dimension k-1   whose support is not contained in  the indeterminacy locus of f^p such that:

    _k(f^p) ⩽ h_(f^p Z) + C _k(f^p).




We adopt the same notation of Section <ref>. 
Fix an integer p, we consider a generically smooth birational model Γ of the graph of f^p.
By Lemma <ref>, we have:

    _k(f) = lim_ϵ→ 0( c_1(π_2^* )^k · Z_1,ϵ) - ∑_j=2^d+2-k∫_Z_j,ϵ()log(||s_j-1,ϵ||) π_2^* ω_X^k ∧ω_1,ϵ^j-2.

For each integer 2 ⩽ j ⩽ d+2-k, Xiao's inequality implies that the (k,k)-form

    j+k-2k∫_Z_j,ϵ()π_2^* ω_X^k ∧π_1^* ω_X^j-2∫_Z_j,ϵ()π_1^* ω_X^k+j-2π_1^* ω_X^k - π_2^*ω_X^k

is represented by a closed positive current on Z_j,ϵ().
Since -log(||s_j-1,ϵ||) ω_1,ϵ^j-2 is a positive (j-2,j-2)-form on Z_j,ϵ(), we deduce that

    -∫_Z_j,ϵ()log(||s_j-1,ϵ||) π_2^* ω_X^k ∧ω_1,ϵ^j-2
    ⩽    -C ∫_Z_j,ϵ()π_2^* ω_X^k ∧π_1^* ω_X^j-2∫_Z_j,ϵ()π_1^* ω_X^k+j-2∫_Z_j,ϵ()log(||s_j-1,ϵ||)π_1^* ω_X^k ∧π_1^* ω_1,ϵ^j-2,

where C = (j+k-2)!k! (j-2)!.

Observe that the term on the right hand side of the above inequality is bounded by 
C' _k(f^p) as ϵ goes to zero where C'  is a positive constant bounded above by C ()/ (H). 
We thus obtain:

    _k(f^p) ⩽ h_(f^p π_1_*(Z_1,ϵ)) + C' _k(f^p),

as required.






§ ON THE FIRST ARITHMETIC DEGREE





 §.§ Proof of Theorem <ref>



Our proof follows closely the recent algebraic approach for the existence of dynamical degrees for dominant rational self-maps due to Dang <cit.>.

We shall use the following inequality on arithmetic divisors  due to Yuan <cit.>.


Let ,  and  be three Hermitian line bundles over an arithmetic variety  of relative dimension d. Assume that  and  are arithmetically ample. Then

    (  + )^1/(d+1)≥()^1/(d+1) + ()^1/(d+1).



An important consequence of the above fact (see <cit.>) is that the class - is arithmetically big if

    ( c_1()^d+1) > (d+1) ( c_1()^d·c_1() ).

This inequality is the key ingredient to prove the submultiplicativity of the sequence (_1(f^p)). 


Fix X, Y , Z three smooth projective varieties of dimension d defined over .
Let f  X Y and g  Y Z be two dominant rational maps.
Fix three regular models , and  of X,Y and Z respectively and three arithmetically ample divisors _, _ and _ respectively.
Take Γ_f,Γ_g the normalization of the graphs of f and g respectively.
Consider also the normalization Γ of the graph of the rational map Γ_f Γ_g induced by f.
We also denote by π_1, π_2 the projection of Γ_f onto the  and , u,v the projection of Γ onto Γ_f and Γ_g, and by π_3, π_4 the projections of Γ_g onto  and  respectively. 
We thus obtain the following diagram.

    Γ[ld]_u [rd]^v        
       Γ_f@–>[rr] [ld]_π_1[rd]^π_2      Γ_g[ld]_π_3[rd]^π_4   
    @–>[rr]_f      @–>[rr]_g      .


We now apply Yuan's estimates to 
= u^* π_2^* _ and = v^* π_4^* _, the class

    (d+1) ( c_1 (u^* π_2^* _)^d ·c_1 (v^* π_4^* _) )( c_1 (u^* π_2^* _)^d+1) u^* π_2^* _ - v^* π_4^* _

is pseudo-effective.
Thus the intersection with the arithmetically nef class u^* π_1^* _ yields that

    _1(g∘ f) ⩽ (d+1) ( c_1 (u^* π_2^* _)^d ·c_1 (v^* π_4^* _) )( c_1 (u^* π_2^* _)^d+1)_1(f).

Since u^* π_2^* = v^* π_3^*, we have:

    ( c_1 (u^* π_2^* _)^d ·c_1 (v^* π_4^* _) )( c_1 (u^* π_2^* _)^d+1) = ( c_1 (v^* π_3^* _)^d ·c_1 (v^* π_4^* _) )( c_1 (v^* π_3^* _)^d+1).

Using the projection formula (cf. <cit.>), we obtain:

    ( c_1 (v^* π_3^* _)^d ·c_1 (v^* π_4^* _) )( c_1 (v^* π_3^* _)^d+1) = ( c_1 (π_3^* _)^d ·c_1 (π_4^* _) )( c_1 (π_3^* _)^d+1) = _1(g)( c_1(_)^d+1).

The above equalities together with the inequality (<ref>) yields:

    _1(g∘ f) ⩽ C _1(f) _1(g),

where C = (d+1) / ( c_1(_)^d+1), and the theorem is proved.





 §.§ Proof of Theorem <ref>


By Theorem <ref> applied to k=1, we have

    α_1(f) ⩾max{λ_1(f), λ_0(f) } = λ_1(f).

For the converse inequality, we apply Theorem <ref>.  There is a constant C>0 such that for any integer p, there exist some points x_i such that ∑ a_i [x_i] = π_1_* (Z_1,ϵ) ∈ Z^n(X) and

    _1(f^p) ⩽∑ a_i h_(f^p x_i) + C _1(f^p).

By Matsuzawa's theorem (cf. <cit.>), the growth rate of the height h_(f^p x_i) of any point x_i is bounded above by λ_1(f), so we obtain:

    α_1(f) ⩽λ_1(f).

We finally conclude that α_1(f) = λ_1(f), as required.









amsalpha






