Algebraic Dynamics, Canonical Heights and Arakelov
Geometry
Xinyi Yuan
Abstract. This expository article introduces some recent applications of Arakelov geometry
to algebraic dynamics. We present two results in algebraic dynamics, and introduce their
proofs assuming two results in Arakelov geometry.
Contents 1. Introduction 1
2. Algebraic theory of algebraic dynamics 2 3. Analytic metrics and measures 6 4.
Analytic theory of algebraic dynamics 10 5. Canonical height on algebraic
dynamical systems 12 6. Equidistribution of small points 15 7. Proof of the rigidity
17 8. Positivity in arithmetic intersection theory 20 9. Adelic line bundles 26 10.
Proof of the equidistribution 30 References 34
1. Introduction Algebraic dynamics studies iterations
of algebraic endomorphisms on algebraic
varieties. To clarify the object of this expository article, we make an early de nition.
Definition 1.1. A polarized algebraic dynamical system over a eld Kconsists
of a triple (X;f;L) where: Xis a projective variety over K, f: X!Xis an algebraic
morphism over K, Lis an ample line bundle on Xpolarizing fin the sense that ffor
some integer q>1. L = Lq
1991 Mathematics Subject Classi cation. Primary 37-02, 14G40; Secondary 14C20, 32Q25. Key
words and phrases. algebraic dynamical system, ampleness, Arakelov geometry, arithmetic intersection theory, hermitian line bundle, bigness, Calabi{Yau theorem, canonical height,
equidistribution, height, equilibrium measure, preperiodic points, volume of line bundle.
The author is supported by a fellowship of the Clay Mathematics Institute.
1
2 XINYI YUAN
In this article, we may abbreviate \polarized algebraic dynamical system" as
\algebraic dynamical system" or \dynamical system".
M
The canonical height hf = hL;f
dim X
a(x)
= fb
0
M
M;f M;LM
M
a
;
b
a(x)
= fb
on X(K), a function naturally de ned using
Arakelov geometry, describes concisely many important properties of an algebraic
dynamical system. For example, preperiodic points are exactly the points of canonical
height zero. Note that the canonical height can be de ned even Kis not a global eld
following the work of Moriwaki and Yuan{Zhang. But we will mainly focus on the
number eld case in this article.
Hence, it is possible to study algebraic dynamics using Arakelov geometry. The
aim of this article is to introduce the relation between these two seemingly di erent
subjects. To illustrate the idea, we are going to introduce mainly the following four
theorems:
(1) Rigidity of preperiodic points (Theorem 2.9), (2) Equidistribution of small
points (Theorem 6.2), (3) A non-archimedean Calabi{Yau theorem, uniqueness
part (Theorem 3.3), (4) An arithmetic bigness theorem (Theorem 8.7).
The rst two results lie in algebraic dynamics, and the last two lie in Arakelov
geometry. The logic order of the implications is:
(2) + (3) =)(1); (4) =)(2): For more details on the arithmetic of
algebraic dynamics, we refer to [Zh06,
Sil07, BR10, CL10a]. For introductions to complex dynamics, we refer to [Mil06,
Sib99, DS10].
Acknowledgements. The article is written for the International Congress of Chinese Mathematicians 2010
held in Beijing. The author would like to thank Matt Baker, Yifeng Liu, Nessim Sibony, Ye Tian, Song Wang
and Shou-wu Zhang for many important communications. The author is very grateful for the hospitality of
the Morningside Center of Mathematics, where the major part of this article was written.
2. Algebraic theory of algebraic dynamics Here we make some basic
de nitions for an algebraic dynamical system, and
state the rigidity of preperiodic points. 2.1. Algebraic dynamical systems. Recall that
an algebraic dynamical system is a triple (X;f;L) over a eld Kas in De nition 1.1. The ampleness of L implies that
the morphism fis nite, and the degree is q. Denote by Kthe algebraic closure of Kas
always.
Definition 2.1. A point x2X(K) is said to be preperiodic if there exist two integers
a>b 0 such that f(x). It is said to be periodic if we can further take b= 0 in the equality.
Denote by Prep(f) (resp. Per(f)) the set of
preperiodic (resp. periodic) points of X( K). Here we take the convention that f(x) =
x. Remark 2.2. The sets Prep(f) and Per(f) are stable under base change.
Namely, for any eld extension Mof K, we have Prep(f ) = Prep(f) and Per(f
represents the dynamical system (X
of points x2X(K) with f
) = Per(f). Here f) over M. More generally, the set Prep(f)(x) is stable under base
change. We leave it as an exercise.
ALGEBRAIC DYNAMICS, CANONICAL HEIGHTS AND ARAKELOV GEOMETRY 3
The following basic theorem is due to Fakhruddin [Fak03]. In the case X= P n
and char(K) = 0, it is a consequence of Theorem 4.8, an equidistribution theorem due
to Briend{Duval [BD99].
Theorem 2.3 (Fakhruddin). The set Per(f) is Zariski dense in X. It implies the
more elementary result that Prep(f) is Zariski dense in X. Now
we give some examples of dynamical systems. Example 2.4. Abelian variety. Let X=
Abe an abelian variety over K,
L = L, which implies [2] L f= [2] be the multiplication by 2, and Lbe a symmetric and
ample line bundle on A. Recall that Lis called symmetric if [1]. It is obvious that Lis not
unique. It is easy to see that Prep(f) = A(K)tor= L4is exactly the set of torsion points.
Example 2.5. Projective space. Let X = Pn, and f : Pbe any nite morphism of
degree greater than one. Then f = (f0;f1n!P; ;fn), where f0;f1; ;fnnare homogeneous
polynomials of degree q >1 without non-trivial common zeros. The polarization is
automatically given by L= O(1) since f O(1) = O(q). We usually omit mentioning the
polarization in this case. The set Prep(f) could be very complicated in general,
except for those related to the square map, the Latt es map, and the Chebyshev
polynomials introduced below.
2
and f(x0; ;xn) = (x2 0
n; ;x; ;xn
0
Example 2.6. Square map. Let X= Pn
1
j
1
t or
1
d-th Chebyshev polynomial Td
d
d
+
Then Td
1
; 8t:
d1
d
d
t+ t
t
d
q=
L
). Set
L= O(1) since fO(1) = O(2). Then Prep(f) is the set of points (x) with each coordinate
xequal to zero or a root of unity. Example 2.7. Latt es map. Let Ebe an elliptic curve
over K, and : E!Pbe any nite separable morphism. For example, can be the natural
double cover rami ed exactly at the 2-torsion points. Then any endomorphism of
Edescends to an endomorphism fon P, and thus gives a dynamical system on Pif the
degree is greater than 1. In that case, Prep(f) = (E(K)). Example 2.8. Chebyshev
polynomial. Let dbe a positive integer. The
be the unique polynomial satisfying cos(d ) = T(cos( )); 8 :
Equivalently, Tis the unique polynomial satisfying t2 = T2
has degree d, integer coe cients and leading term 2. For each d>1, Td: P1!P1gives an
algebraic dynamical system. It is easy to see that the iteration T n d= Tdn , and Prep(Td)
= fcos(r ) : r2Qg: 2.2. Rigidity of preperiodic points. Let Xbe a projective variety over a
L eld K, and Lbe an ample line bundle on X. Denote H(L) = ff: X!Xjffor some
integer q>1g: Obviously H(L) is a semigroup in the sense that it is closed under
compositionof morphisms. For any f 2H(L), the triple (X;f;L) gives a polarized
algebraic dynamical system.
4 XINYI YUAN
Theorem 2.9 ([BDM09], [YZ10a]). Let Xbe a projective variety over any eld K,
and Lbe an ample line bundle on X. For any f;g2H(L), the following are equivalent:
(a) Prep(f) = Prep(g); (b) gPrep(f) Prep(f); (c)
Prep(f) \Prep(g) is Zariski dense in X.
1The theorem is proved independently by Baker{DeMarco [BDM09] in the case X = P,
and by Yuan{Zhang [YZ10a] in the general case. The proofs of these two papers in
the number eld case are actually the same, but the extensions to arbitrary elds are
quite di erent. See x7.2 for some details on the di erence. The major extra work for
the high-dimensional case in [YZ10a] is the Calabi{Yau theorem, which is essentially
trivial in the one-dimensional case.
Remark 2.10. Assuming that K is a number eld, the following are some
previously known results:
(1) Mimar [Mim97] proved the theorem in the case X= P1. (2)
Kawaguchi{Silverman [KS07] obtains the result in the case that f is a
square map on Pn. They also obtain similar results on Latt es maps. Their
treatment gives explicit classi cation of g.
A result obtained in the proof of Theorem 2.9 is the following analytic version:
Theorem 2.11 ([YZ10a]). Let Kbe either C or Cfor some prime number p. Let X be a
projective variety over K, Lbe an ample line bundle on X, and f;g2H(L) be two
polarized endomorphisms with Prep(f) = Prep(g). Then the equilibrium
measures f= gon Xanp. In particular, the Julia sets of fand gare the same in the
complex case.
denotes the equilibrium measure of (X;f;L) on Xan
Here f C(f) = fg2H(L) jg f= f gg:
. See x4.2 for the
de nition. For the notion of Julia sets, we refer to [DS10].
2.3. Morphisms with the same preperiodic points. The conditions of Theorem 2.9
are actually equivalent to the equality of the canonical heights of those two dynamical
systems. We will prove that in the number eld case. Hence, two endomorphisms
satisfying the theorem have the same arithmetic properties. It leads us to the
following result:
Corollary 2.12. Let (X;f;L) be a dynamical system over any eld K. Then the set
H(f) = fg2H(L) : Prep(g) = Prep(f)g is a
semigroup, i.e., g h2H(f) for any g;h2H(f).
Proof. By Theorem 2.9, we can write: H(f) = fg2H(L)
jgPrep(f) Prep(f)g:
Then it is easy to see that H(f) is a semigroup. Example 2.13. Centralizer. For
any (X;f;L), denote
ALGEBRAIC DYNAMICS, CANONICAL HEIGHTS AND ARAKELOV GEOMETRY 5
Then C(f) is a sub-semigroup of H(f). Take any x2Prep(f), and we are going to
prove x2Prep(g). By de nition, there exist a>b 0 such that xlies in
a(y) = f b
Prep(f)a;b = fy2X(K) : f
a;b
c
a;b
c
d
0
n
0
0
nC
tor
n xr n
xr 0
0
0
0
n+1
j
0
1C
d
n)
= (0
2
n+1
n
(with d>1) on P1 C
e
n;
d
n
2;f3
1
(y)g: The commutativity
implies g(x) 2Prep(f)for any integer c 0. It is easy to see that Prep(f)is a nite set. It
follows that g(x) = g(x) for some c6= d, and thus x2Prep(g).We refer to [Er90, DS02]
for a classi cation of commuting morphisms on P. In fact, this was a reason for Fatou
and Julia to study complex dynamical systems.
Example 2.14. Abelian variety. In the abelian case (A;[2];L), the semigroup H(f) =
H(L) is the set of all endomorphisms on Apolarized by L.There is a more precise
description. Assume that the base eld K is algebraically closed for simplicity. Denote
by End(A) the ring of all endomorphisms on Ain the category of algebraic varieties,
and by End(A) the ring of all endomorphisms on Ain the category of group varieties.
Then End(A) consists of elements of End(A) xing the origin 0 of A, and End(A) =
End(L) o A(K)(A)o A(K) naturally. One can check that H(L) = Hwhere H(L) is
semigroup of elements of End(A) polarized by L. The following list is more or less
known in the literature. See [KS07] for the
case of number elds. All the results can be veri ed by Theorem 2.11. Example 2.15.
Square map. Let X= Pand fbe the square map. Then H(f) = fg(x; ;x; ; ) : r>1; root
of unity; 2Sg: Here Sdenotes the permutation group acting on the coordinates.
Example 2.16. Latt es map. If fis a Latt es map on Pdescended from an elliptic curve,
then H(f) consists of all the Latt es maps of degree greater than one descended from
the same elliptic curve.
Example 2.17. Chebyshev polynomial. For the Chebyshev polynomial T
, we have H(f) = f T
: e2Z
n;d
it is known for polynomial maps on P
g: If fis not of the above special cases,
we expect that H(f) is much \smaller". Conjecture 2.18. Fix two integers n 1 and d>1,
and let Mbe the moduli space of nite morphisms f: P!Pof degree d. Then H(f) =
ff;f; g for generic f2M(C ). The conjecture is more or less known in the literature for
n= 1. For example,
by the classi cation of Schmidt{Steinmetz [SS95]. In general, we may expect that
the techniques of [Er90, DS02] can have some consequences on this conjecture.
6 XINYI YUAN
3. Analytic metrics and measures Let Kbe a complete valuation eld,
and denote by j jits absolute value. IfKis archimedean, then it is isomorphic to R or C
by Ostrowski’s theorem. If Kis non-archimedean, the usual examples are:
p-adic eld Qp, nite extensions of Qpp, the
completion Cof algebraic closure Qp, the eld k((t))
of Laurent series over a eld k.
3.1. Metrics on line bundles. Let Kbe as above, Xbe a projective variety over K,
and Lbe a line bundle on X. A K-metric k kon Lis a collection ofa K-norm k kover
the ber L(x) = LK(x) of each algebraic point x2X(K) satisfying the following two
conditions:
The metric is continuous in the sense that, for any section sof Lon a Zariski open
subset Uof X, the function ks(x)kis continuous in x2U(K). The metric is Galois
invariant in the sense that, for any section sas above, one has ks(x)k= ks(x)kfor
any 2Gal(K=K). If K= C , then the metric is just the usual continuous metric of
L(C ) on X(C )
in complex geometry. If K= R , it is a continuous metric of L(C ) on X(C ) invariant
under complex conjugation.
In both cases, we say that the metric is semipositive if its Chern form
c1(L;k k) = @ @ i logksk+ div ( s)
is semipositive in the sense of currents. Here sis any non-zero meromorphic section
of L(C ).
Next, assume that Kis non-archimedean. We are going to de ne the notion of
algebraic metrics and semipositive metrics.
Suppose (X;M) is an integral model of (X;LeK) for some positive integer e, i.e., Xis
an integral scheme projective and at over Owith generic ber Xisomorphic to
X, Mis a line bundle over Xsuch that the generic ber (XK;MKKe) is isomorphic to
(X;L). Any point xof X(K) extends to a point xof X(OKK-lattice of the
one-dimensional K-vector space L) by taking Zariski closure. Then the ber M( x) is
an O0on Le(x). It induces a K-norm k ke(x) by the standard rule that M(x) = fs2L(x) :
ksk01=e0 1g: It thus gives a K-norm k k= k keon L(x). Patching together, we obtain a
continuous K-metric on L, and we call it the algebraic metric induced by (X;M).This
algebraic metric is called semipositive if Mis relatively nef in the sense that Mhas a
non-negative degree on any complete vertical curve on the special ber of X. In that
case, Lis necessarily nef on X.00The induced metrics are compatible with integral
models. Namely, if we further have an integral model (X;M) of (X;Le) dominating
(X;M), then they induce the same algebraic metric on L. Here we say that (X 000;M)
dominates (X;M) if there is a morphism : X!X, extending the identity map on the
generic ber, such that M= M0.
ALGEBRAIC DYNAMICS, CANONICAL HEIGHTS AND ARAKELOV GEOMETRY 7
A K-metric k kon Lis called bounded if there exists one algebraic metric k kon Lsuch
that the continuous function k k=k k00on (the non-compact space) X(K) is bounded. If
it is true for one algebraic metric, then it is true for all algebraic metrics. We leave it as
an exercise. For convenience, in the archimedean case we say that all continuous
metrics are bounded.
Now we are ready to introduce the notion of semipositive metrics in the sense of
Zhang [Zh95b].
mDefinition
3.1. Let K be a non-archimedean and (X;L) be as above. A continuous
K-metric k kon Lis said to be semipositive if it is a uniform limit of semipositive
algebraic metrics on L, i.e., there exists a sequence of semipositive algebraic
K-metrics k kon Lsuch that the continuous function k km=k kon X( K) converges
uniformly to 1.
In the following, we will introduce a measure c1(L;k k)nanon the analytic space
Xassociated to X. We will start with review the de nition of the Berkovich analyti cation
in the non-archimedean case.
an3.2.
Berkovich analyti cation. If K = R ;C , then for any K-variety X, de ne Xto be the
complex analytic space X(C ). It is a complex manifold if X is regular.
anIn the following, let Kbe a complete non-archimedean eld with absolute value j j.
For any K-variety X, denote by Xthe Berkovich space associated to X. It has many
good topological properties in spite of its complicated structure. It is a Hausdor and
locally path-connected topological space with 0(Xan) = 0(X). Moreover, it is compact
if Xis proper.
1)anWe will briey recall the de nition and some basic properties here. For more details,
we refer to Berkovich’s book [Be90]. We also refer to the book Baker{ Rumely [BR10]
for an explicit description of (P. 3.2.1. Construction of the Berkovich analyti cation.
We rst consider the a ne
ancase. Let U = Spec(A) be an a ne scheme of nite type over K, where Ais a nitely
generated ring over K. Then Uis de ned to be the set of multiplicative semi-norms
on Aextending the absolute value of K. Namely, U 0is the set of maps :
A!RKsatisfying: (compatibility) j= j j, (triangle inequality) (a+ b) (a) + (b);
8a;b2A, (multiplicativity) (ab) = (a) (b); 8a;b2A.Any f2Ade nes a map jfj: Uanan!R
; 7!jfj := (f) Endow Uanwith the coarsest topology such that jfjis continuous for all
f2A. For an general K-variety X, cover it by a ne open subvarieties U. Then Xan
in the natural way. Each U an
an
X, and by jXjcan!jXj. Every point of
jXjanc
c ,!X
0
map Uan
is obtained by patching the corresponding
Uan
!jUjis just 7!ker( ). The kernel of : A!R
is an open subspace of Xby de nition. 3.2.2. Maps to the variety.
jXjthe underlying space of the schemethe subset of closed point
a natural surjective map Xhas a unique preimage, and thus there
inclusion jXj. It su ces to describe these maps for the a ne case U
Then the
is a prime ideal of
A
8 XINYI YUAN
by the multiplicativity. Let x2jUjbe a closed point corresponding to a maximal idea
mxof A. Then A=mxcanis a nite eld extension of K, and thus has a unique valuation
extending the valuation of K. This extension gives the unique preimage of xin U.
3.2.3. Shilov boundary. Let Xbe an integral model of X over O KK. That is, Xis an
integral scheme, projective and at over Owith generic ber XanK= X. Further
assume that Xis normal. Then this integral model will determine some special
points on X. Let be an irreducible component of the special ber Xof X. It is a
Weildivisor on Xand thus gives an order function v : K(X) !Z by vanishing order on .
Here K(X) denotes the function eld of X, which is also the function eld of
jK
= eav ( )
an
an
nanwith
1
1(L;k
k)n
W
h
e
n
K
=
R
o
r
K
=
C
,
t
h
e
n
X
total volume Z
= lim c1(L;k km)n
m
c1(L;k k)n
. It de nes a point in Xan, locally represented by the semi-norm
(L;k k)
. Here a>0 Xis the unique constant such th
We call the point 2Xanthe Shilov bounda
Alternatively, one can de ne a (surjective
model. Then is the unique preimage of
under the reduction map. 3.3. Monge{Am
measure. Let Kbe
any valuation eld, Xbe a projective varie
bundle on X, and k kbe a semipositive m
measure con the analytic space X
= degL(X):
c1
Xan
c1
an
= ^nc
an
an
n
is just X(C ). If both Xand k kare smooth, then
(L;k k)(L;k k) is just the usual Monge{Amp ere measure on Xin complex analysis. In
general, the wedge product can still be regularized to de ne a semipositive
measure on X. In the following, assume that Kis non-archimedean. We are going to
introduce
the measure constructed by Chambert-Loir [CL06]. First consider the case that algebraic case on Xan 1(L;k k)n
c
that the metric (L;k k) was inducedby a single normal integral model (X;M) of
(X;Le) over OKthe set of irreducible components of X. Recall that any 2jXj.
Denote by j Xjggcorresponds to a Shilov boundary 2Xan. De ne
c1(L;k k)n =
X 2jXjg degMj ( )
1ne
Here is the Dirac measure supported at . Now let k kbe a general semipositive
metric. It is a uniform limit of semipositivemetrics k kminduced by integral models.
We can assume that these models are normal by passing to normalizations.
Then we de ne
: Chambert-Loir checked that
the sequence converges weakly if Kcontains a countable and dense sub eld. The general case was due to Gubler [Gu08].
Ex
a
m
pl
e
3.
2.
St
an
da
rd
m
A
L
G
E
B
R
A
I
C
D
Y
N
A
M
I
C
S
,
C
A
N
O
N
I
C
A
L
H
E
I
G
H
T
S
A
N
D
A
R
A
K
E
L
O
V
G
E
O
M
E
T
R
Y
9
etr
ic.
As
su
m
e
th
at
(X
;L)
=
(P
n
K;
O(
1)
).
Th
e
st
an
da
rd
m
etr
ic
on
O(
1)
is
gi
ve
n
by
1(L;k k)n
Pn i=00aizi
jg ; s2H0(Pn Kv
;O(1)):
n
n
n
0
nO
=
f(z0; ;zn) 2Pn(C ) : jz0j=
Sn
K
=
1(L;k
k)n
mous conjecture: Let
!2H1;1
= jz
1
and k k2
1be
two semipositive metrics on L. Then
c1
is a constant.
2
Here Pi ks(z0; ;zn)k= j
aixi
njg
: If
Ki
s
no
nar
ch
im
ed
ea
n,
th
en
th
e
m
etr
ic
is
th
e
al
ge
br
ai
c
j maxfjzj; ;jzis the linear form representing the section s. The quotient does not
depend on the choice of the homogeneous coordinate (z; ;z). The metric is
semipositive.
If Kis archimedean, the metric is continuous but not smooth. The Monge{
Amp ere measure con P(C ) is the push-forward of the probability Haar measure
on the standard torus
m
etr
ic
in
du
ce
d
by
th
e
standard integral model (P
f and only if k kk k
R
e
m
ar
k
3.
4.
Th
e
\if"
pa
rt
of
th
e
th
eo
re
K
R ;O(1)) over O
X
= RX
n OK
!n
n
(L;k k1
1 (L;k
i
k2)dim X
. It follows that the Chambert-Loir measure cwhere is the Gauss point, i.e., the
Shilov boundary corresponding to the (irreducible) special ber of the integral
model P. 3.4. Calabi{Yau Theorem. Calabi [Ca54, Ca57] made the following fa-
(X;C ) be a K ahler class on a compact complex manifold X of
dimension n, and be a positive smooth (n;n)-form on Xsuch that. Then there
exists a K ahler form e!in the class !such that e!= : Calabi also proved that the
K ahler form is unique if it exists. The existenceof the K ahler form, essentially a
highly non-linear PDE, is much deeper, and was nally solved by the seminal wor
of S. T. Yau [Yau78]. Now the whole result is called the Calabi{Yau theorem.
We can write the theorem in a more algebraic way in the case that the K ahle
class !is algebraic, i.e., it is the cohomology class of a line bundle L. Then Lis
necessarily ample. There is a bijection
fpositive smooth metrics on Lg=R! fK ahler forms in the class !g
given by >0
k k7!c1(L;k k): Thus the Calabi{Yau theorem
becomes:
Let Lbe an ample line bundle on a projective manifold Xof dimension n, and
be a positive smooth (n;n)-form on Xsuch thatRX = deg1L(X). Then there exists a
positive smooth metric k kon Lsuch that = c(L;k k)n. Furthermore, the metric is
unique up to scalar multiples.
Now it is easy to translate the uniqueness part to the non-archimedean case.
Theorem 3.3 (Calabi{Yau theorem, uniqueness part). Let Kbe a valuation
eld, Xbe a projective variety over K, and Lbe an ample line bundle over X. Let
k k)dim X= c
m
is
tri
vi
al
by
de
ni
tio
n.
10 XINYI YUAN
For archimedean K, the positive smooth case is due to Calabi as we mentioned
above, and the continuous semipositive case is due to Kolodziej [Ko03]. Afterwards
Blocki [Bl03] provided a very simple proof of Kolodziej’s result. The nonarchimedean
case was formulated and proved by Yuan{Zhang [YZ10a] following Blocki’s idea.
The non-archimedean version of the existence part is widely open, though the
one-dimensional case is trivial. In high-dimensions, we only know the case of totally
degenerate abelian varieties by the work of Liu [Liu10]. In general, the di culty lies in
how to formulate the problem, where the complication is made by the big di erence
between positivity of a metric and positivity of its Chambert-Loir measures.
4. Analytic theory of algebraic dynamics Let K be a complete
valuation eld as in x3, and let (X;f;L) be a dynam-ical system over K. In this section,
we introduce invariant metrics, equilibrium measures, and some equidistribution in
this setting.
4.1. Invariant metrics. Fix an isomorphism f L= Lfq. Then there is a unique continuous
K-metric k kon Linvariant under fsuch that f (L;k kf) = (L;k kf)q
is an isometry. We can construct the metric by Tate’s limiting argument as in x5.2. In
fact,take any bounded K-metric k kf0= limon L. Set k km!1((fm) k k01=q)m: Proposition
4.1. The limit is uniformly convergent, and independent of the
choice of k k0. Moreover, the invariant metric k kfis semipositive. Proof. The
archimedean case is classical, and the non-archimedean case is
due to [Zh95b]. Denote k km
= ((f m) k k0 1=q)m
m
<k k1=k k0
1=q))m
1 and k k0
+
m
= ((fm) (k k1=k k0
1
C1
k km+1=k km
C1=qm
: Since both k kare metrics on L,
their quotient is a continuous function on X( K). There exist a constant C>1 such that
<C: In the non-archimedean case, it
follows from the boundedness of the metrics; in the
archimedean case, it follows from the continuity of the quotient and the compactness
of X( K). By
; we have
=k km <C1=qm
<k k
0
0
m
: Thus k kis a uniform Cauchy sequence. Similar argument can prove the
independence on k k. Furthermore, if we take k kto be semipositive (and algebraic),
then each k kis semipositive, and thus the limit is semipositive.
f4.2.
Equi
libriu
m
mea
sure
.
Rec
all
that
A
L
G
E
B
R
A
I
C
D
Y
N
A
M
I
C
S
,
C
A
N
O
N
I
C
A
L
H
E
I
G
H
T
S
A
N
D
A
R
A
K
E
L
O
V
G
E
O
M
E
T
R
Y
1
1
we
have
an
invar
iant
semi
posit
ive
metr
ic
k ko
n L.
By
x3,
one
has
a
semi
posit
ive
mea
sure
c1(L;
k kf)a
ndim
Xon
the
anal
ytic
spac
e X.
It is
the
Mon
ge{A
mp
ere
mea
sure
in
the
archi
med
ean
case
,
and
the
Cha
mbe
rt-Lo
ir
mea
sure
in
the
nonarchi
med
ean
case
.
T
h
e
t
o
t
a
l
v
o
l
u
m
e
o
f
t
h
i
s
m
e
a
s
u
r
e
i
s
d
e
g
(
X
)
,
s
o
w
e
i
n
t
r
o
d
u
c
e
t
h
e
n
o
r
m
a
l
i
z
a
t
i
o
n
It
is
ca
lle
d
th
e
f
=
1
d
e
g
L
(
X
)
c1
L
(
L
;
k
kf
)d
i
m
X
:
eq
uil
ibr
iu
m
m
ea
su
re
as
so
ci
at
ed
to
(X
;f;
L).
It
is
f-i
nv
ari
an
t
in
th
e
se
ns
e
th
at
f
f
= qdim X
f
;f
f
=
rm
f
f
an
f
f
L= Lq
r
f
f
nK
an
= Rr=Zr
r
The
rst
prop
erty
follo
ws
from
the
invar
ianc
e of
f naturally
as its skeleton in the sense that A
an
:
k k,
and
the
seco
nd
prop
erty
follo
ws
from
the
rst
one.
A
n
alter
nativ
e
cons
tructi
on is
to
use
Tate
’s
limiti
ng
argu
men
t
from
any
initia
l
prob
abilit
y
mea
sure
on
Xind
uced
from
a
semi
posit
ive
metr
ic on
L.
Rem
ark
4.2.
The
invar
iant
metr
ic
k kd
epe
nds
on
the
choi
ce of
isom
orph
ism
f.
Di er
ent
isom
orph
isms
may
chan
ge
the
metr
ic by
a
scal
ar.
How
ever
, it
does
not
chan
ge
the
Cha
mbe
rt-Lo
ir
mea
sure
. So
the
equil
ibriu
m
mea
sure
does
not
dep
end
on
the
choi
ce of
the
isom
orph
ism.
Exa
mple
4.3.
Squ
are
map
.
The
metr
ic
and
the
mea
sure
intro
duce
d in
Exampl
e
3.2
are
the
invar
iant
metr
ic
and
mea
sure
for
the
squa
re
map
on
P.
Exa
mple
4.4.
Abel
ian
varie
ty.
Let
X=
Abe
an
abeli
an
varie
ty
over
K,
f=
[2]
be
the
multi
plica
tion
by 2,
and
Lbe
a
sym
metr
ic
and
ampl
e
line
bun
dle
on
A.
We
will
see
that
the
equil
ibriu
m
mea
sure
do
es
not
dep
end
on
L.
(1) If
K=
C,
then
is
exac
tly
the
prob
abilit
y
Haar
mea
sure
on
A(C
). (2)
Ass
ume
that
K is
nonarchi
med
ean.
For
the
gen
eral
case
, we
refer
to
[G
u1
0].
H
er
e
w
e
gi
ve
a
si
m
pl
e
de
sc
rip
tio
n
of
th
e
ca
se
th
at
Ki
s
a
di
sc
ret
e
va
lu
ati
on
el
d
an
d
A
ha
s
a
sp
lit
se
mi
-st
ab
le
re
du
cti
on
,
i.e
.,
th
e
id
en
tit
y
co
m
po
ne
nt
of
th
e
sp
ec
ial
be
r
of
th
e
N
er
on
m
od
el
is
th
e
ex
te
ns
io
n
of
an
ab
eli
an
va
rie
ty
by
Gf
or
so
m
e
0
r
di
m(
A)
.
Th
en
Aa
nc
on
tai
ns
th
e
re
al
tor
us
S
deformation retracts to S . The measure
is the push-forward to A
qnm(fm) x
z:
of the probability Haar measure on Sr. Example 4.5. Good reduction. Assume that=
K is non-archimedean, and(X;f;L) has good reduction over K. Namely, there is a 1qnm
triple (X; ~ f;L) over Oextending (X;f;L) with Xprojective and smooth over O K.
Then fKis the Dirac measure of the Shilov boundary corresponding to the
(irreducible) special ber of X.4.3. Some equidistribution results. Here we introduce
two equidistribution results in the complex setting. We will see that the
equidistribution of small points almost subsumes these two theorems into one
setting if the dynamical system is de ned over a number eld.
We rst introduce the equidistribution of backward orbits. Let (X;f;L) be a
dynamical system of dimension nover C . For any point x2X(C ), consider the
pull-back
z2fmX( x)
12 XINYI YUAN
Here
denotes the Dirac measure, and the summation over f m
z
m) xgm 1
Pm
converges weakly to
1
f
nm
m
jPm
(
x) are counted with multiplicities. Dinh and Sibony
proved the following result:
(fTheorem 4.6 ([DS03, DS10]). Assume that X is
normal. Then there is a proper Zariski closed
subset E of X such that, for any x2X(C ), the
sequence fqif and only if x=2E. Remark 4.7. By a
result of Fakhruddin [Fak03], all dynamical
systems (X;f;L)
can be embedded to a projective space. So the
above statement is equivalent to [DS10, Theorem
1.56]. We refer to [Br65, FLM83, Ly83] for the
case X= P. Now we introduce equidistribution of
periodic points. Let (X;f;L) over C be
as above. For any m 1, denote by Pthe set of
periodic points in X(C ) of period mcounted with
multiplicity. Consider
= 1 z2Pj
z:
Xm
The following result is due to Briend and Duval.
Theorem 4.8 ([BD99]). In the case X = Pn, the
sequence f converges weakly to fPmgm 1. We refer
to [FLM83, Ly83] for historical works on the more
classical case of
X = P1. As remarked in [DS10], one can replace
Pmby the same set without multiplicity, or by the
subset of repelling periodic points of period m.
In the non-archimedean case, both theorems are
proved over P1by Favre and Rivera-Letelier in
[FR06]. The high-dimensional case is not known in
the literature, but it is closely related to the
equidistribution of small points if the dynamical
system is de ned over number elds. See x6.2 for
more details.
5. Canonical height on algebraic
dynamical systems In this section we introduce
the canonical height of dynamical systems
over
number elds using Tate’s limiting argument. We
will also see an expression in the spirit of N eron in
Theorem 5.7. Here we restrict to number eld, but
all results are true for function elds of curves
over nite elds.
For any number eld K, denote by MKthe set of
all places of Kand normalize the absolute values
as follows:
to be the usual absolute value on Kv
v
v
on Kv
v2MYK
v
v
for any a2OKv
If vis a real place, take j jv
= R ; If vis a complex place, take j jto be the square of the usual absolute value
on K’C ; If vis a non-archimedean place, the absolute value j j= (#OKv=a)1is
normalized such that jaj. In that way, the valuation satis es the product formula
jajv = 1; 8a2K :
5.1. Weil’s height machine. We briey recall the
de nition of Weil heights. For a detailed
introduction we refer to [Se89].
Le
t
K
be
a
nu
m
be
r
el
d,
A
L
G
E
B
R
A
I
C
D
Y
N
A
M
I
C
S
,
C
A
N
O
N
I
C
A
L
H
E
I
G
H
T
S
A
N
D
A
R
A
K
E
L
O
V
G
E
O
M
E
T
R
Y
1
3
an
d
Pn
nb
e
th
e
pr
oj
ec
tiv
e
sp
ac
e
ov
er
K.
Th
e
st
an
da
rd
he
ig
ht
fu
nc
tio
n
h:
P(
K)
!R
is
de
n
ed
to
be
h(x0;x1; ;xn
) = 1w2M[E: K] XE logmaxfjx0jw;jx1jw; ;jxmjwg;
h(x) =
where Eis a nite extension of Kcontaining all the coordinates x i, and the
1deg(x)
summation is over all places wof E. By the product formula, the de nition is
independent of both the choice of the homogeneous coordinate and the choice of
E. Then we have a well-de ned function h: P( K) !R . Example 5.1. Height of
algebraic numbers. We introduce a height function h: Q !R by
Here 2Z>0
0 @logja0j+ Xz2C conj to x
A:
a0
1
i
s the
coe
cient
of
the
high
est
term
of
the
n
logmaxfjzj;1g 1
mini
mal
poly
nomi
al of
x in
Z[t],
deg(
x) is
the
degr
ee
of
the
mini
mal
poly
nomi
al,
and
the
sum
mati
on is
over
all
root
s of
the
mini
mal
poly
nomi
al.
In
pa
rti
cu
lar
, if
x=
a=
b2
Q
re
pr
es
en
te
d
by
tw
o
co
pri
m
e
int
eg
er
s
a;
b2
Z,
th
en
h(
x)
=
m
ax
flo
gj
aj;
lo
gj
bj
g:
1Q
It is easy to verify that the height is compatible with the standard height on P
(Q) = Q [f1g. Let Xbe a projective variety over K, and Lbe a
under the identi cation P
bundle on X. Let
be any morphism such that i O(1) = Ld
1
1 and
on X. For k= 1;2, let ik h i: X(K) !R as the composition of i: X(K) !Pn
L;i =
d
A
2
: X!Pnk
1 dh: Pn
k
( 1) 2
L;i
1;i2
1;A2;i1;i2
i: X!Pn
for some d 1. We obtain a height function h(K) and( K) !R . It depends on the
choices of dand i. More generally, let Lbe any line bundle on X. We can always
write L= A Afor two ample line bundles A1
be any two morphisms such that i O(1) = Adkk
for some dk
;i2
= hA1;i1 hA2;i2
L;i
g
i
v
e
s
a
g
r
o
u
p
h
o
m
o
m
o
r
p
h
i
s
m
1.
W
e
ob
tai
n
a
he
ig
ht
fu
nc
tio
n
h:
X(
K)
!R
. It
de
pe
nd
s
on
th
e
ch
oi
ce
s
of
(A
).
H
o
w
ev
er,
th
e
fol
lo
wi
ng
re
su
lt
as
se
rts
th
at
it
is
un
iq
ue
up
to
bo
un
de
d
fu
nc
tio
ns
.
Theorem 5.2 (Weil’s height machine). The above construction L7!h
H: Pic(X) ! ffunctions : X( K) !R gfbounded functions : X(K) !R g :
H
er
e
th
e
gr
ou
p
str
uc
tur
e
on
th
e
rig
htha
nd
si
de
is
th
e
ad
dit
io
n.
In
pa
rti
cu
lar
,
th
e
co
se
t
of
hL;
i1;i2
on
th
e
rig
htha
nd
si
1
de
de
pe
nd
s
on
ly
on
L.
W
e
ca
ll
a
fu
nc
tio
n
h:
X(
K)
!R
a
W
eil
he
ig
ht
co
rr
es
po
nd
in
g
to
Lif
it
lie
s
in
th
e
cl
as
s
of
H(
L)
on
th
e
rig
htha
nd
si
de
of
th
e
ho
m
or
ph
is
m
in
th
e
th
eo
re
m.
It
fol
lo
w
s
th
at
hL;
i1;i2
Lis
a
W
eil
he
ig
ht
co
rr
es
po
nd
in
g
to
L.
A
ba
si
c
an
d
im
po
rta
nt
pr
op
ert
y
of
th
e
W
eil
he
ig
ht
is
th
e
fol
lo
wi
ng
N
ort
hc
ott
pr
op
ert
y.
on the line bundle LKv . The data L= (L;fk k
f
f
)
gi
ve
an
ad
eli
c
lin
e
bu
nd
le
in
th
e
se
ns
e
of
Zh
an
-metric k kf;v over XKv
14 XINYI YUAN
Theorem 5.3 (Northcott pr
variety over K, and Lbe an am
height corresponding to L. The
fx2X( K) : deg(x) <A; hL(x) <Bg
nHere
deg(x) denotes the degr
is easily reduced to the origina
obtained explicitly.
5.2. Canonical height. Let
We are introducing the canoni
Let h: X(K) !R be any Weil heig
X(K) !R with respect to fis de n
qNhL(f(x)): Theorem 5.4. The li
choice of the Weil height h. It h
for any x2X(K), (2) hff(x) 0 for
preperiodic.
The function hf: X(K) !R is the
(1).
Proof. The theorem easily follo
5.5. Square map. For the squa
standard height function h: P(K
standard height function satis
Example 5.6. N eron{Tate
of abelian variety (A;[2];L) as in
height hL;[2], usually denoted ^
with respect to the group law.
5.3. Decomposition to local he
number eld Kas above. Fix an
each vof K, we have an f-invar
f;vgv2MK
v2Mdeg(x)
Xz2O(
XK
x)
f;v:
g
[Z
h9
5b
].
W
e
wil
l
re
vi
e
w
th
e
no
tio
n
of
ad
eli
c
lin
e
bu
nd
le
s
in
x9
.1,
bu
t
w
e
do
no
t
ne
ed
it
he
re
for
th
e
m
o
m
en
t.
Theorem 5.7. For any x2X( K), one has
h(x) = 1
H
er
e
si
s
an
y
logks(z)k
rat
io
na
l
se
cti
on
of
Lr
eg
ul
ar
an
d
no
nva
ni
sh
in
g
at
x,
an
d
O(
x)
=
G
al(
K
=
K)
xi
s
th
e
G
al
oi
s
or
bit
of
xn
at
ur
all
y
vi
e
w
ed
as
a
su
bs
et
of
X(
Kv
)
for
ev
er
y
v.
z
z:
;fCv ;LCv
2
Recall that, associated tox the dynamicalOsystem (X Cv
;
v
v
(
x
)
f;v
v
degL(X) c (L ;k kf;v)
ALGEBRAIC DYNAMICS, CANONICAL HEIGHTS AND ARAKELOV GEOMETRY 15
Proof. We will see that it is a consequence of Lemma 9.6 and Theorem 9.8.
Remark 5.8. The result is a decomposition of global height into a sum of local
heights Pz2O( x)logks(z)kf;v. It is essentially a generalization of the historical work of
N eron [Ne65].
6. Equidistribution of small points 6.1. The equidistribution
theorem. Let (X;f;L) be a polarized algebraic
mdynamical system over a number eld K. Definition 6.1. Let fxgm 1be an in nite
sequence of X( K). (1) The sequence is called generic if any in nite subsequence is
Zariski dense
in
X.
(2)
The
seq
uen
ce
is
call
ed
hf-s
mall
if
hf(x)
!0
as
m!1
. Fix
a
plac
e
vof
K.
For
any
x2X
(K),
the
Gal
ois
orbi
t
O(x
)=
Gal(
deg(x) X
=1
1
C
dim X
is the f-invariant metric depending on an isomorphism fL= Lq . Theorem 6.2. Let
on Xvan C. Here k kf;v(X;f;L) be a polarized algebraic dynamical system over a
number eld K. Let fxmgbe a generic and h-small sequence of X( K). Then for any
place vof K, the probability measure xmfconverges weakly to the equilibrium
measure f;von Xvan C;v. The theorem was rst proved for abelian varieties at
archimedean places v
by Szpiro{Ullmo{Zhang [SUZ97], and was proved by Yuan [Yu08] in the full case.
The function eld analogue was due to Faber [Fa09] and Gubler [Gu08]
independently. For a brief history, we refer to x6.3.
6.2. Comparison with the results in complex case. Theorem 6.2 is closely
related to the equidistribution results in x4.3. Let (X;f;L) be a dynamical system of
dimension nover a number eld K. Fix an embedding K C so that we can also view
it as a complex dynamical system by base change.
We rst look at the equidistribution of periodic points in Theorem 4.8. Recall that
Pmis the set of periodic points in X(K) of period m. It is stable under the action of
Gal(K=K), and thus splits into nitely many Galois orbits. Take one
xm 2Pm for each m. Then fxmgm
is hf -small automatically. Once fxmgm
s
gen
eric,
the
Galo
is
orbit
of
xis
equi
distri
bute
d. A
goo
d
cont
rol
of
the
prop
ortio
n of
the
\non
-gen
eric"
Galo
is
orbit
s in
Pwo
uld
reco
ver
the
equi
distri
butio
n of
P
by
m
m
m
i
Th
eo
re
m
6.
2.
16 XINYI YUAN
mNow
we consider the equidistribution of backward orbits in Theorem 4.6. The
situation is similar. For xed x2X(K), the preimage f(x) is stable under the action of
Gal(K=K), and thus splits into nitely many Galois orbits. Take one
xm 2f m(x) for each m. Then fxmgm
is hf -small since hf (xm) = qmhf
m
mgm
1
v
(x) by the
invariant property of the canonical height. Once fxis generic, its Galois orbit is
equidistributed. To recover the equidistribution of f(x), one needs to control the
\non-generic" Galois orbits in fm(x). For that, we would need to exclude xfrom the
exceptional set E.
6.3. Brief history on equidistribution of small points. In the following we briey
review the major historical works related to the equidistribution. Due to the limitation
of the knowledge of the author and the space of this article, the following list is far
from being complete.
The study of the arithmetic of algebraic dynamics was started with the
introduction of the canonical height on abelian varieties over number elds, also
known as the N eron{Tate height, by N eron [Ne65] and an unpublished work of Tate.
The generalizations to polarized algebraic dynamical systems was treated by
Call{Silverman [CS93] using Tate’s limiting argument and by Zhang [Zh95b] in the
framework of Arakelov geometry. In Zhang’s treatment, canonical heights of
subvarieties are also de ned.
The equidistribution of small points originated in the landmark work of Szpiro{
Ullmo{Zhang [SUZ97]. One major case of their result is for dynamical systems on
abelian varieties. The work nally lead to the solution of the Bogomolov conjecture by
Ullmo [Ul98] and Zhang [Zh98a]. See [Zh98b] for an exposition of the history of the
Bogomolov conjecture.
After the work of [SUZ97], the equidistribution was soon generalized to the
square map on projective spaces by Bilu [Bi97], to almost split semi-abelian varieties
by Chambert-Loir [CL00], and to all one-dimensional dynamical systems by Autissier
[Au01].
All these works are equidistribution at complex analytic spaces with respect to
embeddings : K,!C . From the viewpoint of number theory, the equidistribution is a
type of theorem that a global condition (about height) implies a local result (at ). The
embedding is just an archimedean place of K. Hence, it is natural to seek a parallel
theory for equidistribution at non-archimedean places assuming the same global
condition.
The desired non-archimedean analogue on P, after the polynomial case considered
by Baker{Hsia [BH05], was accomplished by three independent works: Baker{
Rumely [BR06], Chambert-Loir [CL06], and Favre{Rivera-Letelier [FR06]. In the
non-archimedean case, the v-adic manifold X( K) is totally disconnected and
obviously not a good space to study measures. The Berkovich analytic space turns
out the be the right one. In fact, all these three works considered equidistribution over
the Berkovich space, though the proofs are di erent.
Chambert-Loir [CL06] actually introduced equilibrium measures on Berkovich
spaces of any dimension, and his proof worked for general dimensions under a
positivity assumption which we will mention below. Following the line, Gubler
[Gu07] deduced the equidistribution on totally degenerate abelian varieties, and
proved the Bogomolov conjecture for totally degenerate abelian varieties over
function elds as a trophy. Gubler’s equidistribution is on the tropical variety, which
is just the skeleton of the Berkovich space.
ALGEBRAIC DYNAMICS, CANONICAL HEIGHTS AND ARAKELOV GEOMETRY 17
Another problem raised right after [SUZ97] was to generalize the results to
arbitrary polarized algebraic dynamical systems (X;f;L). The proof in [SUZ97] uses a
variational principle, and the key is to apply the arithmetic Hilbert{Samuel formula due
to Gillet{Soule [GS90] to the perturbations of the invariant line bundle. For abelian
varieties, the invariant metrics are strictly positive and the perturbations are still
positive, so the arithmetic Hilbert{Samuel formula is applicable. However, for a
general dynamical systems including the square map, the invariant metrics are only
semipositive and the perturbations may cause negative curvatures. In that case, one
can not apply the arithmetic Hilbert{Samuel formula. The proof of Bilu [Bi97] takes full
advantage of the explicit form of the square map. The works [Au01, CL06] still use the
variational principle, where the key is to use a volume estimate on arithmetic surfaces
obtained in [Au01] to replace the arithmetic Hilbert{Samuel formula.
Finally, Yuan [Yu08] proved an arithmetic bigness theorem (cf. Theorem 8.7),
naturally viewed as an extension of the arithmetic Hilbert{Samuel formula. The
theorem is inspired by a basic result of Siu [Siu93] in algebraic geometry. In
particular, it makes the variational principle work in the general case. Hence, the full
result of equidistribution of small points, for all polarized algebraic dynamical systems
and at all places, was proved in [Yu08].
The function eld analogue of the result in [Yu08] was obtained by Faber [Fa09]
and Gubler [Gu08] independently in slightly di erent settings. It is also worth
mentioning that Moriwaki [Mo00] treated equidistribution over nitely generated elds
over number elds based on his arithmetic height.
7. Proof of the rigidity Now we are ready to treat Theorem
2.9. We will give a complete proof for the
case of number elds, which is easily generalized to global elds. The general case is
quite technical, and we only give a sketch.
7.1. Case of number eld. Assume that Kis a number eld. We are going to prove
the following stronger result:
Theorem 7.1. Let Xbe a projective variety over a number eld K, and Lbe an ample
line bundle on X. For any f;g2H(L), the following are equivalent:(a) Prep(f) =
Prep(g). (b) gPrep(f) Prep(f). (c) Prep(f) \Prep(g) is Zariski dense in X.(d) Lfin the
sense that, at every place v of K, there is a constant c v’Lg>0 such that k kf;v=
cvk kg;v. Furthermore, Qvcv= 1, and c= 1 for all but nitely many v.(e) hf(x) = hgv(x)
for any x2X(K). Remark 7.2. Kawaguchi{Silverman [KS07] proves that (d) ,(e) for
X= P. Remark 7.3. In (d) we do not get the simple equality Lf= Lgnmainly because
the invariant metrics depend on the choice of isomorphisms f L= Lqand gL= Lq0.
One may pose a rigidi cation on Lto remove this ambiguity.
z2O( x) v2MK
h logks(z)kf;v; x2X(K):
f
See Theorem 5.7. Second, (e) )(a) since preperiodic points are exactly points with
canonicalheight equal to zero. Third, (a) )(b) and (a) )(c) since Prep(f) is Zariski
dense by Theorem 2.3. It is also not hard to show (b) )(c). Take any point
x2Prep(f). Considerthe orbit A= fx;g(x);g2(x); g. All points of Aare de ned over the
residue eld K(x) of x. Condition (b) implies A Prep(f), and thus points of Ahas
bounded height. By the Northcott property, Ais a nite set. In another word,
x2Prep(g). It follows that
Prep(f) Prep(g): Then
(c) is true since Prep(f) is Zariski dense in X.
m7.1.2. From (c) to (d). Assume that Prep(f) \Prep(g) is Zariski dense in X. We can
choose a generic sequence fxgin Prep(f)\Prep(g). The sequence is small with
respect to both fand gsince the canonical heights hmm) = 0 for each m.
By Theorem 6.2, for any place v2M, we have xm;v!d f;vK; xm;v!d : It follows that d f;v=
d g;vas measures on Xvan Cg;v, or equivalently c1(L;k kf;v)dim X= c1(L;k kg;v)dim X: By the
Calabi{Yau theorem in Theorem 3.3, we obtain constants cvf;v= cv>0 such that
k kk k: Here cvg;v= 1 for almost all v. To nish the proof, we only need to check that
the product c=Qvcvis 1. In fact, by the height formula in Theorem 5.7 we have
hf(x) = hg(x) logc; 8x2X( K): Then c= 1 by considering any xin Prep(f)
\Prep(g). It nishes the proof.
7.2. Case of general eld. Now assume that Kis an arbitrary eld, and we will
explain some ideas of extending the above proof. Readers who care more about
number elds may skip the rest of this section.
1It is standard to use Lefshetz principle to reduce Kto a nitely generated eld. In
fact, Theorem 2.9 depends only on the data (X;f;g;L). Here Xis de ned by nitely
many homogeneous polynomials, fand gare also de ned by nitely many
polynomials, and Lis determined by a Cech cocycle in H(X;O) which is still given
by nitely many polynomials. Let K0 Xpbe the eld generated over the prime eld
(Q or F) by the coe cients of all these polynomials. Then (X;f;g;L) is de ned over
K0, and we can replace Kby K0in Theorem 2.9.
18 XINYI YUAN
(x) = 1
deg(x) X
X
f
7.1.1. Easy implications. Here we show (d) )(e) )(a)
)(b) )(c):
First, (d) )(e) follows from the formula
(x ) = hg(x
ALGEBRAIC DYNAMICS, CANONICAL HEIGHTS AND ARAKELOV GEOMETRY 19
pIn
the following, assume that Kis a nitely generated eld over a prime eld k= Q or
F. The key of the proof is a good notion of canonical height for dynamical systems
over K.
7.2.1. Geometric height. Let (X;f;L) be a dynamical system over Kas above. Fix a
projective variety Bover kwith function eld K, and x an ample line bundle Hon B.
The geometric canonical height will be a function hH f: X(K) !R . Let ( : X !B;L) be
an integral model of (X;L), i.e., : X !B is amorphism of projective varieties over
kwith generic ber X, and Lis a line bundle on X with generic ber L. It gives a \Weil
height" hH L: X( K) !R by
hH L(x) = 1deg(x) ( H)dim B1
L x:
Here xdenotes the Zariski closure of xin X . Take the convention that h(x) = 0 if d=
0. The canonical height hH fH fis de ned using Tate’s limiting argument
hH f(x) = limN!1 1 hH (f N (x)); x2X(K):
qN L
p, then hIt converges and does not depend on the choice of the integral model. If k=
FH fsatis es Theorem 5.4 (1) (2). The proof of Theorem 2.9 is very similar to the case
of number elds.
If k= Q, then hH f1(x) = 0 does not imply x2Prep(f) due to the failure of Northcott’s
property for this height function. However, it is proved to be true by Baker [Ba09] in
the case that fis non-isotrivial on X= P. With this result, Baker{DeMarco [BDM09]
proves Theorem 2.9 using the same idea of number eld case.
7.2.2. Arithmetic height. Assume that Kis nitely generated over k= Q. We have
seen that the geometric height is not strong enough to satisfy Northcott’s property.
Moriwaki [Mo00, Mo01] introduced a new class of height functions on
X( K) which satis es Northcott’s property. The notion is further re ned by Yuan{
Zhang [YZ10b]. In the following, we will introduce these height functions, and we
refer to x8.1 for some basic de nitions in arithmetic intersection theory.H fFix an
arithmetic variety Bover Z with function eld K, and x an ample hermitian line
bundle Hon B. Moriwaki’s canonical height is a function h: X(K) !R de ned as
follows.Let ( : X!B; L) be an integral model of (X;L), i.e., : X!Bis a morphism of
arithmetic varieties over Z with generic ber X, and Lis a hermtianline bundle on
Xwith generic ber L. It gives a \Weil height" hH L: X(K) !R by
hH L(x) = 1deg(x) ( H)dim B1
Lx
Here xdenotes the Zariski closure of xin X. The the canonical height h H fis still
de ned using Tate’s limiting argument
hH f(x) = limN!1 1 hH N (x)); x2X(K):
qN L(f
It converges and does not depend on the choice of the integral model (X;L). The
height function hH fsatis es Northcott’s property and Theorem 5.4 (1) (2).
20 XINYI YUAN
QWith
this height function, Yuan{Zhang [YZ10a] proves Theorem 2.9. The proof
uses the corresponding results of number eld case by considering the bers of
X!BQ. The proof is very techinical and we refer to [YZ10b]. In the end, we briey
introduce the arithmetic height function of [YZ10b].
Let ( : X!B;L) and x2X(K) be as above. Still denote by f NN(x) the Zariski closure of
f(x) in X. Consider the generically nite map : f(x) !B. If it is at, we obtain a
hermitian line bundle
LjfN( x). Normalize it by LN
=
1 qN
(LjfN
( x) ).
The limiting
height of x.
The idea of [Y
containingc P
the limit by h
not at, we ma
summary, we
height is inde
satis es North
re nes hH fin t
require the p
the linear
itivity propert
notions of lin
refer to Laza
Most of t
setting of Ara
developed by
In this se
More precise
bundles follo
following Zha
[Yu08, Yu09]
8.1. Top
dimension n+
dimension n.
ALGEBRAIC DYNAMICS, CANONICAL HEIGHTS AND ARAKELOV GEOMETRY 21
under any analytic map f : fz2Cn: jzj<1g!X(C ) is smooth in the usual sense.
Pic(X)Now we introduce the top intersection cn+1!R : The intersection number is
de ned to be compatible with birational morphismsof arithmetic varieties, so we can
assume that Xis normal and Xis smooth by pull-back to a generic desingularization
of X.Let L1;L2; ;Ln+1Qbe n+1 hermitian line bundles on X, and let sn+1be any
non-zero rational section of Lon X. The intersection number is de ned inductively by
L1
= L1
L2 Ln div(sn+1 ) Zdiv ( sn+1)( C ) Ln+1 L2
logksn+1kc1(L1) c1(Ln):
The right-hand side depends on div(sn+1) linearly, so it su ces to explain the case
that D= div(sn+1) is irreducible. If Dis horizontal in the sense that it is at over Z, then
L1 L2 Ln D= L1jD
L2jD
LnjD
is an arithmetic intersection on D. The integration on the right-hand side needs
regularization if D(C ) is not smooth.
If Dis a vertical divisor in the sense that it is a variety over F pfor some prime p, then
the integration is zero, and
L1 L2 Ln D= (L1jD
L2jD
LnjD
d
1 L2
d
d j
2
n+1
d deg :
jD D
L
d
j
1
D
1
.
1
1
)logp: Here the
intersection is the usual intersection on the projective variety D.
One can check that the de nition does not depend on the choice of sand is symmetric
and multi-linear. Similarly, we have a multilinear intersection product
c Pic(X) Z(X) !R : Here Zd(X) denotes the group of Chow cycles of codimension
n+1don X(before linear equivalence). For D2Z(X), the intersection L Ld D= L1 L
is interpreted as above. Example 8.1. If dimX= 1, the intersection is just a
degree map
c Pic(X) !R :If Xis normal, then it is isomorphic to Spec(Ois just an O KK) for some
number eld K. Then L-module locally free of rank one. The Hermitian metric k kon
L(C ) = : K!CL (C ) is a collection fk kof metrics on the complex line Ldeg( L(C ). The
degree is just d1) = log#(LHere sis any non-zero element of L=OK g s) X :
K!Clogksk :
22 XINYI YUAN
8.2. Arithmetic linear series and arithmetic volumes. Let Lbe a hermitian line bundle
on an arithmetic variety Xof dimension n+1. The corresponding arithmetic linear
series is the nite set
H0(X; L) = s2HHere ksksup0(X;L) : ksk= supz2X( C
1 :
)ks(z)k sup
is the usual supremum norm on H0(X;L)C . A non-zero element of H00(X;L) is called an e ective section of Lo
called e ective if H(X;L) is nonzero. Denote
h0( L) = log#H0(X;L): The volume of Lis de ned as0
vol(L) =
hn+1(N L)
limsupN!1
N=(n+ 1)! :
vol ( L) = limsupN!10by the convention that we write tensor product additively. To see the
NHere N Ldenotes L
H0(X;L), denote B(L) = fs2H(X;L)R: ksk0 1g: It is the corresponding un
H(X;L)Rsup0of the supremum norm. Then H(X;L) = H00(X;L) \B(L) is th
of a lattice with a bounded set in H(X;L)0R, so it must be nite. To bet
vol, we introduce their \approximations" and vol0. Pick any Haar mea
H(X;L)R. De nevol(H0 (L) = log vol(B( L))(X;L)R=H0(X;L)) ; It is easy to
quotient is independent of the choice of the Haar measure. We furthe
n+1 (N L) N=(n+ 1)! :
0By Minkowski’s theorem on lattice points, we immediately have
h(L) (L) h0(L0)log2: Here h(LQ) = dimH0(XQ;LQQ ) is the usual one. It
follows that vol(L) vol(L): We will see that the other direction is true in
the ample case.
One can verify that both vol and vol n+1are homogeneous of degree n+ 1 in the
sense that vol(m L) = mvol(L) and vol (mL) = mn+1vol(L) for any positive integer m.
Hence, vol and vol extend naturally to functions on c Pic(X)Qby the homogeneity.
See [Mo09, Ik10]. Remark 8.2. Chinburg [Chi91] considered a similar limit de ned
from an adelic set. He called the counterpart of exp(vol(L)) the (inner) sectional
capacity. It measures \how many" algebraic points the adelic set can contain. See
also [RLV00, CLR03].
ALGEBRAIC DYNAMICS, CANONICAL HEIGHTS AND ARAKELOV GEOMETRY 23
08.3.
Ample hermitian line bundles. As in the algebro-geometric case, H(X;NL)
generates NLunder strong positivity conditions on L. Following Zhang [Zh95a], a
hermitian line bundle Lis called ample if it satis esQthe following three conditions: Lis
ample on XQ; Lis relatively semipositive, i.e., the Chern form c C1(L) of Lis
semipositive and Lis relatively nef in the sense that deg(Lj) 0 for any closed curve
Cin any special ber of Xover Spec(Z);dim Y Lis horizontally positive, i.e., the
intersection number L Y>0 for any horizontal irreducible closed subvariety Yof X.One
version of the main result of [Zh95a] is the following arithmetic Nakai{ Moishezon
theorem:
Theorem 8.3 (Arithmetic Nakai{Moishezon, Zhang). Let Lbe an ample hermitian
line bundle on an arithmetic variety Xwith a smooth generic ber X0Q. Then for any
hermitian line bundle Eover X, the Z-module H0(X;E+NL) has a Z-basis contained in
H(X;E+ NL) for Nlarge enough. We also have the following arithmetic version of the
Hilbert{Samuel formula. Theorem 8.4 (Arithmetic Hilbert{Samuel, Gillet{Soul e,
Zhang). Let Lbe aQhermitian line bundle on an arithmetic variety of dimension n+ 1.
(1) If Lis ample and Lis relatively semipositive, then vol n+1(L) = Ln+1. (2) If Lis ample,
then vol(L) = L. In both cases, the \limsup" de ning vol( L) and vol(L) are actually
limits. The result for vol is a consequence of the arithmetic Riemann{Roch theorem of
Gillet{Soul e [GS92] and an estimate of analytic torsions of Bismut{Vasserot [BV89]. It
is further re ned by Zhang [Zh95a]. The result for vol is obtained from that for vol by
the Riemann{Roch theorem on lattice points in Gillet{Soul e [GS92] and the arithmetic
Nakai{Moishezon theorem by Zhang [Zh95a] described above. See [Yu08, Corollary
2.7] for more details.
Remark 8.5. In both cases, we can remove the condition that LQQis ample. Note that
Lis necessarily nef under the assumption that Lis relatively semipositive. The
extensions can be obtained easily by Theorem 8.7 below.
There are three conditions in the de nition of ampleness, where the third one is
the most subtle. The following basic result asserts that the third one can be obtained
from the rst two after rescaling the norm.
QLemma 8.6. Let L= (L;k k) be a hermitian line bundle on an arithmetic variety Xwith
Lample and Lrelatively semipositive. Then the hermitian line bundle L( ) =
(L;k keQ is ample, we can assume there exist sections s 1) is ample for su ciently
large real number . Proof. Since L; ;s0r2 Hi(X;L) which are base-point free over
the generic ber. Let be such that ksksupe Y)dim Y<1 for all i= 1; ;r. We claim that
L( ) is ample. We need to show that ( L( )jjsuch>0 for any horizontal irreducible
closed subvariety Y. Assume that Xis normal by normalization. We can nd an s
24 XINYI YUAN
( L( )jY)dim Y
that div(sj) does not contain Y. Then
Now
the
proo
f can
be
nish
ed
by
indu
ction
on
dim
Y.
8
.4.
Volu
mes
of
her
mitia
n
line
bun
dles.
The
follo
wing
arith
meti
c
versi
on
of a
theo
rem
of
[Siu
93]
impli
es
the
equi
distri
butio
n of
smal
l
= (L( )jdiv ( sj) jY
> (L( )jdiv ( sj) jY
:
)dim Y1
)dim Y1
ZY( C ) log(ksj ke) c1(L)dim Y1
point
s in
alge
braic
dyna
mics
.
Th
eo
re
m
8.
7
(Y
ua
n
[Y
u0
8])
.
Le
t
La
nd
M
be
tw
o
a
m
pl
e
he
rm
iti
an
lin
e
bu
nd
le
s
ov
er
an
ari
th
m
eti
c
va
rie
ty
X
of
di
m
en
si
on
n+
1.
Th
en
vol
1
n+1
n+1(
LM) L
n+1 n(n+
1) L
1 n+1
+ vol( M)
2
Q
d dtj
M: Note that the di erence
of two ample line bundles can be any line bundle, so the
theorem is applicable to any line bundle. It is sharp when Mis \small". Another
interesting property of the volume function is the log-concavity property.
vol(L+ M)
vol(
L)
: The following consequen
equidistribution
theorem. Corollary 8.9 (Di erentia
bundleon an arithmetic variety of
relatively semipositive, then vol(L
Theorem 8.8 (Log-concavity, Yuan [Yu09]). Let Land Mbe two e ective
hermitian line bundles over an arithmetic variety Xof dimension n+ 1. Then
t=0vol n(L+ tM) = (n+ 1)L
M:
Q
+ O(t2
n+1
n
Q
d dtjt=0
(a) Changing Lto L( ) = (L;k ke
Q) = (L+ tM) n+1
n
Q
Q
Q
+ tMQ)n
is ample for small jtj.
Q
); t!0: Therefore,
vol(L+ tM) =((n+ 1)L M: Proof. We prove the results by
the following steps.
) for any real number does not a ect the result in (1). It follows from the
simple facts
Q + tM (L+ tM) + (n+ 1)vol(L+ (n++ tM
n+1
1)(L
+tMQ
vol (L( ) + tM) = vol
); (L( ) + tM): Here the classical volume vol(L) on the generic ber
equals the intersection number (L+tMsince L
2) If Lis ample,
then vol(L+ tM)
= (L+ tM)
1 n+1
ALGEBRAIC DYNAMICS, CANONICAL HEIGHTS AND ARAKELOV GEOMETRY 25
(b) It su ces to prove (1) for Lample. In fact, assume that Lis ample and Lis
relatively semipositive. By Lemma 8.6, L( ) is ample for some realnumber . Then it
follows from (a). (c) Assuming that Lis ample, then Q
2
vol(L+ tM) vol (L+ tM) (L+ tM)n+1
+ O(t2
1
1
1)tA2
vol ( L+ tM) (L+
tA1)= (L+ tM)n+1n+1
vol( L+ tM)1 n+1 + vol(LtM)
n+1= (L)
1 n+1
1 n+1
2
and A2
(n+ 1) (L+ tA+
O(t21)n
n+1
2 vol(L)
+ O(t2 )
L
1 n+1
+ O(t2
+ O(t2) ! :
1
vol( L+ tM)
): It su ces to consider the case t > 0 by replacing Mby M. As inthe classical case,
any hermitian line bundle can be written as the tensor quotient of two ample
hermitian line bundles. Thus M= AAfor ample hermitian line bundles A. It follows
that L+ tM= ( L+tAis the di erence of two ample Q-line bundles. By Theorem 8.7,
): (d) Assuming that Lis ample, then
tA
): In fact, by (c),
( L+ tM) 1 + t
n+1
nn+1 M
1 n+1
L
+ vol( LtM)1 n+1
1 n+1
vol
n+1
n+1
The result remains true if we replace tby t. Take the sum of the estimates for vol(L+
tM) and vol(LtM).(e) Assuming that Lis ample, then vol(L+ tM) 2 vol( L)1 n+1: Note that
(c) implies that both L+ tMand LtMare e ective whenjtjis small. Apply Theorem 8.8
to L+ tMand LtM. (f) The equalities in (d) and (e) are in opposite directions. They
forces the
equality vol( L+ tM) = (L+ tM)
+ O(t2
(L+ tM) = (L+ tM)
+ O(t2):
n+1
n
): It
follows that
Remark 8.10. The inequality in (c) is due to [Yu08], which is su cient for the
equidistribution. The proof of the other direction using the log-concavity is due to
[Che08c]. The abstract idea is that vol(L+ tM)is a concave function in t by the
log-concavity, and (c) asserts that l(t) = t(n+1)L Mis a line supporting the graph of the
concave function, so the line must be a tangent line.
These results are naturally viewed in the theory of arithmetic big line bundles. A
hermitian line bundle Lon Xis called big if vol(L) >0. To end this section, we mention
some important properties of the volume function:
26 XINYI YUAN
Rumely{Lau{Varley [RLV00] proved that \limsup" in the de nition of vol (L) is a
limit if LQis ample. They actually treated the more general setting proposed by
Chinburg [Chi91].
Moriwaki [Mo09] proved the continuity of \vol" at big line bundles. Chen
[Che08a] proved that \limsup" in the de nition of vol( L) is alwaysa limit for any
L. See [Yu09] for a di erent proof using Okounkov bodies. Chen [Che08b] and
Yuan [Yu09] independently proved the arithmeticFujita approximation theorem
for any big L. Chen [Che08c] proved the di erentiability of \vol" at big line
bundles.Corollary 8.9 is just a special case, but it extends to the general case
by the above arithmetic Fujita approximation theorem.
Ikoma [Ik10] claimed a proof of the continuity of \vol ". 9. Adelic
line bundles
The terminology of hermitian line bundles is not enough for the proof of Theorem
6.2. The reason is that the dynamical system can rarely be extended to an
endomorphism of a single integral model over OK. The problem is naturally solved by
the notion of adelic line bundles introduced by Zhang [Zh95a, Zh95b].
9.1. Basic de nitions. Let Kbe a number eld, Xbe a projective variety of
dimension nover K, and Lbe a line bundle on X. Recall that we have considered local
analytic metrics in x3.An adelic metric on Lis a coherent collection fk kvgvof bounded
Kvon LKvover XKvv-metrics k kover all places vof K. That the collection fk kKvgvis
coherent means that, there exist a nite set Sof non-archimedean places of Kand a
(projective and at) integral model (X;L) of (X;L) over Spec(O) S, such that the K v-norm
k kvis induced by (XOK v;LOK v) for all v2Spec(Ov) S. In the above situation, we write L=
(L;fk kgvK) and call it an adelic line bundle on X. We further call Lthe generic ber of L.
Let (X; M) be a hermitian integral model of (X;LKe) for some positive integer e, i.e., Xis
an arithmetic variety over OK, and M= (M;k k) is a hermitian line bundle on X, such
that the generic ber (X;MK) = (X;Lv). Then (X;M) induces an adelic metric fk kgveon L.
The metrics at non-archimedean places are clear from x3. The metrics at
archimedean places are just given by the hermitian metric. In fact, we have
X(C ) = a : K,!CX (C ); M(C ) = a : K,!CM (C ):The left-hand sides denote C -points over
Z, and the right-hand sides denote C points over OKvia : OK0 !C . Then the
hermitian metric on M(C ) becomes a collection of metrics k kon M (C ) = Le 01=e (C
) over all . Then k kgives the archimedean part of the adelic metric. Such an
induced metric is called algebraic, and it is called relatively semipositive if Lis
relatively semipositive.An adelic metric fk kvgvon an ample line bundle Lover Xis
called relatively semipositive if the adelic metric is a uniform limit of relatively
semipositive algebraic adelic metrics. Namely, there exists a sequence ffk km;vof
relatively semipositive algebraic adelic metrics on L, and a nite set Sof
non-archimedean places of K, such that k k= k kvfor any v2Spec(OKm;vgvgmm;v)
Sand any m, and such that k k=k kvconverges uniformly to 1 at all places v. In that
case, we
also
say
that
the
adeli
c
line
bun
dle
L=
A
L
G
E
B
R
A
I
C
D
Y
N
A
M
I
C
S
,
C
A
N
O
N
I
C
A
L
H
E
I
G
H
T
S
A
N
D
A
R
A
K
E
L
O
V
G
E
O
M
E
T
R
Y
2
7
(L;fk
k) is
relati
vely
semi
posit
ive.
It
impli
es
that
the
Kv-m
etric
k kvo
n
LKvvg
vis
semi
posit
ive
in
the
sens
e of
x3.
An
adeli
c
line
bun
dle
is
calle
d
integ
rabl
e if it
is
isom
etric
to
the
tens
or
quoti
ent
of
two
relati
vely
semi
posit
ive
adeli
c
line
bun
dles.
Den
ote
by c
Pic(
X)intt
he
grou
p of
isom
etry
clas
ses
of
integ
rabl
e
adeli
c
line
bun
dles.
Exa
mple
9.1.
Let
(X;f;
L)
be a
dyna
mica
l
syst
em
over
a
num
ber
eld
K.
Fix
an
isom
orph
ism
f L=
Lqas
in
the
de ni
tion.
For
any
plac
e vof
K,
let
k kb
e
the
invar
iant
Kv-m
etric
of
LKvo
n
XKv.
The
n Lf=
(L;fk
kf;vg
vf;v)
is a
relati
vely
semi
posit
ive
adeli
c
line
bun
dle.
It is
f-inv
aria
nt in
the
sens
e
that
f Lf=
Lq f.
9.2.
Exte
nsio
n of
the
arith
meti
c
inter
secti
on
theo
ry.
Let
Xbe
a
projecti
ve
varie
ty of
dime
nsio
n
nove
ra
num
ber
eld
K.
By
Zha
ng
[Zh9
5b],
the
inter
secti
on
of
her
mitia
n
line
bun
dles
in
x8.1
exte
nds
to a
sym
metr
ic
multi
-line
ar
form
c
Pi
c(
X)
n+1
int!
R
:
Le
t
L1
;
;L
n+1
be
int
eg
ra
bl
e
ad
eli
c
lin
e
bu
nd
le
s
on
X,
an
d
w
e
ar
e
go
in
g
to
de
n
e
th
e
nu
m
be
r
L1
L
2
L
n+1
. It
su
c
es
to
as
su
m
e
th
at
ev
er
y
Ljj
on
Li
s
se
mi
po
sit
iv
e
by
lin
ea
rit
y.
Fo
r
ea
ch
j=
1;
;
n+
1,
as
su
m
e
th
at
th
e
m
etr
ic
of
Lj;
m;v
is
th
e
un
ifo
rm
li
mi
t
of
ad
eli
c
m
etr
ic
s
ffk
k
gv
gm
,
an
d
ea
ch
fk
kj;
m;v
gvj
j;mi
s
in
du
ce
d
by
a
se
mi
po
sit
iv
e
int
eg
ral
m
od
el
(X
;M
j;m
)
of
(X
;L
1;m
).
W
e
m
ay
as
su
m
e
th
at
X
=
=
Xn
+1;
mej
f
or
ea
ch
m.
In
fa
ct,
let
Xj;
;m
mm
be
an
int
eg
ral
m
od
el
of
X
do
mi
na
tin
g
Xf
or
all
jin
th
e
se
ns
e
th
at
th
er
e
ar
e
bir
ati
on
al
m
or
ph
is
m
s
j;
m:
X
m!
Xj;
mm
in
du
ci
ng
th
e
id
en
tit
y
m
ap
on
th
e
ge
ne
ric
be
r.
S
uc
h
a
m
od
el
al
w
ay
s
ex
ist
s.
Fo
r
ex
a
m
pl
e,
w
e
ca
n
ta
ke
X
be
th
e
th
e
Za
ris
ki
cl
os
ur
e
of
th
e
im
ag
e
of
th
e
co
m
po
sit
io
n
L1
:
H
er
e
th
e
rs
t
m
ap
is
X,!Xn+1
,!X1;m
OK
X2;m
OK
OK
; Xn+1;m
j;m
n+1
m!1
j;m
j;m
gv
th
e
di
ag
on
al
e
m
be
dd
in
g,
an
d
th
e
se
co
nd
m
ap
is
th
e
inj
ec
tio
n
onto the generic ber. Then replace (X
;M
check that they induce the same adelic metrics.
Go back to the sequence of metrics fk k
1;m M
1e
M
:
2;m
L2 L
= lim
en+1;m M
It is
easy
to
verif
y
that
the
limit
exist
s
and
dep
ends
only
on
L1;
;Ln+1
.
See
Zha
ng
[Zh9
5b].
Pi
c(
1;m
) by (Xm
j;m;v
j;mM
m
j;m).
De ne
). It is easy to
induced by
(X
n+1;m
;M
X)
Si
mi
lar
ly,
w
e
ge
t
an
int
er
se
cti
on
pa
iri
ng
c(
X)
!R
by
se
tti
ng
L1
L
2
L
d+1
d+1
int
Z
D
=
Ld
1jD
L
2jD
Ld
+1j
:
H
er
e
Zd
D(
X)
de
no
te
s
th
e
gr
ou
p
of
ddi
m
en
si
on
al
C
ho
w
cy
cl
es
.
Ex
a
m
pl
e
9.
2.
Le
t
Lb
e
an
int
eg
ra
bl
e
ad
eli
c
lin
e
bu
nd
le
on
X.
Fi
x
a
pl
ac
e
v0
(L( ))n+1
of Lto e k kv0
and a constant 2R . Change the metric k kv0 . Denote the new adelic line bundle by L( ). Then it is easy to check that
n+1= L + (n+ 1)degL(X) :
28 XINYI YUAN
Example 9.3. If X= Spec(K), then the line bundle L1on Xis just a vector space over Kof
dimension one. We simply have
d deg(L1) = XHere sis any non-zero element of L1vlogkskv:, and the degree is
independent of the choice of sby the product formula. One may compare it with
example 8.1.
9.3. Extension of the volume functions. It is routine to generalize the de nitions in
x8.2 to the adelic line bundles.vLet Kbe a number eld, Xbe a projective variety of
dimension nover K, and L= (L;fk kgv0) be an adelic line bundle on X. De ne H(X;L) =
fs2H0(X;L) : ksk0 1; 8vg; h(L) = log#H0(X;L);v;sup
h0n+1(N L)
vol(L) = limsup
N=(n+ 1)! :
N!1
= sup
ks(z)kv
z2X( Kv )
Here for any place vof K, ksk v;sup
is the usual supremum norm on H 0(X;L)Kv . Elements of H0(X;L) are called e ective sections of L.
Similarly, for any place vof K, denote Bv( : kskv;sup
L) = fs2H0(X;L)Kv
vB(L) = Y Bv(L) H0(X;L)AK
:
Here AK
vol(H0
AK=H (X;L)) +
1 h0(L)logjd j;
0
2
=(n+ 1)! :
vol (L) = limsup
N!1
H
e
r
e
AK
d
K
1g: Further denote
is the adele ring of K. Then we introduce
0
K
(L) = log vol(B(L))(X;L)
n+1 (NL) N
0is the discriminant of K. The de nition of (L) does not depend on the choice of a
Haar measure on H(X;L). One easily checks that the de nitions are compatible with
the hermitian case
if the adelic line bundle is algebraic. Furthermore, it is easy to see that h ; ;vol
commute with uniform limit of adelic line bundles. The same is true for vol with some
extra argument using an adelic version of the Riemann{Roch of Gillet{Soul e [GS91].
As in the hermitian case, some adelic version of Minkowski’s theorem gives
vol( L) vol (L): It is easy to generalize Corollary 9.4 for vol to the adelic case,
though we do not know if it is true for vol. We rewrite the statement here, since it
will be used for the equidistribution.
C
or
oll
ar
y
9.
4.
Le
t
X
be
A
L
G
E
B
R
A
I
C
D
Y
N
A
M
I
C
S
,
C
A
N
O
N
I
C
A
L
H
E
I
G
H
T
S
A
N
D
A
R
A
K
E
L
O
V
G
E
O
M
E
T
R
Y
2
9
a
pr
oj
ec
tiv
e
va
rie
ty
of
di
m
en
si
on
no
ve
ra
nu
m
be
r
el
d
K,
an
d
L;
M
be
tw
o
int
eg
ra
bl
e
ad
eli
c
lin
e
bu
nd
le
s
on
X.
As
su
m
e
th
at
Li
s
rel
ati
ve
ly
se
mi
po
sit
iv
e
an
d
th
at
th
e
ge
ne
ric
be
r
Li
s
a
m
pl
e.
Th
en
vol (L+ tM) = (L+ tM)n+1
+ O(t2
g L
a
l
9.4.1. Weil height. Let L= (L;fk kvgv
hL(x) = 1deg(x) L xgal
Here xgal
Ygal
L
Lhdim Y+1(Y)
=L
)
;
t
!
0
:
R
e
m
a
r
k
9
.
5
.
g
e
O
n
e
m
a
y
n
a
t
u
r
a
l
l
y
d
e
n
e
a
m
p
l
e
n
e
s
s
o
f
a
d
e
l
i
c
l
i
n
e
b
u
n
d
l
e
s
t
o
n
e
r
a
l
i
z
e
T
h
e
o
r
e
m
8
.
3
,
8
.
4
,
8
.
7
,
8
.
8
a
n
d
L
e
m
m
a
8
.
6
.
9
.
4
.
H
e
i
g
h
t
s
d
e
n
e
d
b
y
a
d
e
l
i
c
l
i
n
e
b
u
n
d
l
e
s
.
be
an
adeli
c
line
bun
dle
on a
proj
ectiv
e
varie
ty X
over
a
num
ber
eld
K.
De n
e
the
heig
ht
funct
)
ion
h:
X(K)
!R
asso
ciate
d to
Lby
is
th
e
cl
os
ed
po
int
of
Xc
or
re
sp
on
di
ng
to
th
e
al
ge
br
ai
c
po
int
x.
If
Li
s
int
eg
ra
bl
e,
th
e
he
ig
ht
of
an
;
x
2
X
(
K
)
:
y
cl
os
ed
su
bv
ari
et
y
Y
of
XK
as
so
ci
at
ed
to
L
is
de
n
ed
by
(dimY+ 1)deg
Here
Ygal
(
)
Y :
Kis the closed K-subvariety of Xcorresponding to Y, i.e. the image of the
composition Y !X!X. In particular, the height of the ambient variety Xis
h
Lhdim X+1(X) = L(dimX+ 1)degL(X)
L :
W
e
s
e
e
t
h
a
t
h
e
i
g
h
t
s
a
r
e
j
u
s
t
n
o
r
m
a
l
i
z
e
d
a
r
i
t
h
m
e
t
i
c
d
e
g
r
e
e
s
.
n
e
L
e
m
m
a
h
a
s
d
e
g
(
x
)
9
.
6
.
X
F
o
r
a
n
y
x
2
X
(
K
)
,
o
v
2
z
2 M
K
O
(
X
x
)
Here sis any
rational
section of
Lregular and
non-vanishing
at x, and O(x)
= Gal(
K=K)xis the
Galois orbit of
xnaturally
viewed as a
subset of
X(Kv) for
every v.
Proof. It
follows from
the de nitions.
See also
Example
9.3. Theore
m 9.7. For
any adelic
line bundle L,
the height
function hL:
X(K) ! R is a
Weil height
correspondin
g to the line
bundle Lon X.
Proof. We
prove the
result by
three steps.
(1) Let
(X;L) = (Pn
K;O(1)),
and endow
Lwith the
standard
adelic
metric
described
in Example
3.2. By
Lemma
9.6, it is
easy to
check
that
the
heigh
t
functi
on hL:
Pnn(K
) !R is
exactl
y the
stand
ard
heigh
t
functi
on h:
P(K)
!R in
x5.1.
(2)
Go
back
to
gener
al X.
We
claim
that,
for
any
two
adelic
line
bundl
es
l
o
g
k
s
(
z
)
k
v
:
L1
,
th
e
di
er
en
ce
hh
is
bo
un
de
d.
In
fa
ct,
w
e
ha
ve
k
k=
k
k1;
vfo
r
al
m
os
t
all
v,
an
d
k
k=
k
k2;
vis
bo
un
de
d
for
th
e
re
m
ai
ni
ng
pl
ac
es
.
Th
e
cl
L1
= (L1;fk k1;vgv) and L2
L1
1;v =
(L2;fk k2;vgv) with generic ber L1
= L2
2;v
ai
m
fol
lo
w
s
fro
m
Le
m
m
a
9.
6.
30 XINYI YUAN
mbe
an embedding with i(3) By linearity, it su ces to prove the theorem for the
case that Lis very ample. Let i: X,!PO(1) = L. By (2), it su ces to consider the
case L= i O(1) where O(1) is the line bundle O(1) endowed with the standard
metric. Recall that hL;i= h iis the Weil height induced by the embedding. By (1), it
is easy to see that hL;i= hL. It proves the result.
9.4.
2. Canonical height. Now we consider the dynamical case. Let (X;f;L) be
a dynamical system over a number eld K. By x5.2, we have the canonical height
function hf= hL;f: X(K) !R . Recall that Lfis the f-invariant adelic line bundle.
Theorem 9.8. hL;f= hLf. Proof. By Theorem 9.7, hLfis a Weil height corresponding to
L. Furthermore, hLf(f(x)) = qhLf(x) by f Lf= Lq f. Then hLf= hby the uniqueness of
hL;fL;fin Theorem 5.4. Remark 9.9. Following Lemma 9.6, we can decompose
canonical heights of
the height hLf
K
K
(1) hf(Y). (2) hff
2 qf
f
f
f
gv
of subvarieties of X. The following result is a generalization
Proposition 9.10. Let Y be a closed subvariety of X. Then t
p
(f(Y)) = qh
oints in terms of local heights. See Theorem 5.7. By
(Y) 0, and the equality holds if Y is preperiodic in the sens
the theorem, it is meaningful to denote by hf
a nite set. Proof. The arithmetic intersection satis es the p
which
= L . The inequality h
v
f
ives (1) by f Lf
(Y) 0 follows from the construction
g
that Lis a uniform limit of adelic metrics induced
by ample hermitian models. The last property follows from (1).
Remark 9.11. The adelic line bundle Lis \nef" in the sense that it is relatively
semipositive, and it has non-negative intersection with any closed subvariety of X.
Remark 9.12. The converse that h(Y) = 0 implies Y is preperiodic is not true. It was
an old version of the dynamical Manin{Mumford conjecture. See
Ghioca{Tucker{Zhang [GTZ10] for a counter-example by Ghioca and Tucker, and
some recent formulations of the dynamical Manin{Mumford conjecture.
10. Proof of the equidistribution 10.1. Equidistribution in
terms of adelic line bundles. We introduce
an equidistribution theorem for relatively semipositive adelic line bundles, which
includes the dynamical case by taking the invariant adelic line bundle.
Let Xbe a projective variety of dimension nover a number eld Kand let L=
(L;fk k) be an adelic line bundle. Similar to 6.1, we make a de nition of smallness.
ALGEBRAIC DYNAMICS, CANONICAL HEIGHTS AND ARAKELOV GEOMETRY 31
LDefinition
of X( K) is called hL
10.1. An in nite sequence fxmgm 1(xm) !hLL
f
quence is Zariski dense in X. We further recall
that x;v= 1z2Ovdeg(x) X( x) z
is the probability measure on the analytic space Xvan C
= 1degL(X)
;k kv)n
van C
c1(LCv
on Xvan
C
L;v
mL=
(L;fk kvgvgbe a generic
and h
L
i
s
x
YX
vgv
L
L
;
v
m;v
m
e
h
v
L
-small if
h(X) as m!1. Note that we require the limit of the heights to be h(X). It is compatible
with the dynamical case since h(X) = 0 in the dynamical case by Proposition 9.10.
Recall that an in nite sequence of X( K) is called generic if any in nite subse-
at a place vof Kassociated to the
Galois orbit of a point x2X(K). Denote by
the normalized Monge{Amp ere/Chambert-Loir measure on X. Theorem 10.2. Let X
be a projective variety over a number eld Kand let
L) be a relatively semipositive adelic line bundle on Xwith Lample. Let fx-small
sequence of X(K). Then for any place vof K, the probability measure converges
weakly to the measure . The theorem is proved by Yuan [Yu08], and it implies
Theorem 6.2 immediately. The case that vis archimedean and k kis strictly positive was due to Szpiro{
Ullmo{Zhang [SUZ97], which implies their equidistribution on abelian varieties. The
result is further generalized to the case that Lis big and vis archimedean by
Berman{Boucksom [BB10]. If Lis big and vis non-archimedean, a similar result is true
by Chen [Che08c].
10.2. Fundamental inequality. Let Xbe a projective variety of dimension nover a
number eld K, and L= (L;fk k) be an adelic line bundle on X. Following Zhang
[Zh95a, Zh95b], the essential minimum of the height function h
inf
(X) =
x2X( K) Y( K)
sup
L(x
) e
mgm
m!1
fxmgm
generic m!1
alternative de nition eL
hL(x ):
Here the sup is taken over all closed subvarieties Y of X. Let fxbe a generic
sequence in X(K). By de nition, liminf(X): Furthermore, the equality can be attained
by some sequences. Thus we have an
m
(X) = inf
liminf
Theorem 10.3 (fundamental inequality, Zhang). Assume that Lis big. Then
eL(X) vol ( L) (n+ 1)vol(L):
In particular, if Lis relatively semipositive and Lis ample, then e L(X) hL(X):
32 XINYI YUAN
Recall that, in the classical sense, Lis big if
h0n(NL)
vol(L) = lim
N=n! >0:
N!1
The existence of the limit follows from Fujita’s
approximation theorem. See [La04] for example.
It is worth noting that, for each d= 1; ;n, Zhang
[Zh95a, Zh95b] acutally introduced a number eL;d(X)
by restricting codim(Y) din the de nition of the
essential minimum. Then the essential minimum is
just eL(X) = eL;1(X). The fundamental inequality is an
easy part of his theorem on successive minima.
Remark 10.4. In the setting of Theorem 10.2,
the existence of a generic and hL-small sequence is
equivalent to the equality eL(X) = hL(X). The equality
is true in the dynamical case since both sides are
zero, but it is very hard to check in general.
To prove the theorem, we need the following
Minkowski type result: Lemma 10.5. Let K;X;Lbe
as above. Fix a place v0of K. Then for any >0,
there exists a positive integer Nand a non-zero
section s2H(X;NL) satisfying 1N logksk v0;sup( L)
(n+ 1)vol(L) vol + 0
0; 8v6= v0
hL(x) = 1N
Then we immediately have
C.
The
goal
is to
prov
e
van
hL(x)
x 0X
v2MXK
logks(z)kv:
vol (L) (n+ 1)vol(L)
: Proof. It is a consequence of the adelic version of Minkowski’s theo
[BG06, Appendix
a C] for example. Now we prove Theorem 10.3. Let s2
be as in the lemma. By Lemma 9.6, for any x2X(K) jdiv(s)j(K),1 deg(x) z2O
nd logkskv;sup
; 8x2X(K) jdiv(s)j(K):
Let !0. We obtain the theorem. 10.3. Variational principle. Now we are re
prove Theorem 10.2. To
illustrate the idea, we rst consider the archimedean case. We use the va
principle of [SUZ97].
Fix an archimedean place v. Let be a real-valued continuous function on
v X an
C
X an
C
Z
lim
m;v
=Z
L;v:
v
m!1
By density, it su ces to assume is smooth in the
sense that, there exists an embedding of Xvan
Cinto a projective manifold M, such that can be
extended to a smooth function on M.
v=
e De
note
by
O( )
the
trivia
l line
bun
dle
A
L
G
E
B
R
A
I
C
D
Y
N
A
M
I
C
S
,
C
A
N
O
N
I
C
A
L
H
E
I
G
H
T
S
A
N
D
A
R
A
K
E
L
O
V
G
E
O
M
E
T
R
Y
3
3
with
the
trivia
l
metr
ic at
all
plac
es
w6=
v,
and
with
metr
ic
give
n by
k1ka
t v.
Den
ote
L( )
= L+
O( ).
The
smo
othn
ess
of i
mpli
es
that
O( )
is an
integ
rabl
e
adeli
c
line
bun
dle.
Le
t
tb
e
a
s
m
all
po
sit
iv
e
rat
io
na
l
nu
m
be
r.
By
th
e
fu
nd
a
m
en
tal
in
eq
ua
lit
y,
hL( t )(xm) vol (L(t )) (n+
liminf 1)degL(X) :
m!1
H
er
e
w
e
ha
ve
us
ed
th
e
as
su
m
pti
on
th
at
fx
mg
mi
s
ge
ne
ric
.
N
ot
e
th
at
L(t
)
=
L+
tO
()
.
By
th
e
di
er
en
tia
bil
ity
in
C
or
oll
ar
y
9.
4,
vol (L(t )) = (L(t ))n+1
+ O(t2
L( t )(xm) hL( t )(X) + O(t2
h
m!1
L
L
m
X Xvan
C
L;v
xm;v
hL( t )(xm
): It follows that (10.1) liminf): On the other hand, it is easy ) = h
to verify that
):(
10
.3)
hL( t )(X) = h
(X) + t Z va
+ O(t2
n
In
fa
ct,
th
e
rs
t
eq
ua
lit
y
fol
lo
w
s
fro
m
Le
m
m
a
9.
6,
an
d
th
e
se
co
nd
eq
ua
C
(x ) + t Z;(10.2)
lit
y
fol
lo
w
s
fro
m
m!1
O( ) = Z
an C
(x
c1(LCv
;k kv)n:
nL
Note that we have assumed X
lim
hL
m;v
Z an
Z an
:
C
C
liminf
m!1
(X
):
Th
en
(1
0.
1),
(1
0.
2)
an
d
(1
0.
3)
im
pl
y
m
!
1
R
e
p
l
a
c
i
n
g
b
y
i
n
t
h
x
X L;v
v
X
v
v
m)
= hL
e
i
n
e
q
u
a
l
i
t
y
,
w
e
g
e
t
t
h
e
o
t
h
e
r
d
i
r
e
c
t
i
o
n
.
H
e
n
c
e
,
l
i
m
Z
X
v
a
n
C
x
m
;
v
=
Z
X
v
a
n
C
L
;
v
:
It ni
shes
the
proo
f.
10.4
.
Nonarchi
med
ean
case
.
Now
we
cons
ider
The
ore
m
10.2
for
any
nonarchi
med
ean
plac
e v.
The
proo
f is
para
llel
to
the
archi
med
ean
case
,
exce
pt
that
we
use
\mo
del
funct
ions"
to
repl
ace
\smo
oth
funct
ions"
.
The
key
is
that
mod
el
funct
ions
are
dens
e in
the
spac
e of
conti
nuo
us
funct
ions
by a
resul
t of
Gubl
er.
To
int
ro
du
ce
m
od
el
fu
nc
tio
ns
,
w
e
co
ns
id
er
a
sli
gh
tly
ge
ne
ral
se
tti
ng
.
Fi
x
a
pr
oj
ec
tiv
e
va
rie
ty
Y
ov
er
a
no
nar
ch
im
ed
ea
n
el
d
k,
an
d
co
ns
id
er
int
eg
ral
m
od
el
s
of
Y
ov
er
nit
e
ex
te
ns
io
ns
of
k.
Le
t
E
be
a
ni
te
ex
te
ns
io
n
of
k,
an
d
let
(Y
;M
)
be
a
(p
roj
ec
tiv
e
an
d
at)
int
eg
ral
m
od
el
of
(Y
E;
O
YE)
ov
er
O
Mo
n
O
YE.
It
in
du
ce
s
a
m
etr
ic
k
k.
Th
en
lo
gk
1k
1=e
ME
,
for
an
y
po
sit
iv
e
int
eg
er
e,
is
a
fu
nc
tio
n
on
jYj
,
an
d
ex
te
nd
s
to
a
un
iq
ue
co
nti
nu
ou
s
fu
nc
tio
n
on
Ya
n
EE
. It
pu
llba
ck
s
to
a
fu
nc
tio
n
on
Yk
an
Cv
ia
th
e
na
tur
al
pr
oj
ec
tio
n
Yk
an
Ca
n
E!
Y.
H
er
e
Ck
de
no
te
s
th
e
co
m
pl
eti
on
of
th
e
al
ge
br
ai
c
cl
os
ur
e
ko
f
k.
Th
e
re
su
lti
ng
fu
nc
tio
n
on
Yk
an
Ci
s
ca
lle
d
a
m
od
el
fu
nc
tio
n
34 XINYI YUAN
on Ykan C. Alternatively, it is also induced by the integral model (Y) of (YCk;OYC kOC k;MOC
k). Theorem 10.6 (Gubler). The vector space over Q of model functions is uniformly dense in the space of real-valued continuous functions on Ykan C. The result is
stated in [Gu08, Proposition 3.4] in this form. It is essentially a
combination of Gubler [Gu98, Theorem 7.12] and [Yu08, Lemma 3.5]. Now we are
ready to prove Theorem 10.2 when vis non-archimedean. By the
m!1density theorem proved above, it su ces to
L;v:
prove limZXvan C xm;v= ZXvan C
induced by an OE-model (X;M) of (X) for any nite extension Eof KvE;OXE.
for any model function = logk1kM
to nite extensions of K, it is not hard to reduce it to the case that E= Kv.
O( ) the trivial line bundle with the trivial metric at all places
w6= v, and with metric given by k1k v= e Kat v. Denote L( ) = L+ O( ). It is
extend (X;M) to an integral model over Owhich induces the adelic line bu
It follows that O( ) is an integrable adelic line bundle.
Now the proof is the same as the archimedean case. Still consider th
L(t ) = L+ tO( ) for a small rational number t. Note that (10.1) only require
O( ) to be integrable, and (10.2) and (10.3) are true by de nition.
References
[Ar74] S. J. Arakelov, Intersection theory of divisors on an arithmetic surface, Math. USSR
(1974), 1167{1180.
[Au01] P. Autissier, Points entiers sur les surfaces arithm etiques, J. Reine Angew. Math., 5
201{235.
[Ba09] M. Baker, A niteness theorem for canonical heights attached to rational maps over
function elds. J. Reine Angew. Math. 626 (2009), 205{233.
[BB10] R. Berman, S. Boucksom, Growth of balls of holomorphic sections and energy at eq
Invent. Math. 181 (2010), no. 2, 337{394.
[BDM09] M. Baker, L. DeMarco, Preperiodic points and unlikely intersections, to appear in
Math. J., 2009.
[BD99] J.-Y. Briend, J. Duval, Exposants de Liapouno et distribution des points p eriodiqu
endomorphism, Acta Math. 182(2) 143{157 (1999).
[Be90] V.G. Berkovich, Spectral theory and analytic geometry over non-archimedean elds
Monogr. 33, American Mathematical Society, 1990.
[BG06] E. Bombieri, W. Gubler, Heights in Diophantine geometry, New Mathematical Mono
Cambridge University Press, Cambridge, 2006.
[BH05] M. Baker and L. C. Hsia, Canonical heights, trans nite diameters, and polynomial d
Reine Angew. Math. 585 (2005), 61{92.
[Bi97] Y. Bilu, Limit distribution of small points on algebraic tori, Duke Math. J. 89 (1997), n
465{476.
[Bl03] Z. Blocki, Uniqueness and stability for the complex Monge-Amp ere equation on com
manifolds, Indiana Univ. Math. J. 52 (2003), no. 6, 1697{1701.
[BR06] M. Baker, R. Rumely, Equidistribution of small points, rational dynamics, and poten
Ann. Inst. Fourier (Grenoble) 56 (2006), no. 3, 625{688.
[BR10] M. Baker, R. Rumely, Potential Theory and Dynamics on the Berkovich Projective L
Mathematical Surveys and Monographs vol. 159, American Mathematical Society, 2010.
[Br65] H. Brolin, Invariant sets under iteration of rational functions, Ark. Mat. 6, 103{144 (19
J.-M. Bismut, E. Vasserot, The asymptotics of the Ray-Singer analytic torsion associated
with high powers of a positive line bundle, comm. math. physics 125 (1989), 355-367.
ALGEBRAIC DYNAMICS, CANONICAL HEIGHTS AND ARAKELOV GEOMETRY 35
[Ca54] E. Calabi, The space of Kahler metrics, Proc. Int. Congr. Math. Amsterdam 2 (1954), 206{207.
[Ca57] E. Calabi, On K ahler manifolds with vanishing canonical class, Algebraic geometry and topology, A
symposium in honor of S. Lefschetz, Princeton University Press 1957, 78{89.
[Che08a] H. Chen, Positive degree and arithmetic bigness, arXiv: 0803.2583v3 [math.AG]. [Che08b] H.
Chen, Arithmetic Fujita approximation, arXiv: 0810.5479v1 [math.AG], to appear
in Ann. Sci. Ecole Norm. Sup.
[Che08c] H. Chen, Di erentiability of the arithmetic volume function, arXiv:0812.2857v3 [math.AG].
[Chi91] T. Chinburg, Capacity theory on varieties, Compositio Math. 80 (1991), 75{84. [CS93] G. Call and
J.H. Silverman, Canonical heights on varieties with morphisms, Compositio
Math. 89 (1993), 163{205. [CL00] A. Chambert-Loir, Points de petite hauteur sur les vari et es
semi-ab eliennes, Ann. Sci.
Ecole Norm. Sup. (4) 33 (2000), no. 6, 789{821.
[CL06] A. Chambert-Loir, Mesures et equidistribution sur les espaces de Berkovich, J. Reine Angew.
Math. 595 (2006), 215{235.
[CL10a] Th eor emes d’ equidistribution pour les syst emes dynamiques d’origine arithm etique.
Panoramas et synth eses 30 (2010), p. 97{189.
[CL10b] A. Chambert-Loir, Heights and measures on analytic spaces. A survey of recent results, and some
remarks. arXiv:1001.2517v1 [math.NT].
[CLR03] T. Chinburg, C. F. Lau, R. Rumely, Capacity theory and arithmetic intersection theory, Duke Math.
J. 117 (2003), no. 2, 229{285.
[CT09] A. Chambert-Loir, A. Thuillier, Mesures de Mahler et equidistribution logarithmique, Ann. Inst.
Fourier (Grenoble) 59 (2009), no. 3, 977{1014.
[De87] P. Deligne, Le d eterminant de la cohomologie, Current trends in arithmetical algebraic geometry
(Arcata, Calif., 1985), 93{177, Contemp. Math., 67, Amer. Math. Soc., Providence, RI, 1987.
[DS02] T. C. Dinh, N. Sibony, Sur les endomorphismes holomorphes permutables de Pk, Math. Ann. 324,
33{70 (2002).
[DS03] T. C. Dinh, N. Sibony, Dynamique des applications d’allure polynomiale. J. Math. Pures Appl. 82,
367{423(2003).
[DS10] T. C. Dinh, N. Sibony, Dynamics in several complex variables: endomorphisms of projective spaces
and polynomial-like mappings. Holomorphic dynamical systems, 165{294, Lecture Notes in Math., 1998,
Springer, Berlin, 2010.
[Er90] A. E. Eremenko, On some functional equations connected with iteration of rational function,
Leningrad. Math. J. 1 (1990), No. 4, 905{919.
[Fa09] X. Faber, Equidistribution of dynamically small subvarieties over the function eld of a curve. Acta
Arith. 137 (2009), no. 4, 345{389.
[Fak03] N. Fakhruddin, Questions on self maps of algebraic varieties. J. Ramanujan Math. Soc. 18 (2003),
no. 2, 109{122.
[FLM83] A. Freire, A. Lop es, R. Man e, An invariant measure for rational maps. Bol. Soc. Brasil. Mat.
14(1), 45{62 (1983).
[FR06] C. Favre, J. Rivera-Letelier, Equidistribution quantitative des points de petite hauteur sur la droite
projective. Math. Ann. 335 (2006), no. 2, 311{361.
[GS90] H. Gillet, C. Soul e, Arithmetic intersection theory, Publ. Math. IHES, 72 (1990), 93-174. [GS91] H.
Gillet, C. Soul e, On the number of lattice points in convex symmetric bodies and their
duals, Israel J. Math., 74 (1991), no. 2-3, 347{357. [GS92] H. Gillet, C. Soul e, An arithmetic
Riemann-Roch theorem, Inventiones Math., 110 (1992),
473{543. [GTZ10] D. Ghioca, T. Tucker, S. Zhang, Towards a dynamical Manin{Mumford
conjecture, 2010. [Gu98] W. Gubler, Local heights of subvarieties over non-archimedean elds, J. Reine
Angew.
Math. 498 (1998), 61{113. [Gu07] W. Gubler, The Bogomolov conjecture for totally degenerate
abelian varieties. Invent.
Math. 169 (2007), no. 2, 377{400. [Gu08] W. Gubler, Equidistribution over function elds.
Manuscripta Math. 127 (2008), no. 4,
485{510. [Gu10] W. Gubler, Non-Archimedean canonical measures on abelian varieties. Compos.
Math.
146 (2010), no. 3, 683{730.
36 XINYI YUAN
[Ik10] H. Ikoma, Boundedness of the successive minima on arithmetic varieties, preprint. [Ko03] S.
Kolodziej, The Monge-Amp ere equation on compact K ahler manifolds. Indiana Univ.
Math. J. 52 (2003), no. 3, 667{686. [KS07] S. Kawaguchi, J.H. Silverman, Dynamics of projective
morphisms having identical canonical heights. Proc. Lond. Math. Soc. (3) 95 (2007), no. 2, 519{544. [La04] R. Lazarsfeld, Positivity in
algebraic geometry. I. Classical setting: line bundles and linear
series. Ergeb. Math. Grenzgeb.(3) 48. Springer-Verlag, Berlin, 2004. [Liu10] Y. Liu, A
non-archimedean analogue of Calabi-Yau theorem for totally degenerate abelian
varieties, arXiv:1006.2852. [Ly83] M. Ju. Lyubich, Entropy properties of rational endomorphisms of the
Riemann sphere.
P1
[Sib99] N. Sibony, Dynamique des applications rationnelles de P P
217-225.
which lie on a curve. Ph.D. Dissertation, Columbia
[Mil06] E
J. W. Milnor, Dynamics in One Complex Variable, A
rgod. Theor. Dyn. Syst. 3(3), 351{385 (1983). [Mim97] A. Mimar, On the University Press 2006.
preperiodic points of an endomorphism of P1
[Mo00] A. Moriwaki, Arithmetic height functions over nitely
no. 1, 101{142.
[Mo01] A. Moriwaki, The canonical arithmetic height of sub
generated eld, J. Reine Angew. Math. 530 (2001), 33{54.
[Mo09] A. Moriwaki, Continuity of volumes on arithmetic va
407{457.
[Ne65] A. N eron, Quasi-fonctions et hauteurs sur les vari e
249{331.
[PST10] C. Petsche, L. Szpiro, and T. Tucker, A dynamica
2010.
[RLV00] R. Rumely, C. F. Lau, and R. Varley, Existence of
Soc. 145 (2000), no. 690.
[Se89] J. -P. Serre, Lectures on the Mordell-Weil Theorem
1989.
k. Dynamique et g eom etrie complexes (Lyon, 199
Soc. Math. France, Paris, 1999.
[Sil07] J. H. Silverman, The arithmetic of dynamical system
Springer, New York, 2007.
[Siu93] Y.-T. Siu, An e ective Mastusaka Big Theorem, An
1387-1405.
[SS95] W. Schmidt and N. Steinmetz, The polynomials ass
Soc., 27(3), 239{241, 1995.
[SUZ97] L. Szpiro, E. Ullmo, S. Zhang, Equidistribution d
(1997) 337{348. [Ul98] E. Ullmo, Positivit e et discr
Ann. of Math. (2),
147 (1998), 167{179. [Yau78] S. T. Yau, On the Ric
and the complex MongeAmp ere equation. I. Comm. Pure Appl. Math. 31 (1978), no. 3, 339{411.
[Yu08] X. Yuan, Big line bundle over arithmetic varieties. Invent. Math. 173
(2008), no. 3, 603{
649. [Yu09] X. Yuan, On volumes of arithmetic line bundles, Compos.
Math. 145 (2009), no. 6, 1447{
1464. [YZ10a] X. Yuan, S. Zhang, Calabi{Yau theorem and algebraic
dynamics, preprint, 2010. [YZ10b] X. Yuan, S. Zhang, Small points and Berkovich
metrics, preprint, 2010. [Zh95a] S. Zhang, Positive line bundles on arithmetic
varieties, Journal of the AMS, 8 (1995),
187{221. [Zh95b] S. Zhang, Small points and adelic metrics, J. Alg.
Geometry 4 (1995), 281{300. [Zh98a] S. Zhang, Equidistribution of small points
on abelian varieties, Ann. of Math. 147 (1)
(1998) 159{165. [Zh98b] S. Zhang, Small points and Arakelov theory,
Proceedings of ICM, Berlin 1998, Vol II,
ALGEBRAIC DYNAMICS, CANONICAL HEIGHTS AND ARAKELOV GEOMETRY 37
[Zh06] S. Zhang, Distributions in algebraic dynamics. Surveys in Di erential Geometry: A Tribute to
Professor S.-S. Chern, 381-430, Surv. Di er. Geom., X, Int. Press, Boston, MA, 2006.
Department of Mathematics, Columbia University , New York, NY 10027, USA E-mail address:
yxy@math.columbia.edu