Good Height Functions on Quasi-Projective Varieties

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R-line bundle. Worse, in some cases when we can build a metrized line bundle inducing this height function, the continuity of the metric fails [DO] or the metric is not anymore adelic [DMWY]. In the present article, we introduce a notion of good height function on a quasi-projective variety deﬁned over a number ﬁeld and prove an equidistribution of small points for such heights, allowing us for example to prove a general equidistribution statement in families of polarized endomorphisms of projective varieties with marked points.

Good height functions and equidistribution

Let V be a smooth quasi-projective variety defined over a number field K and place v ∈ MK and let h : V (Q¯ ) → R be a function. Definition 1. — We say h is a good height at v if there is:

(a) a sequence (Fi)i of Galois-invariant ﬁnite subsets of V (Q¯ ) which is generic, i.e. for any subvariety Z ⊂ V deﬁned over K, there is i0 such that Fi ∩ Z = ∅ for i ≥ i0, and h-small, 1 i.e. h(Fi) := h(x) → 0, as i → ∞, #Fi x∈Fi (b) a probability measure µ on V an and a constant vol(h) > 0, P v v (c) for any n ≥ 0, a projective model Xn of V together with a birational morphism ψn : Xn → X0 which is an isomorphism above V and a big and nef Q-line bundle Ln on Xn endowed with an adelic semi-positive continuous metrization L¯n, such that the following holds

(1) For any generic and h-small sequence (Fi)i of Galois-invariant finite subsets of V (Q¯ ), the −1 sequence εn({Fi}i) := lim supi hL¯n (ψn (Fi)) − hL¯n (Xn) satisfies εn({Fi}) → 0 as n → ∞, (2) the sequence of volumes vol(Ln) converges to vol(h) as n → ∞, ¯ ¯ an (3) If c1(Ln)v is the curvature form of Ln on Xn,v, then the sequence of finite measures ¯ k an (ψn)∗c1(Ln)v n converges weakly on Vv to the positive measure vol(h) · µv, (4) If k := dim V > 1, for any ample line bundle M on X and any adelic semi-positive contin- 0 0 uous metrization M¯ 0 on M0, there is a constant C ≥ 0 such that ∗ ¯ j ¯ k+1−j ψn(M0) · Ln ≤ C, for any 2 ≤ j ≤ k +1 and any n ≥ 0. We say that vol(h) is the volume of h and that µv is the measure induced by h at the place v. We

finally say h is a good height if it is v-good for all v ∈ MK. In this case, we say {µv}v∈MK is the global measure induced by h. We prove here the following general equidistribution result. Theorem 2 (Equidistribution of small points). — Let V be a smooth quasiprojective variety defined over a number field K, let v ∈ MK and let h be a v-good height on V with induced measure µv. For any h-small sequence (Fm)m of Galois-invariant finite subsets of V (Q¯ ) such that for any hypersurface H ⊂ V defined over K, we have

#(Fn ∩ H)= o(#Fn), as n → +∞, an the probability measure µFm,v on Vv which is equidistributed on Fm converges to µv in the weak C 0 an sense of measures, i.e. for any ϕ ∈ c (Vv ), we have 1 lim ϕ(y)= ϕµv. m→∞ #Fm an y F Vv X∈ m Z This result is inspired by Kühne’s work [K] where he establishes an equidistribution statement in families of abelian varieties and from [YZ], where Yuan and Zhang develop a general theory of adelic line bundles on quasi-projective varieties. Among other results, Yuan and Zhang prove an equidistribution theorem for small point in this context. They also deduce Theorem 3 and Theorem 4 from this general result. The aim of this article is to provide a more naive and independent approach which seems particularly adapted to a dynamical setting. I also have to mention GOOD HEIGHT FUNCTIONS ON QUASI-PROJECTIVE VARIETIES 3 that, even though he focuses on the case of families of abelian varieties in his paper, Kühne has a strategy to generalize his relative equidistribution theorem to a dynamical setting. It is worth mentioning there are many arithmetic equidistribution statement for small points in the past decades see, e.g., [SUZ, Bi, R, T, BR1, CL1, FRL, Y, CLT, BR2, BB3, MY]. It si also worth mentioning that, prior to Kühne’s recent work only the equidistribution results of arithmetic nature from [R] and [BR2] do not rely on the continuity of the underlying metrics and compactness of the variety. Also, in both [R] and [BR2], the results are stated on P1 and each metric is continuous outside a polar set and is bounded. The generalization of their approach remains unexplored in a more general context. It should also be noted that Mavraki and Ye [MY] were the first to get rid of the assumption that the metrization is adelic. Theorem 2 is a general criterion to have such an equidistribution statement without assuming the height function is induced by a metrization on a projective model of the variety, or that the height function is given by an adelic datum. The strategy of the proof of Theorem 2 follows more or less that of Yuan’s result. Let us now quickly sketch the proof. Fix an integer n and let ϕ be a test function at place v with compact an support in Vv and endow the trivial bundle of Xn with the metric induced by ϕn := ϕ ◦ ψn at place v and with the trivial metric at all places w 6= v. As it is classical, we first use the adelic Minkowski’s second Theorem to compare the lim infi hL¯n(ϕ)(Fi) with the arithmetic volume vol(L¯n(ϕ)) of of the metrized Q-line bundle L¯n(ϕ). Then, we rely on the arithmetic version of Siu’s bigness criterion proved by Yuan [Y] to get a lower bound on the expansion of vol(L¯n(tϕ)) −1 ¯ k withc respect to t > 0 of the form thvol(Ln) c1(Ln)v , ϕni + Cn(ϕ, t). When dim V > 1, this is ensured by assumption (4). c

Our main input here is to get an explicit control of the term Cn(ϕ, t) in terms of vol(Ln), sup an |ϕ| only, for all t ∈ (0, 1]. This allows us to ﬁnd C ≥ 1 depending only on ϕ such that for Vv all t ∈ (0, 1], we have c (L¯ )k ε ({F } ) lim sup hµ , ϕi− 1 n v , ϕ ≤ n i i + Ct. Fi,v vol(L ) n t i→∞ n

−1 ¯ k The hypothesis on h-small sequences and the assumption that vol(Ln) (ψn)∗c1(Ln)v converges to µv allow us to conclude.

Applications in families of dynamical systems Our motivation for proving Theorem 2 comes from the study of families of dynamical systems. More precisely, we want to apply Theorem 2 to families of polarized endomorphisms. Let S be a smooth quasi-projective variety of dimension p ≥ 1 and let π : X → S be a family of smooth projective varieties. We say (X ,f,S) is a family of polarized endomorphisms if f : X → X is a morphism with π ◦ f = π and if there is a relatively ample line bundle L on X and an integer ∗ ⊗d d ≥ 2 such that f L ≃ L . Given a family (X , f, L) and a collection a := (a1,...,aq) of sections aj : S → X of π are all defined over Q¯ , we can define a height function on the variety S by letting q ¯ hf,a(t) := hft (aj (t))), t ∈ S(Q), j=1 X b where ft is the restriction of f to the fiber Xt of π : X → S and hft is the canonical height of the endomorphism ft : Xt → Xt, as defined by Call-Silverman [CS]. If v ∈ MK is archimedean, we an can also define a bifurcation current Tf,ai on Sv for each dynamicalb pair (X , f, L,ai) by letting

an Tf,ai := π∗ Tf ∧ [ai(Sv )] . This is a closed positive (1, 1)-current with continuousb potential, see § 3.1 for more details. As an application of Theorem 2, we prove the following. 4 THOMAS GAUTHIER

Theorem 3. — Let (X , f, L) be a family of polarized endomorphisms parametrized by a smooth quasiprojective variety S and let a := (a1,...,aq) be a collection of sections of π : X → S, all deﬁned over a number ﬁeld K. Assume the following properties hold:

1. there exists v ∈ MK archimedean such that the measure dim S µf,a := (Tf,a1 + ··· + Tf,aq ) a an is non-zero with mass volf ( ) := µf,a(Sv ) > 0, 2. there is a generic hf,a-small sequence {Fi}i of ﬁnite Galois-invariant subsets of S(Q¯ ).

Pick any v ∈ MK and let (Fm)m be a h-small sequence of Galois-invariant ﬁnite subsets of S(Q¯) such that for any hypersurface H ⊂ S deﬁned over K, we have

#(Fn ∩ H)= o(#Fn), as n → +∞. an Then the sequence (µFm,v)m of probability measure on Sv converges to the probability measure 1 an µf,a,v in the weak sense of measures on S . volf (a) v

Let us make a few comments on the proof. By Theorem 2, it is sufficient to prove hf,a is a good ¯ height function with associated global measure {µf,a,v}v∈MK . The construction of (Xn, Ln, ψn) is quite easy and just consists in revisiting the convergence −n n hft (aj (t)) = lim d hX ,L(ft (aj (t))). n→∞ ¯ dim S The convergence of measures cb1(Ln)v towards µf,a,v and the upper bound on the local intersection numbers are also not difficult to establish. The two key facts are the convergence of volumes, which relies on the key estimates of [GV], and the fact that for any small generic sequence {Fi}i, the sequence εn({Fi}i) converges to 0. To establish this last point, we use a comparison of hft with hX ,L|Xt established by Call and Silverman [CS], as well as the convergence of the volumes and Siu’s classical bigness criterion. This proof directly inspires from the strategy of [K], whereb the above mentioned arguments replace his use of a deep and recent result on families of abelian varieties due to Gao and Habegger [GH]. † To emphasize the strength of Theorem 3, we finish here with a general equiditribution towards the bifurcation measure µbif of the moduli space Md as introduced by [BB1]. Recall that the moduli space Md of degree d rational maps is the space of P GL(2) conjugacy classes of degree d rational maps, and that it is an irreducible affine variety of dimension 2d − 2 defined over Q, see e.g. [S]. Recall also that a parameter {f} is post-critically finite (or PCF) if its post-critical ◦n set n≥1 f (Crit(f)) is finite.

TheoremS 4. — Fix a sequence (Fn)n of ﬁnite subsets of the moduli space Md(Q¯ ) of degree d 1 rational maps of P such that Fn is Galois-invariant for all n and such that, for any hypersurface H of Md that is deﬁned over Q, then

#(Fn ∩ H)= o(#Fn), as n → ∞. 1 Assume that for any {f} ∈ Fn, the map f is PCF. Then the measure δ n #Fn {f}∈Fn {f} converges weakly to the normalized bifurcation measure µ . S bif P Organization of the paper. — Section 1 is devoted to quantitative height inequalities which are use in the proof of Theorem 2. Section 2 is dedicated to the proof of Theorem 2, and we prove in Subsection 2.3 that quasi-adelic measures induce good height functions. Theorem 3 is proved in Section 3. Finally, in Section 4 we discuss the assumptions of Theorem 3, proving they are sharp and we prove Theorem 4.

Acknowledgment. — I heartily thank Sébastien Boucksom, Charles Favre and Gabriel Vigny. This paper benefited from invaluable comments from them. I also want to thank Xinyi Yuan for very interesting discussions and for pointing out an error in an earlier version of Proposition 10. I also want to thank Laura DeMarco and Lars Kühne for interesting discussions. I finally would like to thank Yuan and Zhang for kindly sharing their draft. GOOD HEIGHT FUNCTIONS ON QUASI-PROJECTIVE VARIETIES 5

1. Quantitative height inequalities

In the whole section, we let X be a smooth projective variety of dimension k deﬁned over a number ﬁeld K.

1.1. Arithmetic intersection and heights

For the material of this section, we refer to [CL2] and [Zha1]. Let L0,...,Lk be Q-line bundle on X. Assume Li is equipped with an adelic continuous metric {k·kv,i}v∈MK and we denote ¯ ¯ Li := (Li, {k·kv}v∈MK ). Assume Li is semi-positive for 1 ≤ i ≤ k.

an Fix a place v ∈ MK. Denote by Xv the Berkovich analytiﬁcation of X at the place v. We also ¯ an let c1(Li)v be the curvature form of the metric k·kv,i on Xv . We will use in the sequel that the arithmetic intersection number L¯0 ··· L¯k is symmetric and multilinear with respect to the Li and that k ¯ ¯ ¯ ¯ −1 ¯ L0 ··· Lk = L1|div(s) ··· Lk|div(s) + log kskv c1(Li)v, Xan v∈MK Z v j=1 X ^ 0 for any global section s ∈ H (X,L0). In particular, if L0 is the trivial bundle and k·kv,0 is the trivial metric at all places but v0, this gives

k ¯ ¯ −1 ¯ L0 ··· Lk = log kskv0,0 c1(Li)v0 . Xan Z v0 j=1 ^ When L¯ is a big and nef Q-line bundle endowed with a semi-positive continuous adelic metric, following Zhang [Zha1], we define hL¯ (X) as k+1 L¯ h ¯ (X) := , L K Q (k + 1)[ : ]vol(L) k where vol(L) = (L) is the volume of the line bundle L (also denoted by degX (L) sometimes). We also define the height of a closed point x ∈ X(Q¯) as ¯ L|x 1 −1 h ¯ (x)= = log ks(σ(x))k , L [K : Q] [K : Q]#O(x) v v∈MK σ K x ֒ C X : (X) → v where O(x) is the Galois orbit of x, for any section s ∈ H0(X,L) which does not vanish at x. ¯ Finally, for any Galois-invariant finite set F ⊂ X(Q), we define hL¯ (F ) as 1 h ¯ (F ) := h ¯ (y). L #F L y F X∈ A fundamental estimate is the following, called Zhang’s inequalities [Zha1]: ¯ Lemma 5 (Zhang). — If L is ample and e(L) = supH infx∈(X\H)(Q¯) hL¯ (x), where the supremum is taken over all hypersurfaces H of X defined over K, then

1 ¯ ¯ e(L)+ k inf hL¯ (y) ≤ hL¯ (X) ≤ e(L). k +1 y∈X(Q¯) ¯ In particular, if hL¯ (x) ≥ 0 for all x ∈ X(Q), then hL¯ (X) ≥ 0.

In particular, we can deduce the following.

Corollary 6. — Assume L is big and nef Q-line bundle on X which is endowed with an adelic ¯ semi-positive continuous metrization {k·kv}v∈MK . Assume hL¯ (x) ≥ 0 for any point x ∈ X(Q), then hL¯ (X) ≥ 0. 6 THOMAS GAUTHIER

¯ Proof. — Let A be an ample hermitian line bundle on X with hA¯ ≥ 0 on X and ǫ > 0 rational. ¯ ¯ ¯ ¯ Then L(ǫ) := L + ǫA is an ample Q-line bundle and hL¯(ǫ) ≥ 0 on X(Q). By Lemma 5, we have hL¯(ǫ)(X) ≥ 0. First, note that k k vol(L(ǫ)) = (L + ǫA)k = (L)j (ǫA)k−j = vol(L)+ O(ǫ). j j=0 X Also, we can compute similarly k+1 k+1 k +1 k+1−j j k+1 L¯(ǫ) = ǫj L¯ A¯ = L¯ + O(ǫ). j j=0 X We thus deduce that

0 ≤ hL¯(ǫ)(X)= hL¯ (X)+ O(ǫ) and the conclusion follows making ǫ → 0.

1.2. Test functions Let L be a big and nef Q-line bundle on X. We equip L with an adelic continuous semi-positive ¯ metric {k·kv}v∈MK and we denote L := (L, {k·kv}v∈MK ). We also ﬁx a Zariski open set V ⊂ X. Let v ∈ MK be any place of the ﬁeld K.

an an Definition 7. — A function ϕ : Xv → R is called a test function on Xv if it is C ∞ an – a -smooth function on Xv if v is archimedean, an – a Q-model function on Xv if v is non-archimedean. an an We also say ϕ is a test function on Vv is it is a test function on Xv and if its support is an compactly contained in Vv . C ∞ an an If v is archimedean, as the space c (Vv ) of smooth functions on Vv with compact support is C 0 an an dense in the space c (Vv ) of continuous functions on Vv with compact support, it is sufficient to test the convergence against smooth functions. If v is non-archimedean, we use a result of an Gubler [Gu, Theorem 7.12]: Q-model functions with compact support in Vv are dense in the C 0 an an space c (Vv ) of continuous functions on Vv with compact support, it is sufficient to test the convergence against compactly supported model functions. In particular, we have the following: C 0 an Lemma 8 (Gubler). — Test functions are dense in in the space c (Vv ) of continuous functions an on Vv with compact support.

1.3. Arithmetic volume and the adelic Minkowski Theorem

We want here to use the arithmetic Minkowski’s second Theorem to relate the limit of hM¯ (Fi) along a generic sequence of finite subsets {Fi}i of X(Q¯ ) and the arithmetic volume of M¯ , for any 0 continuous adelic metrized Q-line bundle. For any v ∈ MK, and any section s ∈ H (X,M), set A K kskv,sup := supx∈X(Q¯ v) ks(x)kv. If K is the ring of adels of and if µ is a Haar measure on the 0 locally compact abelian group H (X,M) ⊗ AK, let 0 0 µ(H (X,M) ⊗ AK/H (X,M)) χsup(M¯ )= − log , µ( v Bv) 0 where Bv is the (closed) unit ball of H (X,M)⊗Kv for theQ norm induced by k·kv,sup, see e.g. [CLT] or [Zha2]. The arithmetic volume volχ(M¯ ) of M¯ is the defined as ¯ ¯ χsup(NM) volχc(M) := lim sup k+1 . N→∞ N /(k + 1)! c an R C ∞ an Choose a place v0 ∈ MK and let ϕ : Xv0 → be a test functions, i.e. ϕ ∈ (Xv0 ) if v0 is archimedean and ϕ is a model function otherwise. We define a twisted metrized line bundle M¯ (ϕ) −ϕ by changing the metric at the place v0 as follows: let k·kv0,ϕ := k·kv0 e . GOOD HEIGHT FUNCTIONS ON QUASI-PROJECTIVE VARIETIES 7

Using the arithmetic Minkowski’s second Theorem, we can prove the next lemma using a classical argument (see, e.g., [BB3]). Lemma 9. — For any big and nef line bundle M¯ endowed with an adelic semi-positive continuous metrization, any , any test function an R and any generic sequence of v0 ∈ MK ϕ : Xv0 → (Fi)i Galois-invariant subsets of X(Q¯), we have

volχ(M¯ (ϕ)) lim inf hM¯ (ϕ)(Fi) ≥ . i→∞ [K : Q](k + 1)vol(M) c Proof. — By Minkowski’s theorem [BG, Theorem C.2.11], as soon as volχ(M¯ (ϕ)) > 0 is large enough, there exists a non-zero small section s ∈ H0(X,NL) such that c vol (M¯(ϕ)) log ksk ≤− χ N + o(N) v0 [K : Q](k + 1)vol(M) c and log kskv ≤ 0 for any other place v ∈ MK, where k·kv is the metrization of NM¯ (ϕ) at the place v induced by that of M¯ . As (Fi)i is generic, there is i0 such that Fi ∩ supp(div(s)) = ∅ for i ≥ i0. We thus can compute the height hNM¯ (ϕ) of Fi using this small section s to ﬁnd

1 volχ(M¯ (ϕ)) h ¯ (F )= h ¯ (F ) ≥ + o(1). M(ϕ) i N NM(ϕ) i [K : Q](k + 1)vol(M) Making N → ∞, this gives the wanted estimate. To conclude,c we remark that for any c > 0, by deﬁnition,

volχ(M¯ (c + ϕ)) = volχ(M¯ (ϕ)) + c(k + 1)vol(M). Also, for any closed point x ∈ X(Q¯) we ﬁnd easily c c c h ¯ (x)= h ¯ (x)+ , M(c+ϕ) M(ϕ) [K : Q] so that the quantity we want to compare are both translated by c/[K : Q] as soon as we add c to

ϕ. Taking c large enough, we can reduce to the case when volχ(M¯ (ϕ)) as large as required.

1.4. A lower bound on the arithmetic volume c Let now L be a big and nef Q-line bundle on X. We equip L with an adelic continuous semi-positive ¯ metric {k·kv}v∈MK and we denote L := (L, {k·kv}v∈MK ). We also ﬁx a Zariski open set V ⊂ X. an an Fix a place v ∈ MK and pick any test function ϕ : Xv → R with compact support in Vv and extend ϕ to X by zero. Let O¯v(ϕ) the trivial line bundle equipped with the trivial metric at all places but v and equipped with the metric induced by ϕ at the place v. For any t ∈ R, we let

L¯(tϕ) := L¯ + tO¯v(ϕ). The key fact of this section is the next proposition which relies on Yuan’s arithmetic bigness criterion `ala Siu [Y, Theorem 2.2] : Proposition 10. — Let M be a big and nef line bundle with non-ampleness locus contained in an an X \ V . For any non-constant test function ϕ : Xv → R compactly supported in Vv , and any decomposition O¯(ϕ) = M¯ + − M¯ − as a diﬀerence of big and nef adelic metrized line bundle with underlying line bundle M and any t> 0, we have ¯ ¯ k volχ(L(tϕ)) t c1(L)v ≥ hL¯ (X)+ ϕ [K : Q](k + 1)vol(L) [K : Q] an vol(L) Vv c Z supV an |ϕ| vol(L + tM) − t v − 1 [K : Q] vol(L) ! k+1 (k +2 − j) k +1 k−j+1 j − L¯ · tM¯ . [K : Q]vol(L) j + j=2 X 8 THOMAS GAUTHIER

an Proof. — Write ϕ = ψ+ − ψ−, where ψ± is a smooth psh metric on Mv . This is possible since M is big and nef and since ϕ is supported outside of the augmented base locus of M. On then can write O¯v(tϕ)= tM¯ + − tM¯ −, where M¯ ± are the induced metrizations on M. Extend both tM¯ ± as adelic metrized line bundles which coincide at all places w 6= v and pick any ample arithmetic line bundle L¯0 on X. As M is nef, for a given q ≥ 1, qM¯ ± + L¯0 is an ample hermitian line bundle. The line bundle qL¯(tϕ) is then the diﬀerence of two ample hermitian line bundles:

qL¯(tϕ)= qL¯ + tqM¯ + − tqM¯ − = qL¯ + t qM¯ + + L¯0 − t qM¯ − + L¯0 . Apply Yuan’s arithmetic bigness criterion [Y, Theorem 2.2] to multiples of NL¯(tϕ), where q divides N and making N → ∞ gives k+1 k (1) volχ(L¯(tϕ)) ≥ L¯ + tM¯ + − (k + 1) L¯ + tM¯ + · tM¯ − ¯ ¯ k+1 ¯ ¯ k ¯ Remark that L + tcM+ − (k + 1) L + tM+ · tM− expands as k+1 k+1 k+1 k−j+1 j−1 L¯ + O¯ (ϕ) · L¯ · tM¯ j v + j=1 X k+1 k +1 k−j+1 j − (k − j) L¯ · tM¯ , j + j=2 X where we used that tM¯ + − tM¯ − = O¯v(tϕ). As the underlying line bundle of O¯v(ϕ) is trivial, we can compute this intersection number with the use of the constant section 1. The term j = 1 is exactly the integral

¯ ¯ k ¯ k ¯ k Ov(tϕ) · L = t ϕc1(L)v = t ϕc1(L)v Xan V an Z v Z v and for j ≥ 2, we can compute

¯ ¯ k−j+1 ¯ j−1 ¯ k−j+1 j−1 Ov(tϕ) · L · tM+ = t ϕc1(L)v ∧ (tc1(M, ψ+)v) Xan Z v ¯ k−j+1 j−1 ≥−t sup |ϕ| c1(L)v ∧ (tc1(M, ψ+)v) Xan an v ZXv ≥−t sup |ϕ|(Lk+1−j · (tM)j−1). an Xv Summing from j =2 to k + 1, we ﬁnd k+1 k+1 k +1 k−j+1 j k +1 k+1−j j−1 O¯v(ϕ) · L¯ · tM¯ + ≥−t sup |ϕ| (L · (tM) ) j Xan j j=2 v j=2 X X k k +1 k ≥−t sup |ϕ| (Lk−j · (tM)j ) Xan j +1 j v j=1 X ≥−t(k + 1) sup |ϕ| (vol(L + tM) − vol(L)) . an Xv an To conclude, it is suﬃcient to notice that since ϕ is compactly supported in Vv , the sup and the an integral can be taken over Vv . The above summarizes as ¯ ¯ k+1 ¯ k volχ(L(tϕ)) ≥ L + (k + 1)t ϕc1(L)v V an Z v c − t(k + 1) sup |ϕ| (vol(L + tM) − vol(L)) an Vv k+1 k +1 k−j+1 j − (k − j) L¯ · tM¯ . j + j=2 X The conclusion follows dividing by (k + 1)[K : Q]vol(L). GOOD HEIGHT FUNCTIONS ON QUASI-PROJECTIVE VARIETIES 9

Remark. — In the non-archimedean case, one can directly relate vol(L¯(tϕ)) to the integral of ¯ k tϕ against c1(L)v and hL¯ (X) using a recent result of Boucksom Gubler and Martin [BGM2] (see also [BGM1]) and the continuity of the volume function on the appropriatec cone (containing the ample cone), instead of the arithmetic Siu bigness criterion. Here, the quantity vol(L¯) is the arithmetic volume defined as 0 ¯ c ¯ log #H (X,NL) vol(L) := lim sup k+1 , N→∞ N (k + 1)! b 0 0 c where H (X, L¯) = {s ∈ H (X,L): kskv,sup ≤ 1, for all v ∈ MK}. Let us sketch briefly how we can proceed in the case when L and M are ample: first, the arithmetic Hilbert-Samuel Theorem k+1 provedb by Zhang [Zha1] gives vol(L¯) = volχ(L¯) = L¯ . Write ϕ = ψ+ − ψ− and ψ± is a continuous psh metric on M. Then Lemma 3.1 of [BGM1] implies obviously that c c k vol(L¯(tϕ)) − vol(L¯) − (k + 1) tϕ(c1(L¯)v + tc1(M, ψ+)v) ≥ Xan Z v c c − (k + 1) sup |ϕ| (vol(L + tM) − vol(L)) . an Xv and we can conclude as above. The difficulty to pursue this approach is that we only have vol(L¯(tϕ)) ≥ volχ(L¯(tϕ)) in general. It could be interesting to be able to use this inequality directly to find a strengthened version of Proposition 10. c c 2. Good height functions and equidistribution

In this section, we prove Theorem 2 using the estimates established in Section 1. Let K be z number field, let V be a smooth quasi-projective variety of dimension k defined over K. Let h be an a v-good height function on V with induced measure µv on Vv and vol(h) > 0 be its volume. Let (Xn, L¯n, ψn) be given by the definition of good height function.

2.1. A preliminary lemma

N Let ι0 : X0 ֒→ P be an embedding deﬁned over K and let M0 be the ample line bundle on X0 given by condition (4). For n ≥ 0, let ¯ ∗ ¯ Mn := ψn(M0).

Since ψn is a birational morphism and since M0 is ample, Mn is big and nef and its non-ampleness locus is contained in Xn \ V . An important ingredient is the following:

Lemma 11. — The sequence vol(Ln + Mn) converges to a limit ℓ> 0. In particular, there exists a constant C ≥ 1 such that 0 ≤ vol(Ln + Mn) ≤ C, for all n ≥ 0.

Proof. — By deﬁnition of the line bundle Mn, the volume of Mn is big and nef, so that vol(Mn)= k k (Mn) = (M0) = vol(M0), hence it is independent of n. By assumption we also have (M )k vol(M ) = (M )k = (M )k = 0 vol(L )+ o(vol(L )) n n 0 vol(h) n n vol(M ) = 0 vol(L )+ o(vol(L )). vol(h) n n

In particular, there exists a constant C1 ≥ 1 independent of n such that vol(Mn) ≤ C1vol(Ln) and thus k k k k vol(Ln + Mn) = (Ln + Mn) ≤ (1 + C1) (Ln) = (1+ C1) vol(Ln).

As vol(Ln) is bounded, the conclusion follows. 10 THOMAS GAUTHIER

2.2. Proof of Theorem 2

an an Let v ∈ MK be such that h is v-good and let ϕ : Vv → R be a test function on Vv . For t > 0, we deﬁne an adelic metrized line bundle by

L¯n(tϕn) := L¯n + O¯(tϕn).

Let (Fi)i be a sequence of generic and h-small Galois-invariant ﬁnite subsets Fi ⊂ V (Q¯ ) and let −1 Fi,n := ψn (Fi). We apply Proposition 10 and Lemma 9 to ﬁnd ¯ k t c1(Ln)v lim inf hL¯n(tϕn)(Fi,n) ≥ hL¯n (Xn) − ϕn i→∞ [K : Q] V an vol(Ln) Z v supV an |ϕ| vol(L + M ) − t2 v n n − 1 [K : Q] vol(Ln) ! k+1 (k − j) k +1 k−j+1 j − L¯ · tM¯ [K : Q]vol(L) j n n,+ j=2 X for any decomposition O¯v(ϕn)= M¯ n,+ −M¯ n,−, where M¯ n,± is a semi-positive continuous metrized an big and nef line bundle on (Xn)v with underlying line bundle Mn. If we write ϕ = ψ+ −ψ− where an ψ± are metrizations on M0,v, then one can obviously write ϕn = ψ+ ◦ ψn − ψ− ◦ ψn. We thus have ¯ ∗ ¯ Mn,+ = ψn(M0,+). As vol(Ln) → vol(h) > 0, by Lemma 11 and hypothesis (4), there exists C1 ≥ 1 independent of n such that for any 0

¯ k 1 c1(Ln)v εn({Fi}i) lim inf ϕ(y) − ϕn ≥− − C1t i→+∞ #Fi an vol(Ln)  t y F Vv X∈ i Z for all 0

¯ k 1 c1(Ln)v εn({Fi}i) lim sup ϕ(y) − ϕn ≤ + C1t, i→+∞ #Fi V an vol(Ln) t y∈Fi Z v X for all 0

¯ k ¯ k c1(Ln)v (ψn)∗c1(Ln)v ϕn = ϕ . an vol(Ln) an vol(Ln) ZVv ZVv Fix now ε> 0 small. We now let t := ε/2C1 > 0. By assumption, we can choose n0 ≥ 1 such that εn({xi}i) we have t ≤ ε/2 for any n ≥ n0. Therefore, for n ≥ n0,

1 (ψ ) c (L¯ )k (2) lim sup ϕ(y) − ϕ n ∗ 1 n v ≤ ε. i→+∞ #Fi V an vol(Ln) y∈Fi Z v X

GOOD HEIGHT FUNCTIONS ON QUASI-PROJECTIVE VARIETIES 11

By assumption (2) of the deﬁnition of good height, we have 1 = 1 + o(1) and there is vol(Ln) vol(h) −1 −1 n1 ≥ 1 such that vol(Ln) − vol(h) ≤ ε for any n ≥ n1. Finally, by assumption (3), there exists n ≥ 1 such that 2

¯ k ϕ (ψn)∗c1(Ln)v − ϕ vol(h)µv ≤ ε. V an V an Z v Z v

Putting together the two inequalities above, we ﬁnd

¯ k (ψn)∗c1(Ln)v ϕ − ϕµv ≤ (1+ kϕkL∞ ) ε. V an vol(Ln) V an Z v Z v

Combined with (2), this completes the proof for generic and h-small sequences.

We now show how to deduce the general case, proceeding as in [FG1, §5.5] . Let us enumerate all irreducible hypersurfaces (Hℓ)ℓ of V that are deﬁned over K. We use the next lemma, see e.g. [FG1, Lemma 5.12].

Lemma 12. — Take a sequence (Fn)n of Galois-invariant ﬁnite subsets of V (Q¯ ) with #(F ∩ H ) lim n ℓ =0, n→∞ #Fn ′ for any ℓ. Then, for any ǫ> 0, there exists a sequence of sets Fn,ǫ ⊂ Fk such that: ′ 1. #Fn,ǫ ≥ (1 − ǫ)#Fn for all n, ′ 2. Fn,ǫ is Galois-invariant, ′ 3. for any ℓ there exists N(ℓ) ≥ 1, such that Fn,ǫ ∩ Hℓ = ∅ for all n ≥ N(ℓ). ′ Fix ǫ> 0. The last condition of Lemma 12 implies Fn,ǫ to be generic. Now pick any continuous C 0 an function ϕ ∈ c (Vv ). The above implies that there is n0 ≥ 1 such that

′ ϕµFn,v − ϕµv ≤ ϕµFn,v − ϕµFn,ǫ,v + ǫ. V an V an V an V an Z v Z v Z v Z v

We infer from Lemma 12 that

1 ϕµFn,v − ϕµv ≤ |ϕ| δx an an #Fn an Vv Vv Vv x F F ′ Z Z Z ∈ n\ n,ǫ X 1 1

+ ′ − |ϕ| δx #F #Fn an n,ǫ Vv x F ′ Z ∈Xn,ǫ ≤ 2 ǫ sup |ϕ| . an Vv

This shows that for n ≥ n0, we have

ϕµFn,v − ϕµv ≤ 1 + 2 sup |ϕ| ǫ, an an an Vv Vv Vv ! Z Z which concludes the proof.

2.3. Quasi-adelic measures on the projective line are good Let K be a number ﬁeld. We prove here that quasi-adelic measures on P1 as deﬁned by Mavraki and Ye [MY], and their induced height functions, satisfy the assumptions of Theorem 2.

Let us deﬁned quasi-adelic measures and their height functions following Mavraki and Ye. Pick + 1 1,an v ∈ MK. Write log | · |v := log max{| · |v, 1} on A (Cv). This function extends to Av and c + 1,an dd log | · |v = δ∞ − λv, as a function from Pv to R+ ∪{+∞}, where λv is the Lebesgue measure 1,an on {|z|v = 1} if v is archimedean, and λv = δG,v is the dirac mass at the Gauß point of Pv otherwise. 12 THOMAS GAUTHIER

1,an c A probability measure on Pv has continuous potential if µv − λv = dd gv for some gv ∈ C 0 1,an 1,an (Pv , R). In this case, there is a unique function gµv : Pv → R ∪{+∞} such that gv = + log | · |v − gµv with the following normalization: if we let

gµv (x/y) + log |y|v for (x, y) ∈ Cv × (Cv \{0}), G (x, y) := log |x| − g (∞) for (x, y) ∈ (C \{0}) ×{0}, µv  v v v  −∞ for (x, y)=(0, 0), C2 then the set Mµv := {(x, y) ∈ v : Gµv (x, y) ≤ 0} has homogeneous logarithmic capacity 1. Note 2 that for α ∈ Cv \{0} and (x, y) ∈ Cv \{(0, 0)}, we have

Gµv (αx, αy)= Gµv (x, y) + log |α|v.

We then deﬁne the inner (resp. outer) radius of µv as ¯ ¯ rin(µv) := sup{r> 0 : Dv(0, r) × Dv(0, r) ⊂ Mµv }, r (µv) := inf{r> 0 : Mµ ⊂ D¯ v(0, r) × D¯v(0, r)} out v 1 Deﬁnition 13. — We say that µ = {µv}v∈MK is a quasi-adelic measure on P if

– for any v ∈ MK, the measure µv has continuous potential, – both series and converge. v∈MK | log rin(µv)| v∈MK | log rout(µv)| The height function h induced by a quasi-adelic measure µ = {µ } is deﬁned as P µ P v v∈MK 1 h (z) := G (σ(x), σ(y))), µ [K : Q]#O(x) µv v M X∈ K σ:K(Xx)֒→Cv where (x, y) ∈ A2(Q¯ ) \{(0, 0)} is any point with z = [x : y].

As the functions Gµv are homogeneous, the product formula implies the height function hµ is well-deﬁned and independent of the choice of (x, y).

We prove here the following.

Proposition 14. — Let K be a number ﬁeld and let µ := {µv}v∈MK be a quasi-adelic measure on 1 1 P with induced height function hµ. Then hµ is a good height function on P with induced global measure {µv}v∈MK .

1 Proof. — Enumerate the places of K as MK := {vn, n ≥ 0} and deﬁne Xn = P and L¯n as OP1 (1) endowed with the adelic continous semi-positive metrization {| · |v,n}v∈MK deﬁned by the following conditions:

– for any j ≥ n + 1, the metric | · |vj ,n is the usual naive metric on OP1 (1) at place vj , – for any j ≤ n, the metric is that induced by g , µvj so that for any z ∈ P1(Q¯) and any (x, y) ∈ A2(Q¯ ) \{(0, 0)} with z = [x : y], we have

n ∞ 1 h ¯ (z)= Gµ (σ(x), σ(y)) + log kσ(x), σ(y))kv . Ln [K : Q]#O(z)  vj ℓ  j=0 ℓ n σ:K(Xx)֒→Cv X =X+1   For any n ≥ 0, we have vol(Ln) = 1 and for all v ∈ MK, we have c1(L¯n)v = µv if n is large enough. All there is left to prove is that there is a h-small sequence and that the condition of pointwise approximation is satisﬁed on P1.

The deﬁnitions of rin(µv) and rout(µv) and the product formula give ∞ ∞ log(rin(µv)) log(rout(µv)) ≤ h ¯ − h ≤ , [K : Q] Ln µ [K : Q] j=n+1 j=n+1 X X on P1(Q¯ ). In particular, if we let ∞ 1 ε(n) := max{| log r (µ )|, | log r (µ )|}, [K : Q] in v out v j=n+1 X GOOD HEIGHT FUNCTIONS ON QUASI-PROJECTIVE VARIETIES 13 then ε(n) → 0 as n → ∞, by assumption, and P1 Q¯ (3) |hL¯n − hµ|≤ ε(n), on ( ).

We are thus left with justifying the existence of a small hµ-sequence to conclude the proof. Let ¯ ¯ P1 Ln be a model of Ln over OK . By the arithmetic Hilbert-Samuel theorem as stated in [RLV, Theorem A], we have n 2 0= − log cap (M )= vol(L¯ )= L¯ , vj µvj n n j=0 X 1 since we assumed log cap (M ) = 0 for all j. In particular,c h ¯ (P ) = 0 and the assumption 1 vj µvj Ln of the definition of good height function is satsified. Finally, for any n ≥ 0, there is a generic sequence (Fi,n)i of Galois invariant finite subsets of P1 Q¯ ( ) such that hL¯n (Fi,n) → 0, as i → ∞. In particular, By inequality (3),

lim sup |hµ(Fi,n)|≤ ε(n), i→∞ and a diagonal extraction argument implies there exists (F˜i)i which is generic and such that ˜ lim supi→∞ |hµ(Fi)| → 0, and the proof is complete.

3. Equidistribution in families of dynamical systems

We give here a couple of applications of Theorem 2 in families of polarized endomorphisms with marked points. We begin with a general result and then we focus on special cases.

3.1. Families of polarized endomorphisms Let S be a smooth quasi-projective variety of dimension p ≥ 1 and let π : X → S be a family of smooth projective varieties of dimension k ≥ p. We say (X , f, L) is a family of polarized endomorphisms if f : X → X is a morphism with π ◦ f = π and if there is a relatively ample line bundle L on X and an integer d ≥ 2 such that f ∗L ≃ L⊗d. When (X , f, L) is defined over Q¯ , following Call and Silverman [CS], for any t ∈ S(Q¯) we can define a canonical height function for −1 the restriction ft : Xt → Xt of f to the fiber Xt := π {t} of π : X → S as 1 ◦n ¯ hft (x) = lim hXt,Lt (ft (x)) , x ∈ Xt(Q). n→∞ dn

If L is a finite extension ofb K such that t ∈ S(L), the canonical height hft is induced by an adelic semi-positive continuous metrization {k·kft,v}v∈ML on the ample line bundle Lt of Xt. We then have b ¯ 1. hft (x) ≥ 0 for all x ∈ X(Q), ¯ 2. hft − hXt,Lt = O(1) on X(Q), and b the function hft is characterized by those two properties. Moreover, by the Northcott property, we haveb b 3. hft (x) = 0 if and only if x is preperiodic under iteration of ft. † Whenb π : X → S, L and f are all defined over C, we can associate to (X , f, L) a closed positive (1, 1)-current on X (C) with continuous potentials which we can define as

1 ◦n ∗ ∗ Tf := lim (f ) (ι ωPN ), n→∞ dn N ∗ where ι : X ֒→ P induces (a largeb power of) L⊗ π (M), where M is ample on a projective model S¯ of S. Moreover, the convergence towards Tf is uniform local for potentials, see, e.g. [GV]. This current Tf restricts on ﬁbers Xt(C) of π as the Green current of the endomorphism ft of Xt(C) b k which is polarized by Lt. Moreover, the closed positive (k, k)-current Tf on X (C) restricts to the b ﬁber Xt(C) as vol(Lt) · µft , where µft is the unique maximal entropy probability measure of ft. b 14 THOMAS GAUTHIER

To a section a : S → X of π which is regular on the quasi-projective variety S, as introduced in [GV] adaptating the classical deﬁnition, we associate a bifurcation current deﬁned as

Tf,a := (π)∗ Tf ∧ [a(S)] . b 3.2. Step 1: reduction to q =1 and deﬁnition of (Bn, L¯n, ψn)

According to Theorem 2, all there is to do is to prove that hf,a is a good height and that, for any an a −1 archimedean place v ∈ MK, the induced measure on Sv is volf ( ) µf,a,v.

[q] Note first that this reduces to proving this holds on the case q = 1. Indeed, if π[q] : X → S is the q-fibered product of π : X → S, let f [q] : X [q] → X [q] be defined by

[q] −1 f (x) = (ft(x1),...,ft(xq)), x = (x1,...,xq) ∈ π[q] {t}.

Then (X [q],f [q], L[q]) is a family of polarized endomorphisms parametrized by S. Moreover, an [q] easy computation gives, for all x = (x1,...,xq) ∈ X with π[q](x)= t,

q [q] (4) h [q] (x1,...,xq)= hft (xj ) ≥ 0 and h [q] ft (x) = d · h [q] (x). ft ft ft j=1 X b b b b [q] Let a : S → X be the section of π[q] induced by a1,...,aq. As before, we deﬁne the bifurcation current of a as

Tf,a := (π[q])∗ Tf [q] ∧ [a(S)] . An easy computation gives b

(5) Tf,a = Tf,a1 + ··· + Tf,aq .

p q Combining equations (5) and (4), we deduce that µf,a,v = Tf,a and that hf,a = j=1 hf,aj . In particular, up to replacing f by f [q] and a ,...,a by a, we can assume q = 1. 1 q P

.Let ι : X ֒→ PM1 × S ֒→ PM1 × PM2 ֒→ PN , where the last embedding is the Segre embedding Let also S¯ be the closure of S in PM2 induced by this embedding. Up to taking a large multiple of L, we can assume the ﬁrst embedding ι1 is induced by L and

hι = hX ,L + hS on X (Q¯ ), where hS is the ample height on S¯ induced by the second embedding. For any n ≥ 1, deﬁne a section an : S → X by a ◦n n(t) := ft (a(t)), t ∈ S.

As an is a section of π : X → S, whenever an(t) = an(s), we have t = π ◦ an(t) = π ◦ an(s) = s, N hence an has finite fibers. In particular, ι ◦ an : S → P is finite. We can assume a extends as a morphism from B0 := S¯ to X¯, which is then generically finite. N If ι ◦ an+1 extends as a morphism Bn → P , let Bn+1 := Bn and ψn+1 = ψn. Otherwise, let pn+1 : Bn+1 → Bn be a finite sequence of blowups such that ι ◦ an+1 ◦ ψn ◦ pn+1 extends as a N N morphism Bn+1 → P . Let then ψn+1 := ψn ◦pn+1. The morphism ι◦an+1 ◦ψn+1 : Bn+1 → P is then generically finite by construction. We equip OPN (1) with its standard metrization and denote it by O¯(1). Define 1 L¯ := (ι ◦ a ◦ ψ )∗O¯(1). n dn n n

As O¯(1) is ample and ι ◦ an ◦ ψn is generically ﬁnite, L¯n is big and nef. Moreover, by construction, the variety Bn is deﬁned over K. GOOD HEIGHT FUNCTIONS ON QUASI-PROJECTIVE VARIETIES 15

3.3. Step 2: convergence of the measures ¯ p an Fix a place v ∈ MK. We now prove that the sequence (c1(Ln)v)n of positive measures on Sv converges weakly to a positive measure µv. Assume ﬁrst v is archimedean. We follow classical arguments we summarize here: we can write −1 ∗ ∗ ∗ c C ∞ an d f (ι ωPN ) − ι ωPN = dd u, where u ∈ (Xv ). An easy induction gives, for any integer n ≥ 1, 1 ◦n ∗ ∗ 1 c ◦n an T − (f ) (ι ω N )= dd g ◦ f on X , f dn P dn v −j ◦j C 0 an a ¯ where g = j≥0 d ·u◦bf ∈ (Xv ). Pulling back by : S → X , we ﬁnd Tf,a −(ψn)∗c1(Ln)v = −n c ◦n an ¯ d · dd g ◦ f on Sv . In particular, (ψn)∗c1(Ln)v converges to Tf,a with a local uniform con- P ¯ p an vergence of potentials. Thus ((ψn)∗c1(Ln)v)n converges weakly on Sv to µf,a,v. When v is non-archimedean, we rely on the work [CD] of Chambert-Loir and Ducros, and we employ freely their vocabulary. We follow the strategy used in the archimedean case, as presented in [BE1]: let k·kv be the naive (Fubini-Study) metric on OPN (1) at the place v so that the metric −n a ∗ −1 ∗ CK on Ln is d (ι◦ n ◦ψn) k·kv. On the space of metrics on L, the map (d f −id) is d -contracting an over any compact subset K of Xv . In particular, we have 1 1 ∗ C (f ◦n)∗ (ι∗k·k ) − f ◦(n+1) (ι∗k·k ) ≤ K , dn v dn+1 v dn+1

over any compact subset K ⋐ X an. In particular, the sequence v 1 (f ◦n)∗ (ι∗k·k ) dn v n an converges uniformly over compact subsets of Xv to a continuous and semi-positive metric k·kf,v on L which satisﬁes 1 C′ (f ◦n)∗ (ι∗k·k ) −k·k ≤ K , dn v f,v dn

⋐ an an over any compact subset K Xv . To conclude, we remark that on Sv , one can write

∗ 1 ◦n ∗ ∗ (ψ ) c (L¯ ) = a (f ) ι c (O¯ N (1)) n ∗ 1 n v 0 dn 1 P v a∗ c = 0 (c1(L, k·kf,v)) + dd un, ⋐ an and the above implies that (un) converges uniformly on any compact subset K Sv to the C 0 an constant function 0 ∈ (Sv ) (which is obviously locally approachable). By [CD, Corollaire (5.6.5)], this implies ¯ p a∗ p (ψn)∗c1(Ln) → µv := ( 0 (c1(L, k·kf,v))) an in the weak sense of measures on Sv , as granted.

3.4. Step 3: convergence of volumes

By the above, L¯n is an ample Q-line bundle equipped with a semi-positive adelic continuous Q¯ metrization. Moreover, we have hL¯n (t) ≥ 0 for all t ∈ Bn( ), so that Corollary 6 implies hL¯n (Bn) ≥ 0. Next, we prove that vol(Ln) → volf (a) as n → ∞ to conclude. To do so, we rely on [GV].

Lemma 15. — There is C1 > 0 depending only on (X , f, L), S, a and ι such that C |vol(L ) − vol (a)|≤ 1 . n f dn

Moreover, for any ample class H on X0, there is a constant C2 > 0 depending only on (X , f, L), S, a, ι and H such that for any n ≥ 1,

∗ p−1 ψnH · (Ln) ≤ C2 16 THOMAS GAUTHIER

an Proof. — Let us prove the claim, by computing masses of currents on Xv with respect to the ∗ K¨ahler form ι (ωPN ). Then, one can interpret vol(Ln) as p 1 a ∗ 1 a an ◦n ∗ ∗ p vol(Ln)= n (ι ◦ n) ωPN = np [ (Sv )] ∧ (f ) ι (ωPN ) . San d d an Z v Z(X )v First, point 1 of Proposition 13 of [GV] implies kTf k is ﬁnite and

n ◦n ∗ ∗ d kTf k−k(f ) b(ι ωPN )k ≤ C0 for some constant C depending only on (X , f, L). Then, the second point of Proposition 13 of 0 b [GV] rereads as

p−1 a ′ n(j−p) ∧j a an ◦n ∗ ∗ ∧(p−j−1) |vol(Ln) − volf ( )|≤ C0 d Tf ∧ [ (Sv )] ∧ ((f ) ι (ωPN )) j=0 X ′ b for some constant C0 > 0 depending only on (X , f, L). This gives p−1 p−j−1 C |vol(L ) − vol (a)|≤ C′ dn(j−p)kT kj · dnkT k + C ≤ 1 , n f 0 f f 0 dn j=0 X by Bezout Theorem. b b

an Let now ωH be a smooth (1, 1)-form on Sv cohomologous to H. Then

p−1 1 an ◦n ∗ ∗ p−1 ∗ a N Hn · (Ln) = n(p−1) [ (Sv )] ∧ (f ) ι (ωP ) ∧ (π ωH ) d an ZXv and as above, for any n ≥ 1, we have

p−2 p−1 ′ n(j−p+1) j a an ◦n ∗ ∗ p−j−2 ∗ Hn · (Ln) ≤ C0 d Tf ∧ [ (Sv )] ∧ ((f ) ι (ωPN )) ∧ (π ωH ) j=0 X p−1 a anb ∗ + Tf ∧ [ (Sv )] ∧ (π ωH ) , so that we deduce as in the proof of the ﬁrst point of the Lemma that b p−1 ′ Hn · (Ln) ≤ C , for some constant C′ > 0 depending only on (X , f, L), a, ι and H.

3.5. Step 4: an upper bound on intersection numbers

Let M¯ 0 be any ample adelic semi-positive line bundle on B0. We prove here the next lemma.

Lemma 16. — There is a constant C ≥ 0 such that for any n ≥ 0 and any 2 ≤ j ≤ p +1, ∗ ¯ j ¯ p+1−j ψn(M0) · Ln ≤ C.

Proof. — Up to changing the initial projective model B0 of S, we can assume B0 \ S is the ∗ ∗ ∗ ∗ support of an eﬀective divisor D of B0 and that E¯ := f (ι O¯(1)) − dι O¯(1) = π (E¯0), for some adelic metrize line bundle E¯0 on B0 which can be represented by a divisor supported on supp(D), where π : X¯ → B0 is the extension of π : X → S. We may also assume a0 extends as a morphism ¯ ¯ ¯ ¯ from B0 to X . Let Xn,1 := X ⊗B0 Bn and there is a normal projective variety Xn, a birational ¯ ¯ ¯ −1 morphism Xn → Xn,1 and a projective morphism πn : Xn → Bn which is ﬂat over ψn (S) such ◦j ¯ ¯ that f extends as a morphism Fj,n : Xn → X for all j ≤ n. Let also a0,(n) be the section of πn ¯ induced by a0. Up to blowing up Bn, we can assume a0,(n) extends as a morphism from Bn to Xn. Let Ψn : X¯n → X¯ be the birational morphism induced by this construction. Note that Ψn is an −1 −1 −1 a a isomorphism from πn (ψn (S)) to π (S) and that π ◦ Ψn = ψn ◦ πn and Ψn ◦ 0,(n) = 0 ◦ ψn. GOOD HEIGHT FUNCTIONS ON QUASI-PROJECTIVE VARIETIES 17

¯ a∗ ¯ ¯ ¯ Let N0 := 0E. By construction of Ln and Ln+1 we can write 1 L¯ − p∗ L¯ = a∗ (F )∗ f ∗ι∗O¯(1) n+1 n+1 n dn+1 0,(n+1) n,n+1 1 ∗ − · a∗ (F ) ι∗O¯(1) dn 0,(n+1) n,n+1 1 ∗ = a∗ (Ψ ) f ∗ι∗O¯(1) − d · ι∗O¯(1) dn+1 0,(n+1) n 1 ∗ = a∗ (Ψ ) (E¯) dn+1 0,(n+1) n 1 = ψ∗ N¯ . dn+1 n+1 0 1−d−n If we let α(n) := d−1 , using that pn+1 ◦ ψn = ψn+1, an easy induction gives ¯ ∗ ¯ ¯ (6) Ln = ψn L0 + α(n) · N0 .

In particular, if N¯0 is a metrized line bundle on B0 which restricts to N¯0 on the special fiber of B O j ∗ ¯ j ¯ p+1−j 0 → Spec( K) andf if we let In := ψn(M0) · Ln , using first the equation (6) and then the projection formula, we find j ∗ ¯ j ∗ ¯ N¯ p+1−j In = ψn(M0) · ψn L0 + α(n) · 0 j p+1−j = M¯ 0 · L¯0 + α(n) · N¯0 p+1−j p+1 − j j ℓ p+1−j−ℓ = (α(n))p+1−j−ℓ M¯ · L¯ · N¯ . ℓ 0 0 0 ℓ=0 X j ℓ p+1−j−ℓ Take C1 ≥ 0 such that M¯ 0 · L¯0 · N¯0 ≤ C1 for any 2 ≤ j ≤ p + 1 and any 0 ≤ ℓ ≤ p +1 − j. As α(n) ≤ 1 for any n ≥ 0, the above gives j ¯ j ¯ p+1−j In ≤ M0 · L0 + C2, where C2 depends only on p and C1. The conclusion of the lemma follows.

3.6. Step 5: end of the proof of Theorem 3

All there is thus left to prove is that, for any given generic and h-small sequence {xi}i, the induced sequence εn({xi}i) := lim sup hL¯n (xi) converges to 0 as n → ∞. We rely on [CS] and again on [GV] and we use Siu’s classical bigness criterion as, e.g., in [GOV, §7]. Deﬁne a line bundle Hn on Bn by letting ∗ Hn := ψn (H) , where H is the ample line bundle on X0 inducing hS. As ψn is a birational morphism, Hn is big and nef. By construction, Ln is also big and nef. Fix n0 ≥ 1 large enough to that Lemma 15 gives 1 vol(L ) ≥ vol (a), for all n ≥ n . n 2 f 0

Fix now an integer M ≥ 1 such that Mvolf (a) > 2pC2. By multilinearity of the intersection number, we ﬁnd M p vol(ML )= M pvol(L ) ≥ vol (a) > pM p−1C ≥ p H · (ML )p−1 , n n 2 f 2 n n so that by the classical bigness criterion of Siu [L, Theorem 2.2.15], MLn −Hn is big. In particular, there exists an integer ℓ ≥ 1 such that ℓ(MLn − Hn) is eﬀective. According to [HS, Theorem B.3.2(e)], this implies there exists a Zariski open subset Un ⊂ Bn such that ¯ hℓ(MLn−Hn) ≥ O(1) on Un(Q). 18 THOMAS GAUTHIER

Also, by functorial properties of Weil heights, see e.g. [HS, Theorem B.3.2(b-c)],

hℓ(MLn−Hn) = hℓMLn − hℓHn + O(1) = ℓ MhL¯n − hHn + O(1) = ℓMh ¯ − ℓh ◦ ψ + O(1), Ln S n so that we have proved that Q¯ MhL¯n ≥ hS ◦ ψn + O(1), on Un( ). −1 Up to removing a Zariski closed subset of Un, we can assume Un ⊂ ψn (S). Since ψn is an isomorphism from Un to Sn := ψn(Un) ⊂ S, this gives −1 Q¯ hS(t) ≤ MhL¯n (ψn (t)) + N, t ∈ Sn( ), for some constant M ≥ 1 independent of n and some constant N which depends a priori on n. In particular, when t ∈ Sn(Q¯ ) and hS(t) → ∞, we ﬁnd −1 h ¯ (ψ (t)) 1 (7) lim inf Ln n ≥ > 0. hS (t)→∞ hS(t) M t∈Sn(Q¯)

According to [CS, Theorem 3.1], there is a constant C3 ≥ 1 depending only on (X , f, L), S, q and ι such that for all t ∈ S(Q¯ ) and all x ∈ X (Q¯ ) with π(x)= t,

∗ hft (x) − hι O¯(1)(x) ≤ C3(hS(t)+1).

Evaluating the above for x = a n(t), we ﬁnd b

a ∗ a hft ( n(t)) − hι O¯(1)( n(t)) ≤ C3(hS (t)+1), which, by invariance of h and by deﬁnition of L¯ , gives ft b n −1 C3 ¯ (8) hbf,a(t) − hL¯ (ψn (t)) ≤ (hS(t)+1), t ∈ S(Q). n dn n We now ﬁx n1 ≥ n0 such that 2MC3 ≤ d for all n ≥ n1. Pick an integer n ≥ n1. Using (7) and (8), we deduce that −1 hf,a(t) hL¯n (ψn (t)) C3 1 C3 1 lim inf ≥ lim inf − n ≥ − n ≥ > 0. hS (t)→∞ hS(t) hS (t)→∞ hS(t) d M d 2M t∈Sn(Q¯) t∈Sn(Q¯) To conclude the proof of Theorem 2, we now rewrite (8) as

−1 C4 h a(t) − h ¯ (ψ (t)) ≤ (1 + h a(t)), t ∈ S (Q¯ ), f, Ln n dn f, n where C4 > 0 is a constant independent of n and t. In particular, if (Fi)i is a generic and hf,a-small sequence of ﬁnite Galois-invariant subsets of S(Q¯ ), then Fi ⊂ Sn(Q¯ ) for i large enough and, using that hL¯n (Bn) ≥ 0, we deduce that

−1 C4 C4 ¯ ¯ lim sup hLn (ψn (Fi)) ≤ n ≤ n − hLn (Bn), i→∞ d d as required. We have proved that hf,a is a good height on S with associated global measure {µf,a,v}v∈K and Theorem 3 follows from Theorem 2.

Note that, on the way, we prove the following in the spirit of [GH, Theorem 1.4]:

Lemma 17. — If volf (a) > 0, there is a non-empty Zariski open subset U ⊂ a(S) and a constant c> 0 depending only on (X , f, L), S, a and ι such that ¯ hS(π(x)) ≤ c 1+ hfπ(x) (x) , x ∈ U(Q). ′ a In particular, there is c > 0 depending only onb (X , f, L), S, and ι such that ′ ¯ hfπ(x) (x) − hX ,L(x) ≤ c 1+ hfπ(x) (x) , x ∈ U(Q).

b b GOOD HEIGHT FUNCTIONS ON QUASI-PROJECTIVE VARIETIES 19

4. Applications and sharpness of the assumptions

We want ﬁnally to explore some speciﬁc cases. First, we consider the case of one parameter families of rational maps of P1. In a second time, we focus on the critical height on the moduli space of rational maps.

4.1. One-dimensional families of rational maps In the case when f : P1 × S → P1 × S is a family of rational maps of P1 parametrized by a smooth quasi-projective curve, Theorem 3 reduces to the following:

Theorem 18. — Let f : P1 × S → P1 × S be a family of degree d ≥ 1 rational maps parametrized by a smooth quasi-projective curve S and let a : S → P1 be a rational function, all defined over a number field K. Assume hfη (aη) > 0, where η is the generic point of S. Then the set ¯ Preper(f,a) := {t ∈ S(Q): hft (a(t))=0} is infinite. Moreover, for any v ∈ MK andb for any non-repeating sequence Fn ⊂ S(Q¯) of Galois-invariant finite sets such that b 1 hft (a(t)) → 0, as n → ∞, #Fn t F X∈ n b an the sequence µFn,v of probability measures on Sv equidistributed on Fn converges weakly to 1 b µf,a,v as n → ∞. hfη (aη)

Proof. — Choose any v ∈ MK archimedean. In this case, the author and Vigny [GV, Theorem B] proved that

hfη (aη)= µf,a,v. San Z v In particular, when S, f and a are definedb over a number field K, the assumption that µf,a,v > 0 for some archimedean v ∈ MK is satisfied whenever hfη (aη) > 0. To be able to apply Theorem 3, Q¯ it is thus sufficient to be able to produce a sequence tn ∈ S( ) such that hftn (a(tn)) = 0 for all n and such that, if O(tn) is the Galois orbit of tn, thenb O(tn) ∩ O(tm)= ∅ for all n 6= m. This is now classical and it can be done using Montel’s Theorem. b an We reproduce here the argument for completeness: let v ∈ MK be archimedean. Let U ⊂ Sv be any euclidean open set such that µf,a,v(U) > 0. By e.g. [De, Theorem 9.1] or [DF, Proposition- Definition 3.1], this is equivalent to the fact that the sequence of rational functions (an)n defined ◦n by an(t) := ft (a(t)) is not a normal family on U. Up to reducing U, by the Implicit Function 1,an Theorem, we can assume there exists N ≥ 3 and holomorphic function z : U → Pv such that N ◦n ◦n is minimal such thatft z(t)= z(t) for all t ∈ U, and z(t) is repelling for ft, i.e. |(ft) (z(t))| > 1. By Montel’s Theorem, one can define inductively tj+1 ∈ U \{tℓ,ℓ ≤ j} such that for all j ≥ 1, ◦2 there is n(j) ≥ 1 and an(j)(tj ) ∈ {z(tj),ftj (z(tj )),ftj (z(tj ))}. We thus have defined an infinite sequence (tn) of parameters for which a(tn) is preperiodic. In particular, we have hftn (a(tn))=0 for all n ≥ 1 and the proof is complete. b 4.2. Sharpness of the assumptions We now explain why the assumptions are sharp. When (X , f, L) is a family of polarized endomorphisms of degree d parametrized by a smooth complex quasi-projective curve S, we still have

hfη (aη)= µf,a = Tf ∧ [a(S(C))] ZS(C) ZX (C) for any regular section a : Sb → X by [GV, Theorem B].b However when the relative dimension of π : X → S is k > 1, the most probable situation is that there are at most ﬁnitely many t ∈ S(Q¯) such that hft (a(t)) = 0.

b 20 THOMAS GAUTHIER

Lemma 19. — There is a family of polarized endomorphisms (X , f, L) parametrized by A1 of relative dimension 2 and a section a : A1 → X , all deﬁned over Q, such that 1 ¯ hfη (aη)=1 and Preper(f,a) := {t ∈ A (Q): hft (a(t))=0} = ∅.

1 2 1 1 2 1 Proof. — let fb: (P ) × A → (P ) × A be defined by b 2 2 1 1 1 ft(z, w) = (z + t, w + t), (z,w,t) ∈ A × A × A . Define now a section a : A1 → A1 × A1 × A1 of the canonical projection π : A1 × A1 × A1 → A1 1 1 2 by letting a(t) := (t, 0, 4) for all t ∈ A . Write | · | := | · |∞. For any t ∈ A , let pt(z) := z + t we define the 1 + ◦n Gt(z) := lim log |pt (z)|, (t,z) ∈ C × C. n→∞ 2n c Then µf,a = ddt (Gt(0) + Gt(4)). Since for u ∈{0;4} we have 1 G (u)= log+ |t| + O(1), as |t| → ∞, t 2 1,an the measure µf,a is a probability measure on A , whence hfη (aη) = 1. Also, by an elementary computation, we have b 1 ¯ hf,a(t)= hpt (0) + hpt (4), t ∈ A (Q).

In particular, if hf,a(t) = 0, then Gt(0) = Gt(4) = 0. The condition Gt(0) = 0 implies |t|≤ 2 and b b by [Bu, Lemma 7] the condition Gt(4) = 0 implies that ◦n – either |t|≤ 2 and |t · pt (4)|≤ 2 for all n ≥ 0, ◦n – or |t| > 2 and |t · pt (4)| < 1 for all n ≥ 0. The second condition is empty since, for n = 0, this implies 2 < 1/4. For n = 0, the ﬁrst condition implies |t| ≤ 1/2 and for n = 1, it gives |t| ≤ 5/32. In particular, the polynomial pt has an attracting ﬁxed point and the only case when hft (0) = 0 is the case t = 0. Finally, for t = 0, G0(4) = log |4| > 0 ending the proof. b We thus can ask whether the condition of existence of a Zariski dense set of small points is reasonable in families with relative dimension > 1. Following the proof of Theorem 0.1 of [Du] exactly as adapted in [Ga], we can prove the next proposition.

Proposition 20. — Let (X , f, L) be a family of polarized endomorphisms of degree d parametrized by a smooth complex quasi-projective variety S with q sections a1,...,aq : S → X , Assume dim S = qk, where k is the relative dimension of X . Assume in addition that µf,a > 0. Then set ¯ Preper(f,a1,...,aq) := {t ∈ S(Q): hft (a1(t)) = ··· = hft (aq(t))=0} is Zariski dense in S(Q¯). In particular, if S, (X , f, L) and a ,...,a are defined over a number b 1 bq field, assumption 2 of Theorem 3 is satisfied.

We omit the proof since it copies verbatim that of [Ga, Theorem 2.2].

Finally, let (X , f, L) be a family of polarized endomorphisms of degree d parametrized by a smooth complex quasi-projective variety S with dim S > 1, given a K¨ahler form ω on S which is cohomologous to c1(M) with M ample on S,[GV, Theorem B] reads as q dim S−1 Tf,a1 + ··· + Tf,aq ∧ ωS = hfη (aj,η). ZS(C) j=1 X b We thus want to emphasize that the hypothesis that µf,a,v > 0 for some archimedean v ∈ MK is q stronger that only assuming j=1 hfη (aj,η) > 0. We can prove the following.

2 Lemma 21. — There is a familyP ofb (X , f, L) of polarized endomorphisms parametrized by A and a section a : A2 → X , all deﬁned over Q such that: GOOD HEIGHT FUNCTIONS ON QUASI-PROJECTIVE VARIETIES 21

2 1. if ωP2 is the Fubini-Study form of P (C), the current Tf,a satisﬁes

Tf,a ∧ ωP2 =1. 2 ZC 2. the bifurcation measure µf,a vanishes identically, 3. the set Preper(f,a) is Zariski dense in A2(Q¯ ).

Proof. — Define a family of degree 4 polynomials f : P1 × A2 → P1 × A2 by letting 1 2 s2 f (z)= z4 − sz3 + z2 + s4, z ∈ A1 and (s,t) ∈ A2, s,t 4 3 2 2 1 2 and define a : A → A × A by a(s,t) = (s, (s,t)). Let | · | := | · |∞. As in the proof of Lemma 19, 2 define G : C × C → R+ by letting

1 + ◦n 2 Gs,t(z) := lim log |fs,t (z)|, (s,t) ∈ C , z ∈ C. n→∞ 4n c c 2 We then have Tf,a = dd Gs,t(s) and µf,a = (dd Gs,t(s)) . We now follows arguments from [DF, §5–6]. Let us ﬁrst justify that µf,a = 0. This follows from c Bedford-Taylor theory. Let g(s,t) := Gs,t(s) so that the current Tf,a is dd g. As g ≥ 0, we have 1 c c g = limn→∞ max{g, n } and since supp(dd g) ⊂{g =0} and supp(dd max{g, 1/n}) ⊂{g =1/n}, we have c c 1 µf,a = lim dd g ∧ dd max g, =0. n→∞ n Since g(s,t) ≤ log+ max{|s|, |t|} + O(1) as k(s,t)k → ∞, we have

Tf,a ∧ ωP2 ≤ 1. 2 ZC 2 In particular, by Siu’s extension Theorem, the trivial extension of Tf,a to P (C) is a closed positive (1, 1)-current S which decomposes at T˜ + α[L∞], where [L∞] is the integration current on the lien at inﬁnity and T˜ ∧ ωP2 gives no mass to L∞. But [BB2, Theorem 4.2] implies that α = 0, whence

Tf,a ∧ ωP2 =1. 2 ZC To prove the last assertion, for n>m ≥ 0, we let 2 ◦n ◦m Preper(n,m) := {(s,t) ∈ C : fs,t (s)= fs,t (s)}. For n>m ≥ 0, the set Preper(n,m) is a (possibly reducible) plane curve of degree 4n which is deﬁned over Q. In particular, the set Preper(n,m)(Q¯ ) is inﬁnite. Also [DF, Theorem 1] implies that for any sequence {m(n)}n with 0 ≤ m(n)

4.3. In the moduli space of degree d rational maps

We finally focus on the case of the moduli space Md of degree d rational maps and we give a very quick proof of Theorem 4. The variety Md is the space of PGL(2) conjugacy classes of rational maps of degree d. By [S], the variety Md is irreducible, affine has dimension 2d − 2, and is defined over Q. cm The good setting to apply Theorem 3 is actually the critically marked moduli space Md . As in [BE2], define first

cm 1 2d−2 Ratd := {(f,c1,...,c2d−2) ∈ Ratd × (P ) : Crit(f)= [cj ]}, j X 22 THOMAS GAUTHIER

cm where Crit(f) stands for the critical divisor of f. The space Ratd is an quasi-projective variety of dimension 2d + 1 which is a finite branched cover of Ratd. We then define the critically marked cm moduli space Md as cm cm Md := Ratd /PGL(2), −1 where PGL(2) acts by φ · (f,c) = (φ ◦ f ◦ φ , φ(c1),...,φ(c2d−2)), and the quotient is geometric in the sense of Invariant Geometric Theory as in [S]. Again, it is an irreducible affine variety defined cm over Q. Moreover, we can directly apply Theorem 3 to the good height function hCrit : Md → R+ defined by 2d−2 hCrit(f,c1,...,c2d−2)= hf (cj ), j=1 X cm ¯ 1 2d−2 cm for all (f,c1,...,c2d−2) ∈ Md (Q). Indeed, we have a naturalb map f : (P ) × Md → 1 2d−2 cm c (P ) ×Md together with a section defined as

c : {(f,c1 ...,c2d−2)} 7→ ((c1,...,c2d−2), {(f,c1 ...,c2d−2)}) .

The current Tf [2d−2],c is then the bifurcation current Tbif of the family.

We now justify quickly why we are in the domain of application of Theorem 3. For any irreducible dim V subvariety V ⊂ Md, the measure µbif,V := Tbif is non zero if and only if V does not coincide 2 with the curve Ld of flexible Lattès maps, by [GOV, Lemma 6.8]. Here Ld is, when d = N , the family of maps induced by the multiplication by N on a non-isotrivial elliptic surface E → S. In 2d−2 cm particular, Tbif > 0 on Md . cm As dim Md =2d − 2, Proposition 20 implies the set of hf,crit-small points form a Zariski dense cm ¯ cm subset of Md (Q). In particular, we are in position to apply Theorem 3 in the family Md . To conclude the proof of Theorem 4, we just need to recall that cm 1. the canonical projection p : Md →Md is a finite branched cover, 2. a conjugacy class {f} is post-critically finite (PCF) iff and only if {f,c1,...,c2d−2} is hf,crit- small, cm 3. the bifurcation measures µbif and µbif,cm respectively of Md(C) and of Md (C) are related ∗ by µbif,cm = p (µbif). In particular, we deduce that for any sequence Fn ⊂ Md(Q¯ ) of Galois-invariant finite sets of post-critically finite parameters such that #(F ∩ H) n → 0, as n → ∞, #Fn for any hypersurface H of Md which is defined over Q, for any place v ∈ MQ the sequence of 1 an −1 probability measures δ on M converges weakly to vol(µ ) µ ,v in the weak #Fn {f}∈Fn {f} d,v bif bif sense of probability measures on Man . P d,v

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Thomas Gauthier, CMLS, Ecole´ Polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau Cedex, France E-mail : [email protected]