Adelic Cartier divisors with base conditions and the continuity of volumes
aa r X i v : . [ m a t h . AG ] F e b ADELIC CARTIER DIVISORS WITH BASE CONDITIONS ANDTHE CONTINUITY OF VOLUMES
HIDEAKI IKOMA
Abstract.
In the previous paper [7], we introduced a notion of pairs of adelic R -Cartier divisors and R -base conditions. The purpose of this paper is topropose an extended notion of adelic R -Cartier divisors that we call an ℓ -adelic R -Cartier divisors, and to show that the arithmetic volume function definedon the space of pairs of ℓ -adelic R -Cartier divisors and R -base conditions iscontinuous along the directions of ℓ -adelic R -Cartier divisors. Contents
1. Introduction 11.1. Notation and terminology 32. Fundamental estimate 52.1. Base conditions 52.2. Comparison of norms 72.3. Main estimate: the case of models 103. Arithmetic volumes of ℓ -adelic R -Cartier divisors 153.1. Preliminaries 153.2. The space of continuous functions 193.3. ℓ -adelic R -Cartier divisors 203.4. Arithmetic volume function 233.5. Continuity of the arithmetic volume function 30Acknowledgement 35References 351. Introduction
In Arakelov geometry, it is essentially important whether or not an adelic linebundle has a nonzero small section. The asymptotic number of the small sectionsof high powers of an adelic line bundle L is encoded in an invariant which we callthe arithmetic volume of L and denote by c vol( L ) . The notion of arithmetic volumewas first introduced by Moriwaki in a series of papers [11, 12, 14], where he provedthat the arithmetic volume has many good properties such as the global continuity,the positive homogeneity, the birational invariance, etc. A purpose of this paperis to give a generalization of Moriwaki’s arithmetic volume function, and study itsfundamental properties. Mathematics Subject Classification.
Primary 14G40; Secondary 11G50.
Key words and phrases.
Arakelov theory, adelic divisors, base conditions, arithmetic volumes.
Let K be a number field, and let O K be the ring of integers of K . Let M fin K be theset of all the finite places of K . For each v ∈ M fin K , K v denotes the v -adic completionof K , and e K v denotes the residue field at v . Let X be a normal projective K -variety,and let Rat( X ) be the field of rational functions on X . For each v ∈ M fin K ∪ {∞} ,let X an v be the associated analytic space over v (see section 3.1.2 for detail). Let D be an R -Cartier divisor on X endowed with a D -Green function g ∞ on X an ∞ . To an O K -model ( X , D ) of ( X, D ) , we can associate an adelic R -Cartier divisor ( D , g ∞ ) ad := D, X v ∈ M fin K g ( X , D ) v [ v ] + g ∞ [ ∞ ] . We then define the ℓ -distance of two such models ( X , D ) and ( X , D ) as X v ∈ M fin K sup x ∈ X an v (cid:12)(cid:12)(cid:12) g ( X , D ) v ( x ) − g ( X , D ) v ( x ) (cid:12)(cid:12)(cid:12) . For example, let v , v , . . . be a sequence in M fin K , and let F i be the fiber of X over v i . The sequence of O K -models X , n X i =1 i log ♯ e K v i F i !! n > is then a Cauchy sequence in the ℓ -distance. However, it does not have a limitin the space of adelic R -Cartier divisors. A basic principle of functional analysistells us that function spaces should be complete, so we decide to extend the notionof adelic R -Cartier divisors so as the above sequence is to converge. For each v ∈ M fin K ∪ {∞} , we put C ( X an v ) as the Banach algebra of R -valued continuousfunctions on X an v endowed with the supremum norm. If v = ∞ , we impose thecondition that the functions in C ( X an ∞ ) are invariant under the complex conjugationmap. We define the space C ℓ ( X ) of continuous functions on X as the ℓ -directsum of the family ( C ( X an v )) v ∈ M fin K ∪{∞} endowed with the ℓ -norm k · k ℓ . We saythat a couple D = (cid:16) D, P v ∈ M fin K ∪{∞} g v [ v ] (cid:17) of an R -Cartier divisor D and an adelic D -Green function P v ∈ M fin K ∪{∞} g v [ v ] is an ℓ -adelic R -Cartier divisor if there existsan O K -model ( X , D ) of ( X, D ) such that (cid:13)(cid:13) D − ( D , g ∞ ) ad (cid:13)(cid:13) ℓ < + ∞ , and denoteby d Div ℓ R ( X ) the R -vector space of all the ℓ -adelic R -Cartier divisors on X .There are several advantages of such an extension. For example, the quotientspace c Cl ℓ R ( X ) of d Div ℓ R ( X ) by the R -subspace generated by principal adelic Cartierdivisors admits an essentially unique norm that makes c Cl ℓ R ( X ) into a Banach space(see section 3.3), which should be a proper arithmetic analogue of the space ofnumerical classes of R -Cartier divisors in algebraic geometry. In particular, anysurjective natural homomorphism c Cl ℓ R ( X ) → c Cl ℓ R ( Y ) is automatically an openmapping. We expect that such a formalism will open a way for applying the pow-erful machinery of functional analysis, such as the duality theory, the semigrouptheory, the spectral theory, etc., to the study of adelic R -Cartier divisors.In the previous paper [7], we introduced a notion of R -base conditions, anddefined the arithmetic volumes for pairs of adelic R -Cartier divisors and R -base HE CONTINUITY OF VOLUMES 3 conditions. An R -base condition V on X is defined as a formal R -linear combination V = X ν ν ( V )[ ν ] such that ν are normalized discrete valuations of Rat( X ) and such that ν ( V ) arezero for all but finitely many ν . A discrete valuation ν assigns to D an order ofvanishing along ν defined as ν ( f ) , where f is a local equation defining D aroundthe center c X ( ν ) of ν on X . We denote by BC R ( X ) the R -vector space of all the R -base conditions on X . Given a pair ( D ; V ) of D ∈ d Div ℓ R ( X ) and V ∈ BC R ( X ) ,we can define b ℓ s (cid:0) D ; V (cid:1) := log (cid:16) ♯ n φ ∈ Rat( X ) × : D + c ( φ ) > , ν ( D + ( φ )) > ν ( V ) , ∀ ν o(cid:17) as a nonnegative real number (see Proposition 3.12), and can define the arithmeticvolume of ( D ; V ) as c vol (cid:0) D ; V (cid:1) := lim m ∈ Z ,m → + ∞ b ℓ s (cid:0) mD ; m V (cid:1) m dim X +1 / (dim X + 1)! (see Proposition 3.13). We will establish the following result (Theorem 3.21). Main Theorem.
Let X be a normal, projective, and geometrically connected K -variety. Let V be a finite-dimensional R -subspace of d Div ℓ R ( X ) , let k · k V be a normon V , let Σ be a finite set of points on X , and let B ∈ R > . Given any ε > , thereexists a δ > such that (cid:12)(cid:12)(cid:12) c vol (cid:0) D + (0 , f ); V (cid:1) − c vol (cid:0) E ; V (cid:1)(cid:12)(cid:12)(cid:12) ε for every D, E ∈ V with max (cid:8)(cid:13)(cid:13) D (cid:13)(cid:13) V , (cid:13)(cid:13) E (cid:13)(cid:13) V (cid:9) B and (cid:13)(cid:13) D − E (cid:13)(cid:13) V δ , f ∈ C ℓ ( X ) with k f k ℓ δ , and V ∈ BC R ( X ) with { c X ( ν ) : ν ( V ) > } ⊂ Σ . This paper comprises two parts. First, in section 2, after showing preliminaryresults on base conditions (section 2.1) and the change of norms (section 2.2), weprove in section 2.3 the fundamental estimate of numbers of small sections of pairs,which is the key step to show Theorem 3.21.Next, section 3 will be devoted to introducing the notion of ℓ -adelic R -Cartierdivisors and showing Theorem 3.21. After recalling basic facts on the adelicallynormed vector spaces (section 3.1.1), the Berkovich analytic spaces (section 3.1.2),and the D -Green functions (section 3.1.3), we will introduce basic definitions onthe ℓ -adelic setting in sections 3.2 and 3.3. We will define the arithmetic volumesof pairs of ℓ -adelic R -Cartier divisors and R -base conditions in section 3.4 and givea proof of Theorem 3.21 in section 3.5.1.1. Notation and terminology. R be a ring and let M be an R -module. Given a subset Γ of M , wedenote by h Γ i R the R -submodule of M spanned by Γ . In this paper, we adopt thedot-product notation, that is, for a = ( a , . . . , a r ) ∈ R r and m = ( m , . . . , m r ) ∈ M r , we write a · m = a m + · · · + a r m r . Moreover, for a = ( a , . . . , a r ) ∈ R r , k a k denotes the ℓ -norm of a : k a k := | a | + · · · + | a r | . HIDEAKI IKOMA normed Z -module M := ( M, k · k ) is a finitely generated Z -module M endowed with a norm k · k on M R = M ⊗ Z R . For such an M , we set b Γ s ( M ) := { m ∈ M : k m ⊗ k } , b ℓ s ( M ) := log ♯ b Γ s ( M ) and b Γ ss ( M ) := { m ∈ M : k m ⊗ k < } , b ℓ ss ( M ) := log ♯ b Γ ss ( M ) . Let ∗ = s or ss. The following properties are fundamental.(a) Let M be a normed Z -module and let → M ′ → M → M ′′ → be an exact sequence of Z -modules. We endow M ′ R (respectively, M ′′ R ) withthe subspace norm k · k sub (respectively, quotient norm k · k quot ) inducedfrom M . One then has(1.1) b ℓ ∗ ( M ) b ℓ ∗ ( M ′ ) + b ℓ ∗ ( M ′′ ) + 3 rk M ′ + 2 log(rk M ′ )! . In fact, if ∗ = s, then the inequality is nothing but [11, Proposition 2.1(4)]and, if ∗ = ss, then it follows from the ∗ = s case by replacing k · k with e ε k · k for ε > and taking ε ↓ .(b) If we replace k · k with e − λ k · k for a λ ∈ R > , then(1.2) b ℓ ∗ ( M, k · k ) b ℓ ∗ ( M, e − λ k · k ) b ℓ ∗ ( M, k · k ) + ( λ + 2) rk M (see the proof of [16, Lemma 2.9]).(c) If M ′ is a Z -submodule of M with M/M ′ torsion, then(1.3) b ℓ ∗ ( M ′ , k · k ) b ℓ ∗ ( M, k · k ) b ℓ ∗ ( M ′ , k · k ) + log ♯ ( M/M ′ ) + 2 rk M (see [14, Lemma 1.3.3, (1.3.3.4)]).(d) Let M be a finitely generated Z -module, and let k · k , k · k be two normson M R . If k · k k · k , then(1.4) b ℓ ∗ ( M, k · k ) > b ℓ ∗ ( M, k · k ) . (e) Let h· , ·i and h· , · , i be two Hermitian inner products on M C = M ⊗ Z C ,and let k · k and k · k be the associated norms on M R , respectively. Let e , . . . , e l be any basis for M C . If k · k k · k , then b ℓ ∗ ( M, k · k ) − b ℓ ∗ ( M, k · k ) − M − M )! (1.5) −
12 log det (cid:0) h e i , e j i (cid:1) i,j det ( h e i , e j i ) i,j (see [11, Proposition 2.1(2)]). The right-hand side does not depend on aspecific choice of e , . . . , e l . The ∗ = ss case follows by the same argumentsas in (a) above.We will also use the elementary inequalities log n ! n log n and log n n for every n ∈ Z > . HE CONTINUITY OF VOLUMES 5 k be a field endowed with a non-Archimedean absolute value | · | . Wewrite(1.6) k ◦ := { a ∈ k : | a | } , k ◦◦ := { a ∈ k : | a | < } , and e k := k ◦ /k ◦◦ . K be a number field and let O K be the ring of integers of K . Let M fin K be the set of all the finite places of K and set(1.7) M K := M fin K ∪ {∞} . Set K ∞ := C and set | α | ∞ := √ αα for α ∈ C . For v ∈ M fin K , we denote by p v theprime ideal of O K corresponding to v , by K ◦ v = proj lim n ∈ Z > O K / p nv the v -adiccompletion of O K , and by K v the quotient field of K ◦ v . We put(1.8) K ◦◦ v := p v K ◦ v and e K v := K ◦ v /K ◦◦ v . We will write a uniformizer of K v by ̟ v . We define the order of an α ∈ K ◦ v as(1.9) ord v ( α ) := ( max { n > α ∈ ( K ◦◦ v ) n } if α = 0 and + ∞ if α = 0 ,and extend it to a map from K v by linearity. The (normalized) v -adic absolutevalue on K v is defined as(1.10) | α | v := (cid:16) ♯ e K v (cid:17) − ord v ( α ) for α ∈ K v . 2. Fundamental estimate
Base conditions. F be a field. A normalized discrete valuation ν on F is a surjective mapfrom F to Z ∪ { + ∞} such that(a) ν ( f ) = + ∞ if and only if f = 0 ,(b) ν ( f · g ) = ν ( f ) + ν ( g ) for f, g ∈ F , and(c) ν ( f + g ) > min { ν ( f ) , ν ( g ) } for f, g ∈ F .We set F ◦ ν := { f ∈ F : ν ( f ) > } and F ◦◦ ν := { f ∈ F : ν ( f ) > } . Since ( F ◦ ν ) × = { f ∈ F : ν ( f ) = 0 } , F ◦◦ ν is a maximal ideal of F ◦ ν . We denote by V ( F ) the set ofall the normalized discrete valuations on F .2.1.2. Let S be a reduced, irreducible, and separated scheme and let F := Rat( S ) be the field of rational functions on S . We assume the condition that,( ⋆ ) for every ν ∈ V ( F ) , there exists a unique point c S ( ν ) ∈ S such that O S,c S ( ν ) ⊂ F ◦ ν and m c S ( ν ) = F ◦◦ ν ∩ O S,c S ( ν ) . We call c S ( ν ) the center of ν on S . By the valuative criterion of properness, if S isproper over Spec( Z ) , then S satisfies the condition ( ⋆ ). Remark . If S is a proper variety over a field k , then we always assume that avaluation ν ∈ V ( F ) is trivial on k . In particular, such a valuation always has aunique center c S ( ν ) on S , and the condition ( ⋆ ) is satisfied. HIDEAKI IKOMA An R -base condition V on S is defined as a finite formal sum V := X ν ∈ V (Rat( S )) ν ( V )[ ν ] with real coefficients ν ( V ) . We denote by BC R ( S ) the R -vector space of all the R -base conditions on S . We write V > if ν ( V ) > for every ν ∈ V (Rat( S )) .2.1.3. Let S be a reduced, irreducible, and projective scheme over a field or Z .Let L be a line bundle on S , let ν ∈ V (Rat( S )) , and let η be a local frame of L around c S ( ν ) . Given any s ∈ H ( L ) \ { } , one can write s c S ( ν ) = f η c S ( ν ) with f ∈ O S,c S ( ν ) \ { } . If η ′ is another local frame of L around c S ( ν ) , then η ′ /η isinvertible in O S,c S ( ν ) . So, if we write s c S ( ν ) = f ′ η ′ c S ( ν ) with f ′ ∈ O S,c S ( ν ) \ { } , then f /f ′ is invertible in O S,c S ( ν ) and ν ( f ) = ν ( f ′ ) . We define(2.1) ν ( s ) := ν ( f ) , which does not depend on a specific choice of η . The following properties areobvious.(a) If s ∈ H ( L ) does not pass through c S ( ν ) , then f is invertible around c S ( ν ) and(2.2) ν ( s ) = 0 . (b) For s, t ∈ H ( L ) and ν ∈ V (Rat( S )) ,(2.3) ν ( s + t ) > min { ν ( s ) , ν ( t ) } . (c) For two line bundles L, M on S , s ∈ H ( L ) , t ∈ H ( M ) , and ν ∈ V (Rat( S )) ,one has(2.4) ν ( s ⊗ t ) = ν ( s ) + ν ( t ) . For a pair ( L ; V ) of a line bundle L and a V ∈ BC R ( S ) , we set(2.5) H ( L ; V ) := (cid:8) s ∈ H ( L ) : ν ( s ) > ν ( V ) for all ν ∈ V (Rat( S )) (cid:9) . k be a field or Z . Let S be a reduced, irreducible, normal, and projec-tive k -scheme, and let K = R , Q , or Z . A K -Cartier divisor on S is an K -linearcombination D = a D + · · · + a r D r such that a i ∈ K and such that D i are Cartier divisors. We denote by Div K ( S ) the K -module of all the K -Cartier divisors on S . If K = Z , we simply write Div( S ) :=Div Z ( S ) as usual.Each ν ∈ V (Rat( S )) can extend to a map ν : Rat( S ) × ⊗ Z R → R by linearity.Given a D ∈ Div R ( S ) and a ν ∈ V (Rat( S )) , we take a local equation f defining D around c S ( ν ) , and define(2.6) ν ( D ) := ν ( f ) , which does not depend on a specific choice of f (see [7, Definition 2.2]). Given apair ( D ; V ) of an R -Cartier divisor D and a V ∈ BC R ( S ) , we set(2.7) H ( D ; V ) := (cid:26) φ ∈ Rat( S ) × : D + ( φ ) > and ν ( D + ( φ )) > ν ( V ) for all ν ∈ V (Rat( S )) (cid:27) ∪ { } , HE CONTINUITY OF VOLUMES 7 and define(2.8) vol( D ; V ) := lim sup m ∈ Z ,m → + ∞ rk k H ( mD ; m V ) m dim S / (dim S )! . X be a projective arithmetic variety over Spec( Z ) ; namely, X is areduced and irreducible scheme projective and flat over Spec( Z ) . Let X ( C ) be thecomplex analytic space associated to X C := X × Spec( Z ) Spec( C ) . A continuousHermitian line bundle on X is a couple ( L , | · | L ) of a line bundle L on X anda continuous Hermitian metric | · | L on L ( C ) . Definition 2.1.
Let ( L ; V ) be a pair of a continuous Hermitian line bundle L on X and a V ∈ BC R ( X ) . The Z -module H ( L ; V ) = (cid:8) s ∈ H ( L ) : ν ( s ) > ν ( V ) for all ν ∈ V (Rat( X )) (cid:9) is endowed with the supremum norm k · k L sup defined as(2.9) k s k L sup := sup x ∈ X ( C ) | s | L ( x ) for s ∈ H ( L ) . We will abbreviate(2.10) b ℓ ∗ (cid:0) L ; V (cid:1) := b ℓ ∗ (cid:16) H ( L ; V ) , k · k L sup (cid:17) for ∗ = s and ss (see Notation and terminology 1.1.2).2.2. Comparison of norms.
Let T be a finite disjoint union T = l [ i =1 T i of compact complex Kähler manifolds T i of pure dimension d . Let ω be a Kählerform on T and let Ω = ω ∧ d be the volume form on T associated to ω . Let M =( M, h M ) be a line bundle M on T endowed with a C ∞ -Hermitian metric h M . The supremum norm of s ∈ H ( M ) is defined as k s k M sup := sup t ∈ T | s | M ( t ) , where | s | M ( t ) := q h M ( s, s )( t ) . The L -inner product of s , s ∈ H ( M ) with respect to Ω is defined as h s , s i ML := Z T h M ( s , s )( t ) Ω , and the L -norm of s is k s k ML := q h s, s i ML . In the rest of this subsection, we studythe effects of the change of norms to the numbers of small sections.2.2.1. The first one (Proposition 2.3) gives us a direct (not optimal) relation be-tween the supremum norms and the subspace norms induced by a fixed nonzerosection. Lemma 2.2 (see [11, Lemma 1.1.4]) . Let M = ( M , . . . , M r ) be C ∞ -Hermitianline bundles on T and let U be an open subset of T . Assume that U ∩ T i are HIDEAKI IKOMA nonempty for all i . There then exists a positive constant C > depending only on M , U , and T such that sup t ∈ U | s | a · M ( t ) k s k a · M sup C k a k sup t ∈ U | s | a · M ( t ) for every a ∈ Z r > and s ∈ H ( a · M ) . Proposition 2.3.
Let M = ( M , . . . , M r ) and E be C ∞ -Hermitian line bundleson T . Fix an s ∈ H ( E ) \ { } . The C -vector space H ( a · M ) is endowed with thetwo norms k · k a · M sup and k · k a · M + bE sup , sub( s ⊗ b ) , where k · k a · M + bE sup , sub( s ⊗ b ) is the subspace norminduced from (cid:16) H ( a · M + bE ) , k · k a · M + bE sup (cid:17) via H ( a · M ) ⊗ s ⊗ b −−−→ H ( a · M + bE ) .There then exists a constant C > depending only on M , ( E, s ) , and T suchthat k · k a · M sup C k a k + b k · k a · M + bE sup , sub( s ⊗ b ) for every a ∈ Z r > and b ∈ Z > .Proof. We choose a nonempty open subset U of T such that δ := inf t ∈ U | s | E ( t ) > and such that T i ∩ U = ∅ for all i . By Lemma 2.2, there is a C > such that k s k a · M sup C k a k sup t ∈ U | s | a · M ( t ) for every a ∈ Z r > and s ∈ H ( a · M ) . Hence k s k a · M sup δ − b C k a k sup t ∈ U | s ⊗ s ⊗ b | a · M + bE ( t ) max { δ − , C } k a k + b k s ⊗ s ⊗ b k a · M + bE sup for every a ∈ Z r > , b ∈ Z > , and s ∈ H ( a · M ) . (cid:3) T and Ω be as above, and consider the L -norms with respect to Ω .Let M = ( M, | · | M ) be a C ∞ -Hermitian line bundle on T , let V be a linear seriesbelonging to M , and let e , . . . , e l be an L -orthonormal basis for V . We define the Bergman distortion function β ( V ; M , Ω) as(2.11) β ( V ; M ,
Ω)( x ) := l X i =1 | e i | M ( x ) for x ∈ T . It is easy to see that β ( V ; M , Ω) does not depend on a specific choice of e , . . . , e l . If V = H ( M ) , then we abbreviate(2.12) β ( M ) := β ( H ( M ); M , Ω) for simplicity. The distortion function has the following elementary properties.(a) If W is a linear series containing V , then β ( V ; M , Ω) β ( W ; M , Ω) .(b) For a c ∈ R > , β ( V ; M , c
Ω) = c − β ( V ; M , Ω) . HE CONTINUITY OF VOLUMES 9
Proposition 2.4 ([12, Theorem 1.2.1]) . Let A and B = ( B , . . . , B r ) be C ∞ -Hermitian line bundles on T such that A and B are all ample and such that theHermitian metrics are all pointwise positive definite. Suppose that the volume formis given as Ω := c ( A ) ∧ d . There then exists a constant C > such that (cid:13)(cid:13) β (cid:0) aA − b · B (cid:1)(cid:13)(cid:13) sup C a d for every a ∈ Z > and b ∈ Z r > . X be a projective arithmetic variety over Spec( Z ) , let M and A becontinuous Hermitian line bundles on X , and fix an s ∈ H ( A ) \ { } . The Z -module H ( M ; V ) is endowed with the supremum norm k · k M sup and the L -norm k · k M L . Let k · k M + A sup , sub( s ) (respectively, k · k M + A L , sub( s ) ) be the subspace norm inducedvia H ( M ; V ) ⊗ s −−→ H ( M + A ) ; namely,(2.13) k s k M + A sup , sub( s ) := k s ⊗ s k M + A sup (respectively, k s k M + A L , sub( s ) := k s ⊗ s k M + A L )for s ∈ H ( M ; V ) . For ∗ = s and ss, we write(2.14) b ℓ ∗ sub( s ) (cid:0) M ; V (cid:1) := b ℓ ∗ (cid:16) H ( M ; V ) , k · k M + A sup , sub( s ) (cid:17) for short. The next one plays a key role in showing the main estimate in section 2.3. Theorem 2.5.
Let X be a projective arithmetic variety of dimension d + 1 over Spec( Z ) . We assume that the generic fiber X Q is smooth over Spec( Q ) . Let LLL =( L , . . . , L r ) and A be C ∞ -Hermitian line bundles on X , and fix an s ∈ b Γ s ( A ) \{ } . If the Hermitian metrics of L + A , . . . , L r + A , and A are all pointwisepositive definite, then there exists a constant C > depending only on LLL , ( A , s ) ,and X such that b ℓ ∗ sub( s ⊗ b ) (cid:16) a · LLL ; V (cid:17) b ℓ ∗ (cid:16) a · LLL ; V (cid:17) + C k a k d ( b + log k a k ) for ∗ = s , ss , a ∈ Z r > with k a k > , b ∈ Z > , and V ∈ BC R ( X ) .Proof. We set the volume form as
Ω := c (cid:0) L + · · · + L r + r A (cid:1) ∧ d , and consider the L -norms with respect to Ω . By Proposition 2.4, there exists aconstant D > such that (cid:13)(cid:13)(cid:13) β (cid:16) H ( a · LLL ; V ) ; a · LLL , Ω (cid:17)(cid:13)(cid:13)(cid:13) sup (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) β k a k ( L + · · · + L r + r A ) − r X i =1 ( k a k − a i )( L i + A ) − k a k A !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup D k a k d for every a ∈ Z r > with k a k > and V ∈ BC R ( X ) .We fix an L -orthonormal basis e , . . . , e l for H ( a · LLL ; V ) in which the Hermitianform, ( s, t ) (cid:10) s ⊗ s ⊗ b , t ⊗ s ⊗ b (cid:11) a · LLL + b A L , is diagonalized. By [11, Corollary 1.1.2], we can change the supremum norms tothe L -norms up to error term O ( k a k d log k a k ) (see [12, Proof of Lemma 1.3.3]).Since k · k a · LLL + b A L , sub( s ⊗ b ) k · k a · LLL L · (cid:16) k s k A sup (cid:17) b k · k a · LLL L , we can apply the inequality (1.5) and find a constant D > such that b ℓ ∗ sub( s ⊗ b ) (cid:16) a · LLL ; V (cid:17) − b ℓ ∗ (cid:16) a · LLL ; V (cid:17) − D k a k d log k a k (2.15) −
12 log det (cid:16) h e i ⊗ s ⊗ b , e j ⊗ s ⊗ b i a · LLL + b A L (cid:17) i,j det (cid:16) h e i , e j i a · LLL L (cid:17) i,j = − l X i =1 log Z X ( C ) (cid:16) | e i | a · LLL (cid:17) · (cid:16) | s | A (cid:17) b Ω for every a ∈ Z > with k a k > , b ∈ Z > , and V ∈ BC R ( X ) .Since R X ( C ) (cid:16) | e i | a · LLL (cid:17) Ω = 1 for each i and s does not vanish identically on eachconnected component of X ( C ) , one can apply Jensen’s inequality [15, Theorem 3.3]to the right-hand side of (2.15) and obtain b ℓ ∗ sub( s ⊗ b ) (cid:16) a · LLL ; V (cid:17) − b ℓ ∗ (cid:16) a · LLL ; V (cid:17) − D k a k d log k a k l X i =1 Z X ( C ) (cid:16) | e i | a · LLL (cid:17) · ( − b log | s | ) Ω D Z X ( C ) − log | s | Ω ! · k a k d b. (cid:3) Main estimate: the case of models.
Let X be a projective arithmeticvariety over Spec( Z ) of dimension d + 1 . Given a family LLL := ( L , . . . , L r ) ofcontinuous Hermitian line bundles on X and an a = ( a , . . . , a r ) ∈ Z r , we write a · LLL := a L + · · · + a r L r , a · LLL := a L + · · · + a r L r , and k a k := | a | + · · · + | a r | as in Notation and terminology 1.1.1. The purpose ofthis section is to show the following estimate. Theorem 2.6.
Let X be a projective arithmetic variety of dimension d + 1 over Spec( Z ) . Assume that the generic fiber X Q is smooth over Spec( Q ) . Let LLL :=( L , . . . , L r ) be a family of C ∞ -Hermitian line bundles on X and let Σ be afinite set of points on X . Let A be any continuous Hermitian line bundle on X .There then exists a constant C > depending only on X , LLL , A , and Σ such that b ℓ ∗ (cid:16) a · LLL + b A ; V (cid:17) − b ℓ ∗ (cid:16) a · LLL ; V (cid:17) C (cid:0) ( k a k + | b | ) d | b | + k a k d log k a k (cid:1) for every a ∈ Z r with k a k > , b ∈ Z , and V ∈ BC R ( X ) with { c X ( ν ) : ν ( V ) > } ⊂ Σ . HE CONTINUITY OF VOLUMES 11
Proof.
We divide the proof into five steps.Step 1. We may assume V > . By considering ± L , . . . , ± L r , and ± A , onecan observe that it suffices to show the theorem for a ∈ Z r > with k a k > and b ∈ Z > . Moreover, if the theorem is true for an A , then it is also true for any A ′ with A ′ A in place of A . Hence we can assume without loss of generality that A has the following four properties.(a) A is ample on X .(b) The Hermitian metric of A is C ∞ , and the Hermitian metrics of L + A , . . . , L r + A , and A are all pointwise positive definite.(c) For every n ≫ , h b Γ ss ( n A ) i Z = H ( n A ) (see Notation and terminol-ogy 1.1.1 and [8, Lemma 5.3]).(d) There is a nonzero small section s ∈ b Γ s ( A ) such that div( s ) Q is smoothand such that s does not pass through any point in Σ .Step 2. For each k ∈ Z > , we set(2.16) k Y := div( s ⊗ k ) . For a ∈ Z r and b ∈ Z , we consider the Z -module(2.17) H X | k Y ( a · LLL + b A ; V ):= Image (cid:0) H ( a · LLL + b A ; V ) → H (( a · LLL + b A ) | k Y ) (cid:1) endowed with the quotient norm k · k a · LLL + b A sup , quot( X | k Y ) induced from (cid:16) H ( a · LLL + b A ; V ) , k · k a · LLL + b A sup (cid:17) . By abuse of notation, we will abbreviate, for • = sub( — ) , quot( X | — ) , etc., b ℓ ∗ • (cid:16) a · LLL + b A ; V (cid:17) := b ℓ ∗ (cid:0) H ( a · LLL + b A ; V ) , k · k ?sup , • (cid:1) for simplicity, which in practice will cause no confusion.By Snapper’s theorem [9, page 295], one can find a constant C > dependingonly on LLL , A , X , and Y such that(2.18) h ( a · LLL + b A ) C ( k a k + b ) d and h (( a · LLL + b A ) | Y ) C ( k a k + b ) d − for every a ∈ Z r > and b ∈ Z > with k a k + b > .In the rest of the proof, the constant C will be fittingly changed without explicitmentioning of it. Claim 2.7.
There exists a constant
C > depending only on LLL , A , and Y suchthat b ℓ ∗ quot( X | Y ) (cid:16) a · LLL + b A ; V (cid:17) C ( k a k + b ) d for every a ∈ Z r > and b ∈ Z > with k a k + b > and V ∈ BC R ( X ) .Proof. It suffices to show the estimate b ℓ ∗ (cid:16) ( a · LLL + b A ) | Y (cid:17) C ( k a k + b ) d for a ∈ Z r > and b ∈ Z > with k a k + b > . Let Y horiz be the horizontal part of Y ,that is, the Zariski closure of Y Q in X . Let I (respectively, I horiz ) be the ideal sheafdefining Y (respectively, Y horiz ) in X . By the properties (a) and (c) of Step 1, one finds an n ∈ Z > and nonzero small sections t i ∈ b Γ s ( n A − L i ) for i = 1 , . . . , r suchthat each t i does not pass through any associated point of O X / I horiz and I horiz / I .First, one finds a constant C > such that b ℓ ∗ (cid:16) ( a · LLL + b A ) | Y horiz (cid:17) b ℓ ∗ (cid:0) ( n k a k + b ) A | Y horiz (cid:1) (2.19) C ( k a k + b ) d for every a ∈ Z r > and b ∈ Z > with k a k + b > (see for example [3, Theorem 2.8]).Next, I horiz / I is a torsion sheaf having support of dimension d . So, by Snap-per’s theorem, one has log ♯H (( a · LLL + b A ) ⊗ I horiz / I ) log ♯H (( n k a k + b ) A ⊗ I horiz / I ) (2.20) C ( k a k + b ) d for every a ∈ Z r > and b ∈ Z > with k a k + b > .Applying (1.1) to the exact sequence → ( a · LLL + b A ) ⊗ ( I horiz / I ) → ( a · LLL + b A ) | Y → ( a · LLL + b A ) | Y horiz → , one obtains, by (2.19) and (2.20), b ℓ ∗ (cid:16) ( a · LLL + b A ) | Y (cid:17) b ℓ ∗ (cid:16) ( a · LLL + b A ) | Y horiz (cid:17) + log ♯H (( a · LLL + b A ) ⊗ ( I horiz / I )) C ( k a k + b ) d for every a ∈ Z r > and b ∈ Z > with k a k + b > . (cid:3) Step 3. We begin the estimation with the following claim.
Claim 2.8.
Let k ∈ Z > , let M be a line bundle on X , and let V ∈ BC R ( X ) suchthat { c X ( ν ) : ν ( V ) > } ⊂ Σ .(1) Tensoring by s ⊗ k induces a homomorphism H ( M ; V ) → H ( M + k A ; V ) .(2) The sequence → H ( M ; V ) ⊗ s ⊗ k −−−→ H ( M + k A ; V ) q −→ H X | k Y ( M + k A ; V ) → is exact.Proof. (1): Let t ∈ H ( M ; V ) . By the property (d) of Step 1, one has ν ( s ) = 0 for every ν with ν ( V ) > . By (2.4), ν ( t ⊗ s ⊗ k ) = ν ( t ) + kν ( s ) ( = ν ( t ) > ν ( V ) if ν ( V ) > and > if ν ( V ) = 0 for every ν ∈ V (Rat( X )) ; hence t ⊗ s ⊗ k ∈ H ( M + k A ; V ) .(2): Suppose that t ∈ H ( M + k A ; V ) satisfies q ( t ) = 0 ; hence one finds a t ∈ H ( M ) such that t = t ⊗ s ⊗ k . By (2.4) and the property (d) of Step 1, ν ( t ) ( = ν ( t ) > ν ( V ) if ν ( V ) > and > if ν ( V ) = 0 for every ν ∈ V (Rat( X )) ; hence t ∈ H ( M ; V ) . (cid:3) HE CONTINUITY OF VOLUMES 13 If b = 0 , then the theorem is obvious, so that we can assume b > . We apply(1.1) to the exact sequence(2.21) → H ( a · LLL ; V ) ⊗ s ⊗ b −−−→ H ( a · LLL + b A ; V ) −→ H X | b Y ( a · LLL + b A ; V ) → , and obtain, by (2.18) and Theorem 2.5, b ℓ ∗ (cid:16) a · LLL + b A ; V (cid:17) (2.22) b ℓ ∗ sub( s ⊗ b ) (cid:16) a · LLL ; V (cid:17) + b ℓ ∗ quot( X | b Y ) (cid:16) a · LLL + b A ; V (cid:17) + C k a k d (1 + log k a k ) b ℓ ∗ (cid:16) a · LLL ; V (cid:17) + b ℓ ∗ quot( X | b Y ) (cid:16) a · LLL + b A ; V (cid:17) + C k a k d ( b + log k a k ) for every a ∈ Z r > with k a k > and b ∈ Z > .Step 4. We are going to estimate the middle term b ℓ ∗ quot( X | b Y ) (cid:16) a · LLL + b A ; V (cid:17) in the right-hand side of (2.22). For each k ∈ Z > , we identify − k A with an idealsheaf of O X via the morphism − k A ⊗ s ⊗ k −−−→ O X . The inclusions − ( k + 1) A ⊗ s −−→− k A ⊗ s ⊗ k −−−→ O X induce an injective morphism σ k : − k A | Y = Coker (cid:16) − ( k + 1) A ⊗ s −−→ − k A (cid:17) → Coker (cid:18) − ( k + 1) A ⊗ s ⊗ ( k +1)0 −−−−−−→ O X (cid:19) = O ( k +1) Y . Claim 2.9.
For each k ∈ Z > , σ k induces a homomorphism H X | Y ( a · LLL + ( b − k ) A ; V ) → H X | ( k +1) Y ( a · LLL + b A ; V ) . Proof.
This is obvious because σ k induces a homomorphism H (( a · LLL + ( b − k ) A ) | Y ) → H (cid:0) ( a · LLL + b A ) | ( k +1) Y (cid:1) and the diagram H (( a · LLL + ( b − k ) A ) | Y ) σ k / / H (cid:0) ( a · LLL + b A ) | ( k +1) Y (cid:1) H ( a · LLL + ( b − k ) A ; V ) O O ⊗ s ⊗ k / / H ( a · LLL + b A ; V ) O O is commutative. (cid:3) A commutative diagram of O X -modules: / / − k A | Y σ k / / O ( k +1) Y / / O k Y / / / / − k A O O ⊗ s ⊗ k / / O X O O / / O k Y / / , yields a commutative diagram of Z -modules: / / H X | Y ( a · LLL + ( b − k ) A ; V ) σk / / H X | ( k +1) Y ( a · LLL + b A ; V ) / / H X | k Y ( a · LLL + b A ; V ) / / / / H ( a · LLL + ( b − k ) A ; V ) ⊗ s ⊗ k / / O O H ( a · LLL + b A ; V ) / / O O H X | k Y ( a · LLL + b A ; V ) / / . Since the right vertical arrow is an identity and the lower horizontal sequence isexact (see Claim 2.8), one sees that the upper horizontal sequence of the diagramis also exact. Applying (1.1) to the upper horizontal sequence, one obtains b ℓ ∗ quot( X | ( k +1) Y ) (cid:16) a · LLL + b A ; V (cid:17) (2.23) b ℓ ∗ quot( X | ( k +1) Y ) , sub( σ k ) (cid:16) a · LLL + ( b − k ) A ; V (cid:17) + b ℓ ∗ quot( X | k Y ) (cid:16) a · LLL + b A ; V (cid:17) + C ( k a k + b ) d − (1 + log( k a k + b )) for every a ∈ Z r > and b, k ∈ Z > with k a k + b > and k b (see (2.18)).Step 5. By applying [11, Lemma 3.4(2)] to the right square of the commutativediagram / / H ( a · LLL + ( b − k − A ; V ) ⊗ s ⊗ ( k +1)0 / / H ( a · LLL + b A ; V ) / / H X | ( k +1) Y ( a · LLL + b A ; V ) / / / / H ( a · LLL + ( b − k − A ; V ) ⊗ s / / H ( a · LLL + ( b − k ) A ; V ) / / ⊗ s ⊗ k O O H X | Y ( a · LLL + ( b − k ) A ; V ) / / σk O O , and by using Proposition 2.3, one can get a constant D with < D such that k · k a · LLL + b A sup , quot( X | ( k +1) Y ) , sub( σ k ) = k · k a · LLL + b A sup , sub( s ⊗ k ) , quot( X | Y ) > D k a k + b k · k a · LLL +( b − k ) A sup , quot( X | Y ) on H X | Y ( a · LLL + ( b − k ) A ; V ) , where k · k a · LLL + b A sup , quot( X | ( k +1) Y ) , sub( σ k ) is the subspacenorm induced from (cid:16) H X | ( k +1) Y ( a · LLL + b A ; V ) , k · k a · LLL + b A sup , quot( X | ( k +1) Y ) (cid:17) via σ k , and k · k a · LLL + b A sup , sub( s ⊗ k ) , quot( X | Y ) is the quotient norm induced from (cid:16) H ( a · LLL + ( b − k ) A ; V ) , k · k a · LLL + A sup , sub( s ⊗ k ) (cid:17) . Hence, by (1.2), (1.4), (2.18), and Claim 2.7, one gets a constant
C > such that(2.24) b ℓ ∗ quot( X | ( k +1) Y ) , sub( σ k ) (cid:16) a · LLL + ( b − k ) A ; V (cid:17) b ℓ ∗ quot( X | Y ) (cid:16) a · LLL + ( b − k ) A ; V (cid:17) + C ( k a k + b ) d C ( k a k + b ) d for a ∈ Z r > and b, k ∈ Z > with k a k + b > and k b .By (2.23) and (2.24),(2.25) b ℓ ∗ quot( X | ( k +1) Y ) (cid:16) a · LLL + b A ; V (cid:17) b ℓ ∗ quot( X | k Y ) (cid:16) a · LLL + b A ; V (cid:17) + C ( k a k + b ) d . HE CONTINUITY OF VOLUMES 15
By summing up (2.25) for k = 1 , , . . . , b − and by using Claim 2.7 again, one has b ℓ ∗ quot( X | b Y ) (cid:16) a · LLL + b A ; V (cid:17) b ℓ ∗ quot( X | Y ) (cid:16) a · LLL + b A ; V (cid:17) + C ( k a k + b ) d b C ( k a k + b ) d b. Therefore, by (2.22), one obtains b ℓ ∗ (cid:16) a · LLL + b A ; V (cid:17) b ℓ ∗ (cid:16) a · LLL ; V (cid:17) + C (cid:0) ( k a k + b ) d b + k a k d log k a k (cid:1) for every a ∈ Z r > with k a k > and b ∈ Z > . (cid:3) Arithmetic volumes of ℓ -adelic R -Cartier divisors Preliminaries.
In this section, we recall definitions and basic properties ofadelically normed vector spaces (section 3.1.1), Berkovich analytic spaces (sec-tion 3.1.2), and adelic Green functions (section 3.1.3).3.1.1. Let K be a number field. Let V := (cid:16) V, ( k · k Vv ) v ∈ M K (cid:17) be a couple of afinite-dimensional K -vector space V and a collection ( k · k Vv ) v ∈ M K such that each k · k Vv is a ( K v , | · | v ) -norm on V K v = V ⊗ K K v and such that, if v ∈ M fin K , then k · k Vv is non-Archimedean. For such a V , we set(3.1) b Γ f ( V ) := n s ∈ V : k s k Vv for every v ∈ M fin K o , (3.2) b Γ s ( V ) := n s ∈ b Γ f ( V ) : k s k V ∞ o , b Γ ss ( V ) := n s ∈ b Γ s ( V ) : k s k V ∞ < o , and b ℓ ∗ ( V ) := log ♯ b Γ ∗ ( V ) for ∗ = s and ss. Note that b Γ f ( V ) is a O K -submodule of V and b ℓ ∗ ( V ) may be infinite.We set, for λ ∈ R , F λ ( V ) := D s ∈ b Γ f ( V ) : k s k V ∞ e − λ E K (see Notation and terminology 1.1.1), and set(3.3) e max ( V ) := sup (cid:8) λ ∈ R : F λ ( V ) = 0 (cid:9) . Proposition 3.1.
Let V = (cid:16) V, ( k · k Vv ) v ∈ M K (cid:17) be a couple of a finite-dimensional K -vector space V and a collection ( k · k Vv ) v ∈ M K such that each k · k Vv is a ( K v , | · | v ) -norm on V K v and such that, if v ∈ M fin K , then k · k Vv is non-Archimedean.(1) The following are equivalent.(a) For each s ∈ V , k s k Vv for all but finitely many v ∈ M K .(b) b Γ f ( V ) contains an O K -submodule E of V satisfying E K = V .(2) Suppose that V satisfies the equivalent conditions of (1). The following arethen equivalent.(a) b Γ f ( V ) is a finitely generated O K -module.(b) b Γ s ( V ) is finite.(c) b Γ ss ( V ) is finite.(d) e max ( V ) < + ∞ . Proof. (1) (a) ⇒ (b): For each s ∈ V , one can find an n > such that k ns k Vv = | n | v k s k Vv for every v ∈ M K by the condition (a). Thus ns ∈ b Γ f ( V ) , whichimplies V = b Γ f ( V ) K .(b) ⇒ (a): For each s ∈ V , there exists an α ∈ O K such that αs ∈ E . Hence k αs k Vv = k s k Vv for all but finitely many v ∈ M K .For the assertion (2), we refer to [3, Proposition 2.4] and [2, Proposition C.2.4]. (cid:3) Definition 3.1. An adelically normed K -vector space is a couple (cid:16) V, ( k · k Vv ) v ∈ M K (cid:17) satisfying the all conditions in Proposition 3.1(1),(2). Notice that here the existenceof an O K -model of V that defines k · k Vv except for finitely many v is not assumedwhile it is in the classical definition in [17, (1.6)] and in [4, Definition 3.1].Let λ ∈ R and let v ∈ M K . We define an adelically normed K -vector space V ( λ [ v ]) = (cid:16) V, ( k · k V ( λ [ v ]) w ) w ∈ M K (cid:17) as(3.4) k · k V ( λ [ v ]) w := ( k · k Vw if w = v and e − λ k · k Vv if w = v . Lemma 3.2.
Let λ ∈ R > and let v ∈ M fin K . If we set p Z := p v ∩ Z , then b ℓ ∗ (cid:0) V ( λ [ v ]) (cid:1) − b ℓ ∗ (cid:0) V (cid:1) (cid:18)(cid:24) λ − log | p | v (cid:25) log( p ) + 2 (cid:19) dim Q V. Proof.
Set n λ := (cid:24) λ − log | p | v (cid:25) . We are going to show(3.5) p n λ b Γ f (cid:0) V ( λ [ v ]) (cid:1) ⊂ b Γ f (cid:0) V (cid:1) . Suppose that s ∈ b Γ f (cid:0) V ( λ [ v ]) (cid:1) . Then k p n λ s k Vw = | p | n λ w k s k Vw ( if w = v and e λ | p | n λ v if w = v .Since λ + n λ log | p | v , we have p n λ s ∈ b Γ f (cid:0) V (cid:1) .We apply (1.3) to the inclusion b Γ f (cid:0) V (cid:1) ⊂ b Γ f (cid:0) V ( λ [ v ]) (cid:1) , and obtain b ℓ ∗ (cid:0) V ( λ [ v ]) (cid:1) b ℓ ∗ (cid:0) V (cid:1) + log ♯ (cid:16)b Γ f (cid:0) V ( λ [ v ]) (cid:1) / b Γ f (cid:0) V (cid:1)(cid:17) + 2 dim Q V b ℓ ∗ (cid:0) V (cid:1) + log ♯ (cid:16)b Γ f (cid:0) V ( λ [ v ]) (cid:1) /p n λ b Γ f (cid:0) V ( λ [ v ]) (cid:1)(cid:17) + 2 dim Q V b ℓ ∗ (cid:0) V (cid:1) + ( n λ log( p ) + 2) dim Q V by (3.5). (cid:3) X be a normal, projective, and geometrically connected K -variety.For v = ∞ , we denote by X an ∞ the complex analytic space associated to X C := X × Spec( Q ) Spec( C ) . For v ∈ M fin K , we denote by ( X an v , ρ v : X an v → X K v ) theBerkovich analytic space associated to X K v (see [1]). For each x ∈ X an v , we denote HE CONTINUITY OF VOLUMES 17 by κ ( x ) the residue field of ρ v ( x ) ∈ X K v and by | · | x the corresponding norm on κ ( x ) . Given a local function f on X K v defined around ρ v ( x ) , we write(3.6) | f | ( x ) := | f ( ρ v ( x )) | x . An O K -model of X is a reduced, irreducible, projective, and flat O K -schemewith generic fiber X K ≃ X . Given an O K -model X of X , we set(3.7) f X v := X × Spec( O K ) Spec( e K v ) . For each x ∈ X an v , the morphism ρ v ( x ) : Spec( κ ( x )) → X K ◦ v uniquely extends to amorphism Spec( κ ( x ) ◦ ) → X K ◦ v by the valuative criterion of properness. We define r X v ( x ) as the image of the closed point of Spec( κ ( x ) ◦ ) .Let U = Spec( A ) be an affine open subscheme of X K ◦ v with U ∩ f X v = ∅ , andset U = U K v = Spec( A ) . We put(3.8) U an v, U := { x ∈ U an v : | f | ( x ) for all f ∈ A } . Lemma 3.3. (1) U an v, U = (cid:0) r X v (cid:1) − (cid:16) U ∩ f X v (cid:17) .(2) U an v, U is compact.Proof. (1): If x ∈ U an v, U , then the image of the homomorphism A → κ ( x ) is in κ ( x ) ◦ , so r X v ( x ) ∈ U . Conversely, if r X v ( x ) ∈ U ∩ f X v , then ρ v ( x ) ∈ U and x ∈ ρ − v ( U ) = U an v . Since r X v ( x ) ∈ U , the image of the morphism Spec( κ ( x ) ◦ ) → X K ◦ v is in U , so f ( ρ v ( x )) ∈ κ ( x ) ◦ for every f ∈ A .(2): The map u : U an v → I := Y f ∈ A R > , x ( | f | ( x )) f ∈ A , is injective and continuous, where I is endowed with the product topology. ByTychonoff’s theorem, J := Q f ∈ A [0 , is a compact subset of I , and U an v, U = u − ( J ) .Thus it suffices to show that u is a closed map. Suppose that ( u ( x α )) α is a net in I that converges to ( λ f ) f ∈ A ∈ I . For each f ∈ A , we set | f | x := λ f . Claim 3.4. | · | x extends to a multiplicative seminorm on A whose restriction to K v is | · | v .Proof of Claim 3.4. Since, for every α , | · | x α satisfies the conditions: • | a | ( x α ) = | a | v for a ∈ K v , • | f − g | ( x α ) | f | ( x α ) + | g | ( x α ) for f, g ∈ A , and • | f g | ( x α ) = | f | ( x α ) | g | ( x α ) for f, g ∈ A ,we know that the limit |·| x is a multiplicative seminorm on A . For a general f ∈ A ,we can take an n > such that ̟ nv f ∈ A , and define | f | x := | ̟ v | − nv | ̟ nv f | x , which does not depend on a specific choice of n > . Then | · | x is a multiplicativeseminorm on A . (cid:3) By Claim 3.4, | · | x corresponds to a point x ∈ U an v . Since | f | x α → | f | x for every f ∈ A , the net ( x α ) α converges to x in the Gel ′ fand topology, and ( λ f ) f ∈ A = u ( x ) .It implies that u is a closed map. (cid:3) Let f X v, gen be the set of all the generic points of irreducible components of f X v .For each ξ ∈ f X v, gen , (cid:0) r X v (cid:1) − ( ξ ) consists of a single point x ξ given as(3.9) | φ | x ξ := (cid:16) ♯ e K v (cid:17) − ord ξ ( φ )ord ξ ( ̟v ) for φ ∈ Rat( X ) . We set Γ( X an v ) := n x ξ : ξ ∈ f X v, gen o (see also [1, Proposition 2.4.4and Corollary 2.4.5]). Lemma 3.5.
Suppose that A is integrally closed in A . Then, for each f ∈ A , max x ∈ U an v, U {| f | ( x ) } = max x ∈ Γ( X an v ) ∩ U an v, U {| f | ( x ) } . Proof.
Since U ∩ f X v = ∅ , we have Γ( X an v ) ∩ U an v, U = ∅ . The inequality > is obvious,so that we are going to show the reverse. Choose a ξ ∈ f X v, gen such that | f | ( x ξ ) = max x ∈ Γ( X an v ) ∩ U an v, U {| f | ( x ) } . If we set n := ord ξ ( ̟ v ) and l := ord ξ ( f ) , then ord ξ ( ̟ − lv f n ) > for every ξ ∈ f X v, gen . By [8, Lemma 2.3(3)], it implies ̟ − lv f n ∈ A . Hence | f | ( x ) | ̟ v | ln v = | f | ( x ξ ) for every x ∈ U an v, U . (cid:3) X be a normal, projective, and geometrically connected K -variety, let K = R , Q , or Z , and let D be a K -Cartier divisor on X . The support of D is aZariski closed subset defined as(3.10) Supp( D ) := [ Z : prime Weil divisor, ord Z ( D ) =0 Z (see [7, Notation and terminology 2]). Let v ∈ M K . A D -Green function on X an v is a continuous map g v : ( X \ Supp( D )) an v → R such that, for each x ∈ X an v ,(3.11) g v ( x ) + log | f | ( x ) extends to a continuous function around x , where f ∈ Rat( X ) × ⊗ Z K is a localequation defining D around ρ v ( x ) (see [14, Definition 2.1.1]). If v = ∞ , we assumethat a D -Green function is invariant under the complex conjugation map. We thenset(3.12) d Div tot K ( X ) := ( D, X v ∈ M K g Dv [ v ] ! : D ∈ Div K ( X ) and g Dv is a D -Greenfunction on X an v for each v ∈ M K ) . An element D ∈ d Div tot R ( X ) is called effective if(3.13) D > and ess . inf x ∈ X an v g Dv ( x ) > , ∀ v ∈ M K , and, for D, E ∈ d Div tot R ( X ) , we write D E if E − D is effective. Each g Dv definesthe supremum norm on H ( D ) as(3.14) k φ k Dv, sup := sup x ∈ X an v | φ | ( x ) exp (cid:18) g Dv ( x ) (cid:19) HE CONTINUITY OF VOLUMES 19 for φ ∈ H ( D ) (see [14, Proposition 2.1.3]).In the following, we impose on ν ∈ V (Rat( X )) a condition that the restrictionof ν to K is trivial (see section 2.1). Given a D ∈ d Div tot R ( X ) and a V ∈ BC R ( X ) ,we set(3.15) b Γ ∗ (cid:0) D ; V (cid:1) := b Γ ∗ (cid:16) H ( D ; V ) , ( k · k Dv, sup ) v ∈ M K (cid:17) for ∗ = f, s, and ss, and set b ℓ ∗ (cid:0) D ; V (cid:1) := log ♯ b Γ ∗ (cid:0) D ; V (cid:1) for ∗ = s and ss (see sec-tion 3.1.1 and (2.10)). An O K -model of a couple ( X, D ) is a couple ( X , D ) suchthat X is a normal O K -model of X and such that D is an R -Cartier divisor on X satisfying D | X = D . Given an O K -model ( X , D ) of ( X, D ) and a v ∈ M fin K , wedefine the D -Green function associated to ( X , D ) as(3.16) g ( X , D ) v ( x ) := − log | f ′ | ( x ) , where f ′ is a local equation defining D around r X v ( x ) .Let K := R , Q , or Z . A couple D = ( D , g D ∞ ) on X such that ( X , D ) is an O K -model of ( X, D ) with D ∈ Div K ( X ) and such that g D ∞ is a D -Green functionon X an ∞ is called an arithmetic K -Cartier divisor on X . If X is smooth and g D ∞ is of C ∞ -type, then D is said to be of C ∞ -type (see [13, section 2.3]). We denoteby d Div K ( X ) (respectively, d Div K ( X ; C ∞ ) ) the K -module of all the arithmetic K -Cartier divisors (respectively, arithmetic K -Cartier divisors of C ∞ -type) on X . If K = Z and — = a blank or C ∞ , we will abbreviate d Div( X ; — ) := d Div Z ( X ; — ) asusual.Given a couple ( D ; V ) of a D ∈ d Div R ( X ) and a V ∈ BC R ( X ) , we abbreviate(3.17) b ℓ ∗ (cid:0) D ; V (cid:1) := b ℓ ∗ (cid:16) H ( D ; V ) , k · k D ∞ , sup (cid:17) for ∗ = s and ss (see Notation and terminology 1.1.2 and (2.10)), and define(3.18) c vol (cid:0) D ; V (cid:1) := lim sup m ∈ Z ,m → + ∞ b ℓ s (cid:0) m D ; m V (cid:1) m dim X / (dim X )! . Moreover, the adelization of D ∈ d Div R ( X ) is defined as(3.19) D ad := D, X v ∈ M fin K g ( X , D ) v [ v ] + g ∞ [ ∞ ] , which belongs to d Div tot R ( X ) .3.2. The space of continuous functions.
Let K be a number field, and let X bea projective and geometrically connected K -variety. For each v ∈ M K , we denoteby C ( X an v ) the space of R -valued continuous functions on X an v that are assumed tobe invariant under the complex conjugation if v = ∞ . We endow C ( X an v ) with thesupremum norm: k f k sup := sup x ∈ X an v | f ( x ) | ∞ for f ∈ C ( X an v ) , where | f ( x ) | ∞ denotes the usual absolute value of the real number f ( x ) (see Notation and terminology 1.1.4). By elementary arguments, ( C ( X an v ) , k · k sup ) is a Banach algebra for every v ∈ M K . We denote by(3.20) C tot ( X ) := Y v ∈ M K C ( X an v ) = ( f = X v ∈ M K f v [ v ] : f v ∈ C ( X an v ) ) the algebraic direct product of the family ( C ( X an v )) v ∈ M K , and by(3.21) C ( X ) := M v ∈ M K C ( X an v ) the algebraic direct sum of ( C ( X an v )) v ∈ M K . The ℓ -norm of an f ∈ C tot ( X ) is(3.22) k f k ℓ := X v ∈ M K k f v k sup , where the sum is taken with respect to the net indexed by all the finite subsets of M K , and the ℓ -direct sum of ( C ( X an v )) v ∈ M K is given as C ℓ ( X ) := { f = ( f v ) v ∈ M K : k f k ℓ < + ∞} endowed with the ℓ -norm. For f , g ∈ C tot ( X ) , we write f g if f v g v for every v ∈ M K . If f , g ∈ C ℓ ( X ) , then the entrywise product f g satisfies k f g k ℓ X v ∈ M K k f v k sup k g v k sup sup v ∈ M K {k f v k sup } · k g k ℓ k f k ℓ · k g k ℓ , so f g ∈ C ℓ ( X ) . By the same arguments as in [15, page 67, Theorem 3.11], oneverifies that ( C ℓ ( X ) , k · k ℓ ) is a Banach algebra. Note that C tot (Spec( K )) is just R M K and C ℓ (Spec( K )) = ℓ ( M K ) is just the ℓ -sequence space indexed by M K .We will identify C tot (Spec( K )) with the space of constant functions in C tot ( X ) . Lemma 3.6.
Let f ∈ C ℓ ( X ) . Given any ε > , there exists a h ∈ C ( X ) such that h f and k f − h k ℓ ε. Proof.
Since P v ∈ M K k f v k sup < + ∞ , there is a finite subset S ⊂ M K such that X v ∈ ( M K \ S ) k f v k sup ε. Hence h := P v ∈ S f v [ v ] satisfies the required conditions. (cid:3) ℓ -adelic R -Cartier divisors. Let X be a normal, projective, and geometri-cally connected K -variety. The natural homomorphisms(3.23) C tot ( X ) → d Div tot R ( X ) , f (0 , f ) , and(3.24) ζ : d Div tot R ( X ) → Div R ( X ) , D = D, X v ∈ M K g Dv [ v ] ! ζ ( D ) = D, form an exact sequence(3.25) → C tot ( X ) → d Div tot R ( X ) ζ −→ Div R ( X ) → . Let K and K ′ be either R , Q , or Z . Given a D ∈ d Div tot K ( X ) , we set(3.26) [ Mod K ′ ( D ) := ( ( X , ( D , g ∞ )) : ( X , D ) is an O K -model of ( X, D ) , ( D , g ∞ ) ∈ d Div K ′ ( X ) , and D ad D ) . HE CONTINUITY OF VOLUMES 21
We call D ∈ d Div tot K ( X ) an adelic K -Cartier divisor if there exists an ( X , D ) ∈ [ Mod R ( D ) such that D − D ad ∈ C ( X ) . Denote by d Div K ( X ) the K -module of allthe adelic K -Cartier divisors on X . As before, we will write d Div( X ) := d Div Z ( X ) .For a D ∈ d Div R ( X ) , there are a nonempty open subset U of Spec( O K ) and an ( X , D ) ∈ [ Mod R ( D ) such that g Dv = g ( X , D ) v for every v ∈ U . In this case, we callthe couple ( X U , D U ) a U -model of definition for D (see [14, Definition 4.1.1] and[7, Notation and terminology 4]). Given a D ∈ d Div R ( X ) and a V ∈ BC R ( X ) , wedefine(3.27) c vol (cid:0) D ; V (cid:1) := lim sup m ∈ Z ,m → + ∞ b ℓ s (cid:0) mD ; m V (cid:1) m dim X +1 / (dim X + 1)! , which is finite as in [7, section 2.5]. Proposition 3.7.
Let K = R or Q . For any D ∈ d Div tot K ( X ) , the following areequivalent.(1) There exists an ( X , D ) ∈ [ Mod K ( D ) such that (cid:13)(cid:13)(cid:13) D − D ad (cid:13)(cid:13)(cid:13) ℓ < + ∞ .(2) For any ( X , D ) ∈ [ Mod R ( D ) , (cid:13)(cid:13)(cid:13) D − D ad (cid:13)(cid:13)(cid:13) ℓ < + ∞ .(3) For any ε > , there exists an ( X ε , D ε ) ∈ [ Mod K ( D ) such that (cid:13)(cid:13)(cid:13) D − D ad ε (cid:13)(cid:13)(cid:13) ℓ ε .(4) For any ε > , there exists an D ε ∈ d Div K ( X ) such that ζ ( D ε ) = ζ ( D ) , D ε D , and (cid:13)(cid:13) D − D ε (cid:13)(cid:13) ℓ ε .Proof. The implications (2) ⇒ (1) and (3) ⇒ (1) are obvious. The equivalence (3) ⇔ (4) results from the approximation theorem (see [14, Theorem 4.1.3]).(1) ⇒ (2): It suffices to show that for any ( X ′ , D ′ ) ∈ [ Mod R ( D ) (3.28) (cid:13)(cid:13)(cid:13) D ad − D ′ ad (cid:13)(cid:13)(cid:13) ℓ < + ∞ . Let X ′′ be a normal O K -model of X that dominates both X and X ′ . Let µ : X ′′ → X and µ ′ : X ′′ → X ′ be the dominant morphisms. Then (cid:0) µ ∗ D (cid:1) ad = D ad and (cid:0) µ ′∗ D (cid:1) ′ ad = D ′ ad (see [14, Proposition 2.1.4]). We write D = a Z + · · · + a r Z r with a i ∈ R and prime Weil divisors Z i . Let Z i be the Zariski closure of Z i in X ′′ .Since µ ∗ D − r X i =1 a i Z i and µ ′∗ D ′ − r X i =1 a i Z i are both vertical, one can find a nonempty open subset U ⊂ Spec( O K ) such that ( µ ∗ D ) U = ( µ ′∗ D ′ ) U . Hence we have (3.28).(1) ⇒ (4): Set (0 , f ) := D − D ad . By Lemma 3.6, there exists an f ε ∈ C ( X ) such that f ε f and such that k f − f ε k ℓ ε . Set D ε := D ad + (0 , f ε ) . Then D ε ∈ d Div K ( X ) , D ε D , and (cid:13)(cid:13) D − D ε (cid:13)(cid:13) ℓ = k f − f ε k ℓ ε . (cid:3) Definition 3.2.
Let K = R , Q , or Z . We call an element D ∈ d Div tot K ( X ) an ℓ -adelic K -Cartier divisor on X if there exists an ( X , D ) ∈ [ Mod R ( D ) such that (cid:13)(cid:13)(cid:13) D − D ad (cid:13)(cid:13)(cid:13) ℓ < + ∞ . We denote by d Div ℓ K ( X ) the K -module of all the ℓ -adelic K -Cartier divisors on X . If K = Z , then the subscript Z will be omitted as usual.Moreover, we set(3.29) d D iv ℓ K , R ( X ) := d Div ℓ K ( X ) × BC R ( X ) . Let
Pic
X/K be the Picard scheme of X and let Pic X/K be the neutral componentof
Pic
X/K . Let
Pic( X ) = Pic
X/K ( K ) be the Picard group of X , and let(3.30) NS( X ) := Pic
X/K ( K ) / Pic X/K ( K ) be the Néron–Severi group of X . By Severi’s theorem of the base, NS( X ) is afinitely generated Z -module, and, since Pic X/K is an abelian variety over K (seefor example [10, Theorem 5.4]), Pic X/K ( K ) is also a finitely generated Z -moduleby the Mordell-Weil theorem. Since Pic X/K ( K ) ∩ Pic
X/K ( K ) = Pic X/K ( K ) , we obtain an exact sequence(3.31) → Pic X/K ( K ) → Pic( X ) → NS( X ) . Hence
Pic( X ) is also a finitely generated Z -module.Let b P R ( X ) (respectively, P R ( X ) ) be the R -subspace of d Div ℓ R ( X ) (respectively, Div R ( X ) ) generated by the principal divisors c ( φ ) (respectively, ( φ ) ) for φ ∈ Rat( X ) × .Let Pic R ( X ) := Pic( X ) ⊗ Z R be the R -vector space of R -line bundles on X . By [6,Proposition II.6.15], the sequence(3.32) → P R ( X ) → Div R ( X ) O X −−→ Pic R ( X ) → is exact. So, if we set(3.33) Cl R ( X ) := Div R ( X ) /P R ( X ) , then Cl R ( X ) = Pic R ( X ) is a finite-dimensional R -vector space. Definition 3.3.
We define c Cl ℓ R ( X ) := d Div ℓ R ( X ) / b P R ( X ) . Lemma 3.8.
The sequence → C ℓ ( X ) → c Cl ℓ R ( X ) ζ −→ Cl R ( X ) → is exact.Proof. Obviously, the sequence → C ℓ ( X ) → d Div ℓ R ( X ) ζ −→ Div R ( X ) → is exact. If ζ ( D ) ∈ P R ( X ) , then D = ( φ ) for a φ ∈ Rat( X ) × ⊗ Z R or D = 0 . Hence ζ − ( P R ( X )) = b P R ( X ) ⊕ C ℓ ( X ) , which infers the required result. (cid:3) HE CONTINUITY OF VOLUMES 23
We fix a section ι : Cl R ( X ) → c Cl ℓ R ( X ) of ζ and a norm k · k on the finite-dimensional R -vector space Cl R ( X ) . We can then define a norm on c Cl ℓ R ( X ) as(3.34) (cid:13)(cid:13) D (cid:13)(cid:13) ι, k·k := k D k + (cid:13)(cid:13) D − ι ( D ) (cid:13)(cid:13) ℓ for D ∈ c Cl ℓ R ( X ) , where we regard D − ι ( D ) ∈ C ℓ ( X ) . Proposition 3.9.
Let ι : Cl R ( X ) → c Cl ℓ R ( X ) be a section of ζ and let k · k be anorm on Cl R ( X ) .(1) (cid:18)c Cl ℓ R ( X ) , k · k ι, k·k (cid:19) is a Banach space.(2) Let ι ′ : Cl R ( X ) → c Cl ℓ R ( X ) be another section and let k · k ′ be another norm.Then k · k ι ′ , k·k ′ is equivalent to k · k ι, k·k .Proof. (1): If (cid:0) D n (cid:1) n > is a Cauchy sequence in c Cl ℓ R ( X ) , then (cid:0) ζ ( D n ) (cid:1) n > is aCauchy sequence in Cl R ( X ) , and converges to an E ∈ Cl R ( X ) . Set (0 , f n ) := D n − ι ( ζ ( D n )) . The sequence ( f n ) n > is then a Cauchy sequence in C ℓ ( X ) , andconverges to a g ∈ C ℓ ( X ) . The sequence (cid:0) D n (cid:1) n > then converges to ι ( E ) + (0 , g ) .(2): It suffices to show k · k ι ′ , k·k ′ C k · k ι, k·k for a C > . We choose a basis A , . . . , A l for Cl R ( X ) and set k a A + · · · + a l A l k := | a | + · · · + | a l | . We can find a constant C > such that k ·k ′ C k ·k and such that k ·k C k ·k .We set (0 , f i ) := ι ( A i ) − ι ′ ( A i ) for each i , and set C := max i l {k ι ( A i ) − ι ′ ( A i ) k ℓ , } . Then, for any D ∈ c Cl ℓ R ( X ) with D = a A + · · · + a l A l , (cid:13)(cid:13) D (cid:13)(cid:13) ι ′ , k·k ′ = k D k ′ + (cid:13)(cid:13) D − ι ′ ( D ) (cid:13)(cid:13) ℓ C k D k + k D − ι ( D ) k ℓ + l X i =1 | a i |k ι ( A i ) − ι ′ ( A i ) k ℓ C k D k + k D − ι ( D ) k ℓ + C k D k C C k D k ι, k·k . (cid:3) Arithmetic volume function.
The following is a key idea to introduce thenotion of ℓ -adelic R -Cartier divisors. Lemma 3.10.
Let X be a normal, projective, and geometrically connected arith-metic variety over Spec( O K ) , and let D ∈ d Div R ( X ) . Suppose that every irre-ducible component of D is Cartier. Let U = U ( X , D ) be a nonempty open subset of Spec( O K ) having the following properties.(a) π U : X U → U is geometrically reduced and geometrically irreducible.(b) For every v ∈ U , ord π − U ( v ) ( D ) = 0 .Then, for every v ∈ U and φ ∈ H ( D ) \ { } , one has inf x ∈ X an v n g ( X , D ) v ( x ) − log | φ | ( x ) o ∈ (2 log ♯ e K v ) Z . Proof.
By assumption, every irreducible component of D | X K ◦ v is Cartier, so we canwrite D | X K ◦ v = a D + · · · + a r D r with a i ∈ R and prime Cartier divisors D i .We choose a finite affine open covering ( U λ ) λ of X K ◦ v such that U λ ∩ f X v = ∅ and D i ∩ U λ is principal with equation f i,λ for each λ . We set U λ = Spec( A λ ) withfinitely generated and integrally closed K ◦ v -algebra A λ , and set U λ := Spec( A λ ⊗ K ◦ v K ) . We then have X an v = [ λ ( U λ ) an v, U λ and ψ λ := φ · f ⌊ a ⌋ ,λ · · · f ⌊ a r ⌋ r,λ ∈ A λ for every φ ∈ H ( D ) \ { } and λ .By Lemma 3.5, the function ( U λ ) an v, U λ → R , x
7→ | ψ λ | ( x ) · | f ,λ | a −⌊ a ⌋ ( x ) · · · | f r,λ | a r −⌊ a r ⌋ ( x ) , attains its maximum at the single point in Γ( X an v ) ∩ ( U λ ) an v, U λ that corresponds tothe fiber f X v . Let ̟ v be a uniformizer of K v . Since ord f X v ( ̟ v ) = 1 and ord f X v ( f ,λ ) = · · · = ord f X v ( f r,λ ) = 0 , we have
12 inf x ∈ ( U λ ) an v, U λ n g ( X , D ) v ( x ) − log | φ | ( x ) o = ord f X v (cid:16) φ · f a ,λ · · · f a r r,λ (cid:17) ord f X v ( ̟ v ) · log ♯ e K v = ord f X v ( φ ) log ♯ e K v ∈ (log ♯ e K v ) Z for every λ . We have thus proved the lemma. (cid:3) Proposition 3.11.
Let X be a normal, projective, and geometrically connected K -variety and let µ : e X → X be a resolution of singularities of X . Let D ∈ d Div R ( X ) ,and let a = P v ∈ M K a v [ v ] ∈ C tot (Spec( K )) with a > .(I) Let U be a nonempty open subset of Spec( O K ) over which a model of defi-nition for D exists.We choose an O K -model ( f X , e D ) of ( e X, µ ∗ D ) such that ( f X U , e D U ) gives a U -modelof definition for µ ∗ D and such that every irreducible component of e D is Cartier.(II) Let U ( f X , e D ) be a nonempty open subset of U such that π : f X U ( f X , e D ) → U ( f X , e D ) is smooth and such that ord π − ( v ) ( e D ) = 0 for every v ∈ U ( f X , e D ) .(III) U a := n v ∈ M fin K : a v < ♯ e K v o .We set a ′ := X v / ∈ U ( f X , e D ) ∩ U a a v [ v ] . Then the following holds.(1) If a b ∈ C tot (Spec( K )) satisfies b > a , then U b ⊂ U a .(2) For any V ∈ BC R ( X ) , one has b Γ f (cid:0) D + (0 , a ); V (cid:1) = b Γ f (cid:0) D + (0 , a ′ ); V (cid:1) . HE CONTINUITY OF VOLUMES 25 (3) If ♯ ( M fin K \ U a ) is finite (in particular, if a is a bounded sequence), then b ℓ ∗ (cid:0) D + (0 , a ); V (cid:1) is finite for every V ∈ BC R ( X ) and ∗ = s, ss.Proof. The assertion (1) is obvious.(2): Since a > , the inclusion ⊃ is obvious. Suppose v ∈ U ( f X , e D ) ∩ U a ; hence,in particular,(3.35) > − a v > − ♯ e K v . If φ ∈ H ( µ ∗ D ; V ) \ { } = H ( D ; V ) \ { } satisfies g Dv ( x ) + a v − log | φ | ( x ) > for every x ∈ X an v , then g ( f X , e D ) v ( x ′ ) − log | φ | ( x ′ ) > − a v for every x ′ ∈ e X an v . By Lemma 3.10 and (3.35), we have inf x ′ ∈ e X an v n g ( f X , e D ) v ( x ′ ) − log | φ | ( x ′ ) o = inf x ∈ X an v n g Dv ( x ) − log | φ | ( x ) o > . Hence φ ∈ b Γ f (cid:0) D + (0 , a ); V (cid:1) implies φ ∈ b Γ f (cid:0) D + (0 , a ′ ); V (cid:1) .If M K \ U a is finite, then so is M K \ ( U ( f X , e D ) ∩ U a ) . Hence, the assertion (2)implies the assertion (3) (see [14, Proposition 4.3.1(3)]). (cid:3) Proposition 3.12.
Let X be a normal, projective, and geometrically connected K -variety and let ∗ = s or ss.(1) To each D ∈ d Div R ( X ) , one can assign a constant δ ( D ) > , which dependsonly on D and X , such that b ℓ ∗ (cid:0) D + (0 , f ); V (cid:1) − b ℓ ∗ (cid:0) D ; V (cid:1) (cid:18) k f k ℓ + δ ( D ) (cid:19) dim Q H ( D ; V ) . for every f ∈ C ℓ ( X ) and V ∈ BC R ( X ) . Moreover, one can assume that δ ( tD ) = δ ( D ) holds for every t ∈ R \ { } .(2) For any ( D ; V ) ∈ d D iv ℓ R , R ( X ) , b Γ ∗ (cid:0) D ; V (cid:1) is a finite set.(3) For any ( D ; V ) ∈ d D iv ℓ R , R ( X ) , (cid:16) H ( D ; V ) , ( k · k Dv, sup ) v ∈ M K (cid:17) is an adelicallynormed K -vector space.Proof. (1): Set(3.36) a := X v ∈ M K k f v k sup [ v ] ∈ C ℓ (Spec( K )) . For each v ∈ M fin K , we denote by p v the prime number satisfying p v Z = p v ∩ Z .Let µ : e X → X be a resolution of singularities of X and let U be a nonemptyopen subset of Spec( O K ) over which a model of definition for D exists. Let ( f X , e D ) be an O K -model of ( e X, µ ∗ D ) such that ( f X U , e D U ) gives a U -model of definition for µ ∗ D and such that every irreducible component of e D is Cartier.We choose the two nonempty open subsets U ( f X , e D ) and U a as in Proposition 3.11;namely, • U ( f X , e D ) is chosen to satisfy that U ( f X , e D ) ⊂ U , that π : f X U ( f X , e D ) → U ( f X , e D ) is smooth, and that ord π − ( v ) ( e D ) = 0 for every v ∈ U ( f X , e D ) , and • U a := n v ∈ M fin K : a v < ♯ e K v o .We divide M fin K into three disjoint subsets: S := U ( f X , e D ) ∩ U a , S := n v ∈ M fin K : 2 log ♯ e K v a v and log( p v ) o , and S := M fin K \ ( S ∪ S ) . Note that only S is an infinite subset and S iscontained in a finite subset(3.37) S ′ := (cid:16) M K \ U ( f X , e D ) (cid:17) ∪ { v ∈ M fin K : log( p v ) < } , which is determined only by U ( f X , e D ) . Put(3.38) a ′ := X v ∈ S a v [ v ] + X v ∈ S a v [ v ] . By Proposition 3.11(2), Lemma 3.2, and (1.2), we have b ℓ ∗ (cid:0) D + (0 , f ); V (cid:1) − b ℓ ∗ (cid:0) D ; V (cid:1) (3.39) b ℓ ∗ (cid:0) D + (0 , a ); V (cid:1) − b ℓ ∗ (cid:0) D ; V (cid:1) = b ℓ ∗ (cid:0) D + (0 , a ′ ); V (cid:1) − b ℓ ∗ (cid:0) D ; V (cid:1) X v ∈ S ∪ S (cid:18)(cid:24) k f v k sup − | p v | v (cid:25) log( p v ) + 2 (cid:19) + k f ∞ k sup ! dim Q H ( D ; V ) . We can estimate the sum with respect to S as X v ∈ S (cid:18)(cid:24) k f v k sup − | p v | v (cid:25) log( p v ) + 2 (cid:19) (3.40) X v ∈ S k f v k sup v ( p v )[ e K v : F p v ] + 2 log( p v ) ! X v ∈ S k f v k sup v ( p v )[ e K v : F p v ] + k f v k sup [ e K v : F p v ] ! X v ∈ S k f v k sup and the sum with respect to S as(3.41) X v ∈ S (cid:18)(cid:24) k f v k sup − | p v | v (cid:25) log( p v ) + 2 (cid:19) X v ∈ S (cid:18) k f v k sup + log( p v ) + 2 (cid:19) . Hence, if we set p ∞ := 1 and(3.42) δ ( D ) := X v ∈ S ′ (log( p v ) + 2) , then we obtain b ℓ ∗ (cid:0) D + (0 , f ); V (cid:1) − b ℓ ∗ (cid:0) D ; V (cid:1) (cid:18) k f k ℓ + δ ( D ) (cid:19) dim Q H ( D ; V ) HE CONTINUITY OF VOLUMES 27 by (3.39), (3.40), and (3.41). Since the constant δ ( D ) depends only on U ( f X , e D ) , wehave δ ( tD ) = δ ( D ) for every t ∈ R \ { } .The assertion (2) is obvious from the assertion (1). The assertion (3) follows fromthe assertion (2) and the fact that b Γ f ( D ; V ) contains H ( D ; V ) for any ( X , D ) ∈ [ Mod( D ) . (cid:3) Proposition 3.13.
Let ∗ = s or ss. For any ( D ; V ) ∈ d D iv ℓ R , R ( X ) , lim sup m ∈ Z ,m → + ∞ b ℓ ∗ (cid:0) mD ; m V (cid:1) m dim X +1 / (dim X + 1)! is finite.Proof. Take a D ∈ d Div R ( X ) such that ζ ( D ) = ζ ( D ) and D D . By Proposi-tion 3.12(1), we have lim sup m ∈ Z ,m → + ∞ b ℓ ∗ (cid:0) mD ; m V (cid:1) m dim X +1 / (dim X + 1)! lim sup m ∈ Z ,m → + ∞ b ℓ ∗ (cid:0) mD ; m V (cid:1) m dim X +1 / (dim X + 1)!+ (dim X + 1) lim sup m ∈ Z ,m → + ∞ (cid:18) (cid:13)(cid:13) D − D (cid:13)(cid:13) ℓ + δ ( mD ) m (cid:19) dim Q ( mD ; m V ) m dim X / (dim X )! c vol( D ; V ) + 32 (dim X + 1)[ K : Q ] (cid:13)(cid:13) D − D (cid:13)(cid:13) ℓ vol( D ; V ) < + ∞ . (cid:3) Definition 3.4.
Given a ( D ; V ) ∈ d D iv ℓ R , R ( X ) , we define(3.43) c vol( D ; V ) := lim sup m ∈ Z ,m → + ∞ b ℓ s (cid:0) mD ; m V (cid:1) m dim X +1 / (dim X + 1)! . By Proposition 3.13, c vol( D ; V ) is finite and(3.44) c vol( D ; V ) − c vol( D ; V )
32 (dim X + 1)[ K : Q ] vol( D ; V ) · (cid:13)(cid:13) D − D (cid:13)(cid:13) ℓ for every D ∈ d Div R ( X ) with ζ ( D ) = ζ ( D ) and D D . Moreover, we can easilyobserve(3.45) c vol( D ; V ) = lim sup m ∈ Z ,m → + ∞ b ℓ ss (cid:0) mD ; m V (cid:1) m dim X +1 / (dim X + 1)! . Proposition 3.14.
Let X be a normal, projective, and geometrically connected K -variety, let D = (cid:16) D, P v ∈ M K g Dv [ v ] (cid:17) ∈ d Div ℓ R ( X ) , and let x ∈ X ( K ) . The infinitesum ∆ := X v ∈ M fin K X w ∈ M fin κ ( x ) ,w | v [ κ ( x ) w : K v ] g Dv ( x w ) + X σ : κ ( x ) → C g D ∞ ( x σ ) then converges, where the limit is taken with respect to the net indexed by all thefinite subsets of M fin K , x w ∈ X an v is a point corresponding to ( κ ( x ) , | · | w ) , and x σ ∈ X an ∞ is a point defined as Spec( C ) σ −→ Spec( κ ( x )) x −→ X .Proof. Let ( X , D ) ∈ [ Mod( D ) and let (0 , f ) := D − D ad . We write D = a D + · · · + a r D r with a i ∈ R and effective Cartier divisors D i . Then ∆ − X σ : κ ( x ) → C g D ∞ ( x σ )= X v ∈ M fin K X w ∈ M fin κ ( x ) ,w | v [ κ ( x ) w : K v ] g ( X , D ) v ( x w ) + X v ∈ M fin K X w ∈ M fin κ ( x ) ,w | v [ κ ( x ) w : K v ] f v ( x w )= 2 r X i =1 a i log ♯ ( O κ ( x ) ( D i ) /O κ ( x ) ) + X v ∈ M fin K X w ∈ M fin κ ( x ) ,w | v [ κ ( x ) w : K v ] f v ( x w ) (see [14, section 2.3]). Let ε > . Since f ∈ C ℓ ( X ) , one can find a finite subset S ⊂ M fin K such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X v ∈ S X w ∈ M κ ( x ) ,w | v [ κ ( x ) w : K v ] f v ( x w ) − X v ∈ S X w ∈ M κ ( x ) ,w | v [ κ ( x ) w : K v ] f v ( x w ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ [ κ ( x ) : K ] X v ∈ M fin K \ S k f v k sup ε for every finite subsets S , S of M fin K such that S ⊃ S and S ⊃ S . So, bycompleteness of R , ∆ converges. (cid:3) Definition 3.5. An ℓ -adelic R -Cartier divisor D on X determines a height func-tion h D : X ( K ) → R by h D ( x ) := 1[ κ ( x ) : Q ] X v ∈ M fin K X w ∈ M fin κ ( x ) ,w | v [ κ ( x ) w : K v ] g Dv ( x w ) + 12 X σ : κ ( x ) → C g D ∞ ( x σ ) , which is well-defined by Proposition 3.14 above, and belongs, up to O (1) , to the Weilheight function corresponding to D . Moreover, from the proof of Proposition 3.14,one deduces(3.46) sup x ∈ X ( K ) (cid:12)(cid:12) h D ( x ) − h D ′ ( x ) (cid:12)(cid:12) (cid:13)(cid:13)(cid:13) D − D ′ (cid:13)(cid:13)(cid:13) ℓ for every D, D ′ ∈ d Div ℓ R ( X ) with ζ ( D ) = ζ ( D ′ ) .We abbreviate e max (cid:0) D ; V (cid:1) := e max (cid:16) H ( D ; V ) , ( k · k Dv, sup ) v ∈ M K (cid:17) HE CONTINUITY OF VOLUMES 29 (see (3.3)), and define the essential minimum of D as(3.47) ess . min x ∈ X ( K ) h D ( x ) = sup Y ( X inf x ∈ ( X \ Y )( K ) h D ( x ) , where the supremum is taken over all the closed proper subvarieties of X . Lemma 3.15.
For any D ∈ d Div ℓ R ( X ) , we have lim m ∈ Z ,m → + ∞ e max (cid:0) mD (cid:1) m ess . min x ∈ X ( K ) h D ( x ) < + ∞ . Proof.
Note that e max ( D ) = min n λ ∈ R : b Γ s ( D + (0 , λ [ ∞ ])) = { } o and lim m ∈ Z ,m → + ∞ e max ( mD ) m = sup m ∈ Z > e max ( mD ) m by Fekete’s lemma. Let λ ∈ R > , let φ ∈ b Γ s ( mD + (0 , λ [ ∞ ])) \ { } , and let Z := Supp( mD + ( φ )) . For every x ∈ ( X \ Z )( K ) , we have h D ( x ) > m inf x ∈ ( X \ Z ) an ∞ g mD ∞ ( x ) > λm . Hence we have the first inequality.To show the second inequality, we write D = a D + · · · + a r D r + (0 , f ) such that a i ∈ R , D i ∈ d Div( X ) , f ∈ C ℓ ( X ) , and D i are all effective (see [13,Proposition 2.4.2(1)]). We set D ′ := ⌈ a ⌉ D + · · · + ⌈ a r ⌉ D r and Σ := r [ i =1 Supp( D i ) . By [3, Proposition 2.6], (cid:8) x ∈ ( X \ Σ)( K ) : h D ′ ( x ) C (cid:9) is Zariski dense in X fora constant C .If x ∈ ( X \ Σ)( K ) , then h D ( x ) h D ′ ( x ) + k f k ℓ . Hence (cid:8) x ∈ X ( K ) : h D ( x ) C + k f k ℓ (cid:9) ⊃ (cid:8) x ∈ ( X \ Σ)( K ) : h D ′ ( x ) C (cid:9) , and the left-hand side is also Zariski dense in X . It implies that the essentialminimum is bounded from above by C + k f k ℓ . (cid:3) Lemma 3.16.
For any ( D ; V ) ∈ d D iv R , R ( X ) , one has c vol( D ; V ) (dim X + 1)[ K : Q ] vol( D ; V ) max lim m ∈ Z ,m → + ∞ e max ( mD ; m V ) m , . Proof.
By Gillet–Soulé’s formula [5, Proposition 6], we have b ℓ s (cid:0) mD ; m V (cid:1) max (cid:8) e max ( mD ; m V ) , (cid:9) · rk H ( mD ; m V )+ 2 (cid:0) rk H ( mD ) + log(rk H ( mD ))! (cid:1) for every m ∈ Z > . Therefore, c vol( D ; V ) (dim X + 1)[ K : Q ] max lim m ∈ Z ,m → + ∞ e max ( mD ; m V ) m , · lim sup m ∈ Z ,m → + ∞ dim K H ( mD ; m V ) m dim X / (dim X )!= (dim X + 1)[ K : Q ] vol( D ; V ) max lim m ∈ Z ,m → + ∞ e max ( mD ; m V ) m , . (cid:3) Lemma 3.17.
Let ( D ; V ) ∈ d D iv ℓ R , R ( X ) . Let (cid:0) D n (cid:1) n > be an increasing sequence in d Div ℓ R ( X ) such that ζ ( D n ) = D and such that (cid:13)(cid:13) D − D n (cid:13)(cid:13) ℓ → as n → + ∞ . Onethen has c vol (cid:0) D ; V (cid:1) = lim n → + ∞ c vol (cid:0) D n ; V (cid:1) . Proof.
Since (cid:0) D n (cid:1) n > is an increasing sequence, we can assume D n ∈ d Div R ( X ) forevery n > by Proposition 3.7. Hence, by (3.44), (cid:12)(cid:12)(cid:12) c vol( D ; V ) − c vol( D n ; V ) (cid:12)(cid:12)(cid:12)
32 (dim X + 1)[ K : Q ] vol( D ; V ) · (cid:13)(cid:13) D − D n (cid:13)(cid:13) ℓ → as n → + ∞ . (cid:3) Proposition 3.18.
Let ( D ; V ) ∈ d D iv ℓ R , R ( X ) . For any f ∈ C ℓ ( X ) , we have (cid:12)(cid:12)(cid:12) c vol (cid:0) D + (0 , f ); V (cid:1) − c vol (cid:0) D ; V (cid:1)(cid:12)(cid:12)(cid:12)
12 (dim X + 1)[ K : Q ] vol( D ; V ) · k f k ℓ . Proof.
Let (cid:0) D n (cid:1) n > be an increasing sequence in d Div R ( X ) such that ζ ( D n ) = D and such that (cid:13)(cid:13) D − D n (cid:13)(cid:13) ℓ → as n → + ∞ , and let ( f n ) n > be an increasingsequence in C ( X ) such that k f − f n k ℓ → as n → + ∞ . By the same argumentsas in [14, Proposition 5.1.3], Lemma 3.2 implies (cid:12)(cid:12)(cid:12) c vol (cid:0) D n + (0 , f n ); V (cid:1) − c vol (cid:0) D n ; V (cid:1)(cid:12)(cid:12)(cid:12)
12 (dim X + 1)[ K : Q ] vol( D ; V ) · k f n k ℓ . By taking n → + ∞ , we have the required assertion by Lemma 3.17. (cid:3) Continuity of the arithmetic volume function.
The purpose of this sec-tion is to establish the global continuity of the arithmetic volume function over d D iv ℓ R , R ( X ) along the directions of ℓ -adelic R -Cartier divisors (see Theorem 3.21).To begin with, we show the homogeneity of the arithmetic volume function in thefollowing form. Lemma 3.19.
Let X be a projective arithmetic variety of dimension d + 1 havingsmooth generic fiber X Q . Let D ∈ d Div Q ( X ; C ∞ ) and let V ∈ BC R ( X ) with V > .For any p ∈ Z > , one has c vol (cid:0) p D ; p V (cid:1) = p dim X +1 c vol (cid:0) D ; V (cid:1) . Proof.
First, we note the following.
HE CONTINUITY OF VOLUMES 31
Claim 3.20.
It suffices to show that, to each D ∈ d Div Q ( X ; C ∞ ) , one can assigna q D ∈ Z > such that the equality is true for all multiples of q D .Proof of Claim 3.20. For any p ∈ Z > , one has c vol (cid:0) p D ; p V (cid:1) = 1( q D q p D ) dim X +1 c vol (cid:16) ( pq D q p D ) D ; ( pq D q p D ) V (cid:17) = p dim X +1 c vol (cid:0) D ; V (cid:1) . (cid:3) By Claim 3.20, it suffices to show the equality for every p ∈ Z > with D ′ := p D ∈ d Div( X ; C ∞ ) . We fix an E ∈ d Div( X ) such that E > and E ± D ′ > . By Theorem 2.6, there isa constant C > such that b ℓ s (cid:16) O X ( m D ′ + E ); n V (cid:17) − b ℓ s (cid:16) O X ( m D ′ − E ); n V (cid:17) Cm dim X (1 + log( m )) for m ∈ Z > and n ∈ Z > . Hence, for each r = 1 , , . . . , p − , we obtain lim sup m ∈ Z ,m → + ∞ b ℓ s (cid:0) ( pm + r ) D ; ( pm + r ) V (cid:1) ( pm + r ) dim X +1 / (dim X + 1)! lim sup m ∈ Z ,m → + ∞ b ℓ s (cid:16)(cid:16) m + rp (cid:17) D ′ ; pm V (cid:17) ( pm ) dim X +1 / (dim X + 1)!= lim sup m ∈ Z ,m → + ∞ b ℓ s (cid:16) O X ( m D ′ ); pm V (cid:17) ( pm ) dim X +1 / (dim X + 1)! . Therefore, c vol (cid:0) D ; V (cid:1) = max r
Let X be a normal, projective, and geometrically connected K -variety. Let V be a finite-dimensional R -subspace of d Div ℓ R ( X ) endowed with a norm k · k V , let Σ be a finite set of points on X , and let B ∈ R > . For any ε ∈ R > ,there exists a δ ∈ R > such that (cid:12)(cid:12)(cid:12) c vol (cid:0) D + (0 , f ); V (cid:1) − c vol (cid:0) E ; V (cid:1)(cid:12)(cid:12)(cid:12) ε for every D, E ∈ V with max (cid:8)(cid:13)(cid:13) D (cid:13)(cid:13) V , (cid:13)(cid:13) E (cid:13)(cid:13) V (cid:9) B and (cid:13)(cid:13) D − E (cid:13)(cid:13) V δ , f ∈ C ℓ ( X ) with k f k ℓ δ , and V ∈ BC R ( X ) with { c X ( ν ) : ν ( V ) > } ⊂ Σ . We need the following.
Proposition 3.22.
Let X be a projective arithmetic variety of dimension d + 1 such that X Q is smooth. Let V = ( V, k · k V ) be a couple of a finite-dimensional R -subspace V of d Div R ( X ; C ∞ ) and a norm k · k V on V , and let Σ be a finite setof points on X . There then exists a positive constant C V , Σ > such that (cid:12)(cid:12)(cid:12) c vol (cid:0) D ; V (cid:1) − c vol (cid:16) D ′ ; V (cid:17)(cid:12)(cid:12)(cid:12) C V , Σ max (cid:26)(cid:13)(cid:13) D (cid:13)(cid:13) dV , (cid:13)(cid:13)(cid:13) D ′ (cid:13)(cid:13)(cid:13) dV (cid:27) · (cid:13)(cid:13)(cid:13) D − D ′ (cid:13)(cid:13)(cid:13) V . for every D , D ′ ∈ V and V ∈ BC R ( X ) with { c X ( ν ) : ν ( V ) > } ⊂ Σ .Proof. By extending ( V, k · k V ) if necessary, we may assume that V has a basis A , . . . , A r ∈ d Div( X ; C ∞ ) such that A , . . . , A r are all effective. We set (cid:13)(cid:13) a A + · · · + a r A r (cid:13)(cid:13) := | a | + · · · + | a r | for a , . . . , a r ∈ R , and set D = a · AAA , D ′ = a ′ · AAA , and A := A + · · · + A r . If a ′ = 0 , then we can see c vol (cid:0) D ; V (cid:1) C k a k d +11 for(3.48) C := max n , c vol (cid:0) A (cid:1)o by using Lemma 3.19, so that we can assume that both a and a ′ are nonzero.First, we assume a , a ′ ∈ Z r and b := a ′ − a > . By Theorem 2.6, we get aconstant C ′ > C depending only on AAA , Σ , and X such that b ℓ s (cid:0) m O X ( D ); m V (cid:1) − b ℓ s (cid:0) m O X ( D ); m V (cid:1) b ℓ s (cid:16) O X ( m a · AAA + m max i { b i } A ); m V (cid:17) − b ℓ s (cid:16) O X ( m a · AAA ); m V (cid:17) C ′ m d ( k a k + k b k ) d ( m k b k + log( m k a k )) for every m ∈ Z > . Hence(3.49) c vol (cid:0) D ; V (cid:1) c vol (cid:0) D ; V (cid:1) c vol (cid:0) D ; V (cid:1) + C ′ ( k a k + k b k ) d k b k . For general a , a ′ ∈ Z r , we set a ′′ := max { a , a ′ } and D ′′ := a ′′ · AAA . By (3.49) (cid:12)(cid:12)(cid:12) c vol (cid:0) D ; V (cid:1) − c vol (cid:16) D ′ ; V (cid:17)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) c vol (cid:16) D ′′ ; V (cid:17) − c vol (cid:0) D ; V (cid:1)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) c vol (cid:16) D ′′ ; V (cid:17) − c vol (cid:16) D ′ ; V (cid:17)(cid:12)(cid:12)(cid:12) C ′ ( k a k + k a ′′ − a k ) d k a ′′ − a k + C ′ ( k a ′ k + k a ′′ − a ′ k ) d k a ′′ − a ′ k d C ′ max (cid:8) k a k d , k a ′ k d (cid:9) k a − a ′ k . Therefore, by using Lemma 3.19, we can verify that the estimate is also true forevery a , a ′ ∈ Q r .Next, we show the estimate for every a , a ′ ∈ R r . Claim 3.23.
Let (cid:0) p ( n ) (cid:1) n > be a sequence in Q r that converges to a ∈ R r . Then lim n → + ∞ c vol (cid:16) p ( n ) · AAA ; V (cid:17) = c vol (cid:16) a · AAA ; V (cid:17) . HE CONTINUITY OF VOLUMES 33
Proof of Claim 3.23.
Let (cid:0) b ( n ) (cid:1) n > and (cid:0) c ( n ) (cid:1) n > be two sequences in Q r such that b (1) i b (2) i . . . b ( n ) i . . . a i . . . c ( n ) i . . . c (2) i c (1) i and lim n → + ∞ (cid:12)(cid:12)(cid:12) c ( n ) i − b ( n ) i (cid:12)(cid:12)(cid:12) = 0 for i = 1 , , . . . , r . We then have c vol (cid:16) b (1) · AAA ; V (cid:17) c vol (cid:16) b (2) · AAA ; V (cid:17) . . . c vol (cid:16) b ( n ) · AAA ; V (cid:17) . . . c vol (cid:16) a · AAA ; V (cid:17) . . . c vol (cid:16) c ( n ) · AAA ; V (cid:17) . . . c vol (cid:16) c (2) · AAA ; V (cid:17) c vol (cid:16) c (1) · AAA ; V (cid:17) and lim n → + ∞ (cid:16) c vol (cid:16) c ( n ) · AAA ; V (cid:17) − c vol (cid:16) b ( n ) · AAA ; V (cid:17)(cid:17) = 0 by the above arguments. Hence we have the required claim. (cid:3) We choose two sequences (cid:0) p ( n ) (cid:1) n > and (cid:0) q ( n ) (cid:1) n > in Q r such that lim n → + ∞ p ( n ) = a and lim n → + ∞ q ( n ) = a ′ , respectively. Then (cid:12)(cid:12)(cid:12) c vol (cid:16) p ( n ) · AAA ; V (cid:17) − c vol (cid:16) q ( n ) · AAA ; V (cid:17)(cid:12)(cid:12)(cid:12) C max (cid:26)(cid:13)(cid:13)(cid:13) p ( n ) (cid:13)(cid:13)(cid:13) d , (cid:13)(cid:13)(cid:13) q ( n ) (cid:13)(cid:13)(cid:13) d (cid:27) (cid:13)(cid:13)(cid:13) p ( n ) − q ( n ) (cid:13)(cid:13)(cid:13) for every n > by the previous argument. Taking n → + ∞ , we obtain the requiredresult. (cid:3) Proof of Theorem 3.21.
We may assume that X is smooth. In fact, let µ : e X → X be a resolution of singularities of X , and regard V as an R -subspace of d Div ℓ R ( e X ) via d Div ℓ R ( X ) → d Div ℓ R ( e X ) . Since X is normal, we have c vol (cid:0) µ ∗ D ; V (cid:1) = c vol (cid:0) D ; V (cid:1) for every D ∈ V and V ∈ BC R ( X ) . Let A , . . . , A r ∈ d Div ℓ R ( X ) be a basis for V ,put (cid:13)(cid:13) a A + · · · + a r A r (cid:13)(cid:13) := | a | + · · · + | a r | for a , . . . , a r ∈ R , and suppose that k · k V is given as k · k . We can easily find aconstant B ′ ∈ R > such that vol( D ) B ′ for every D ∈ V with k D k B .We put(3.50) δ ′ := ε X + 1)[ K : Q ] B ′ ( B + 1) , and fix, for each i , ( X , A i ) ∈ [ Mod R ( A i ) such that A i ∈ d Div R ( X ; C ∞ ) and suchthat (cid:13)(cid:13)(cid:13) A i − A ad i (cid:13)(cid:13)(cid:13) ℓ δ ′ by using the Stone–Weierstrass theorem and Proposi-tion 3.7. Proposition 3.18 implies that(3.51) (cid:12)(cid:12)(cid:12) c vol (cid:0) a · A + (0 , f ); V (cid:1) − c vol (cid:16) a · AAA ad ; V (cid:17)(cid:12)(cid:12)(cid:12)
12 (dim X + 1)[ K : Q ] B ′ ( k a k + 1) δ ε holds for every a ∈ R r with k a k B , f ∈ C ℓ ( X ) with k f k ℓ δ ′ , and V ∈ BC R ( X ) .Thanks to Proposition 3.22, there is a constant C AAA , Σ > such that (cid:12)(cid:12)(cid:12) c vol (cid:16) a · AAA ad ; V (cid:17) − c vol (cid:16) a ′ · AAA ad ; V (cid:17)(cid:12)(cid:12)(cid:12) C AAA , Σ max (cid:8) k a k d , k a ′ k d (cid:9) k a − a ′ k for every a , a ′ ∈ R r and V ∈ BC R ( X ) with { c X ( ν ) : ν ( V ) > } ⊂ Σ , so, if we set(3.52) δ := min (cid:26) δ ′ , ε C AAA , Σ B d (cid:27) , then(3.53) (cid:12)(cid:12)(cid:12) c vol (cid:16) a · AAA ad ; V (cid:17) − c vol (cid:16) a ′ · AAA ad ; V (cid:17)(cid:12)(cid:12)(cid:12) ε for every a , a ′ ∈ R r with max {k a k , k a ′ k } B and k a − a ′ k δ . All in all, wehave (cid:12)(cid:12)(cid:12) c vol (cid:0) a · A + (0 , f ); V (cid:1) − c vol (cid:0) a ′ · A ; V (cid:1)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) c vol (cid:0) a · A + (0 , f ); V (cid:1) − c vol (cid:16) a · AAA ad ; V (cid:17)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) c vol (cid:16) a · AAA ad ; V (cid:17) − c vol (cid:16) a ′ · AAA ad ; V (cid:17)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) c vol (cid:16) a ′ · AAA ad ; V (cid:17) − c vol (cid:0) a ′ · A ; V (cid:1)(cid:12)(cid:12)(cid:12) ε as required. (cid:3) Theorem 3.21 implies the following corollaries.
Corollary 3.24.
For a ( D ; V ) ∈ d D iv ℓ R , R ( X ) , the following are equivalent.(1) c vol (cid:0) D ; V (cid:1) > .(2) For any A ∈ d Div ℓ R ( X ) with c vol (cid:0) A (cid:1) > , there exists a t ∈ R > such that ( D − tA ; V ) > . Corollary 3.25.
For any ( D ; V ) ∈ d D iv ℓ R , R ( X ) and p ∈ R > , one has c vol (cid:0) pD ; p V (cid:1) = p dim X +1 c vol (cid:0) D ; V (cid:1) . Proof.
We may assume that X is smooth. Let V be a finite-dimensional R -subspaceof d D iv ℓ R , R ( X ) such that V has a basis A , . . . , A r ∈ d Div ℓ Q ( X ) and such that D = a · A ∈ V for an a ∈ R r . Let (cid:0) b ( n ) (cid:1) n > be a sequence in Q r that converges to a . By the Stone–Weierstrass theorem and Proposition 3.7, one finds, for each i , asequence (cid:0) ( X n , A in ) (cid:1) n > in [ Mod Q ( A i ) such that A in ∈ d Div Q ( X n ; C ∞ ) , such that A ad i A ad i . . . , and such that (cid:13)(cid:13)(cid:13) A i − A ad in (cid:13)(cid:13)(cid:13) ℓ → as n → + ∞ . By Lemma 3.19, c vol (cid:16) p b ( n ) · AAA ad n ; p V (cid:17) = p dim X +1 c vol (cid:16) b ( n ) · AAA ad n ; V (cid:17) for p ∈ Q > and n > . Taking n → + ∞ (Theorem 3.21), we obtain the equalityfor every p ∈ Q > .To show the corollary, we note that the inequality is obvious. We choose andecreasing sequence ( q n ) n > in Q > that converges to p . Then c vol (cid:0) q n D ; p V (cid:1) > c vol (cid:0) q n D ; q n V (cid:1) = q dim X +1 n c vol (cid:0) D ; V (cid:1) for n > . By taking n → + ∞ , we conclude the proof by Theorem 3.21. (cid:3) Corollary 3.26.
For any ( D ; V ) ∈ d D iv ℓ R , R ( X ) and φ ∈ Rat( X ) × ⊗ Z R , one has c vol (cid:16) D + c ( φ ); V (cid:17) = c vol (cid:0) D ; V (cid:1) . HE CONTINUITY OF VOLUMES 35
Proof.
We write φ = φ a · · · φ a r r with a i ∈ R and φ i ∈ Rat( X ) . Let V be the R -subspace of d Div ℓ R ( X ) generated by φ , . . . , φ r . For each i , we choose a sequence (cid:16) b ( n ) i (cid:17) n > in Q such that b ( n ) i → a i as n → + ∞ . By homogeneity (Corollary 3.25),we have c vol D + r X i =1 b ( n ) i d ( φ i ); V ! = c vol (cid:0) D ; V (cid:1) for every n > . Taking n → + ∞ , we obtain the required assertion by Theo-rem 3.21. (cid:3) Corollary 3.27.
For each V ∈ BC R ( X ) , the arithmetic volume function induces acontinuous function c Cl ℓ R ( X ) → R > , D c vol (cid:0) D ; V (cid:1) .Proof. By using Corollary 3.26, we can obtain the required map. To show thecontinuity, let q : d Div ℓ R ( X ) → c Cl ℓ R ( X ) be the natural projection and fix a section ι ′ : Cl R ( X ) → d Div ℓ R ( X ) of ζ . Let V be the image of ι ′ and let k · k be a norm on Cl R ( X ) . Set (cid:13)(cid:13) D (cid:13)(cid:13) ι ′ , k·k := (cid:13)(cid:13) ζ ( D ) (cid:13)(cid:13) + (cid:13)(cid:13) D − ι ′ ◦ ζ ( D ) (cid:13)(cid:13) ℓ for D ∈ V ⊕ C ℓ ( X ) , and set ι := q ◦ ι ′ . We then have (cid:13)(cid:13) D (cid:13)(cid:13) ι ′ , k·k = (cid:13)(cid:13) q ( D ) (cid:13)(cid:13) ι, k·k forevery D ∈ V . Hence the assertion results from Theorem 3.21. (cid:3) Acknowledgement
This work was supported by JSPS KAKENHI Grant Number 16K17559. Theauthor is grateful to Professors Namikawa, Yoshikawa, and Moriwaki and KyotoUniversity for the financial supports.
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Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan
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