Introduction

Let K be a global field, and X/K a projective variety. Let C be an ample

line bundle on X with adelically normed sections: an ample invertible sheaf C with

norms || \\v on the spaces of sections

KV®K

r(X,

£®n),

for each place v of K. The

sectional capacity S1{£) is an asymptotic measure of the volume of the set of adelic

sections of

C®n

with norm at most 1 at all places v. The quantity — log(57(£))

generalizes the top arithmetic self-intersection number of a metrized line bundle

in Arakelov theory, and its existence is equivalent to a very general arithmetic

Hilbert-Samuel theorem.

The purpose of this paper is to prove the existence of the sectional capacity

under mild hypotheses on the norms and the variety X, and to study its functoriality

properties. For each n 0, let

•F(Z®n)

= {/ e AK ®K T(£®")) : ||/„||„ 1 for all v} .

where A^ is the adele ring of K. Fix a Haar measure vol on

AK®KF(C®n),

and let

covol(r(£^n))

be the volume of a fundamental domain for

(AK®Kr(C®n))/r(C®n).

Then S7(C) is defined by:

-log(S

7

(£)) = lim ^±Mlog(vol(^(Z 0 "))/covol(r(/:^)) )

provided the limit exists.

There are at least three settings where such limits arise.

The first is in a generalization of the classical logarithmic capacity proposed

by Chinburg ([Ch]). Suppose X/K is geometrically integral. Let D be an ample,

If-rational Cartier divisor on X, and for each v let Ev C X(CV) be a nonempty set,

disjoint from the support of the positive part D and stable under all continuous

automorphisms of Cv/Kv. (Here Cv is the completion of the algebraic closure of

Kv.) Put C = Ox(D). Dehomogenizing at D, we can identify sections of

£®n

with

functions in K (X):

T(C®n) = T(nD) = {/ € K{X) : div(/) + nD 0} .

After base change to Kv, an analogous statement holds for Kv 0 T(nD). We take

the norms \\f\\v to be the sup norms \\f\\Ev of \f(x)\

v

over Ev, where I I

v

on Cv is

the unique extension of the canonical absolute value on Kv given by the modulus

of additive Haar measure. For each n 0, put ^(Ey^nD) = {/ G Kv (g T(nD) :

| | / l k ! } • Write E = UvEv and put

^(E,nD ) - ( A K ® r ( r i D ) ) n J ] : ^ ( £

v

, n D ) .

Received by the editor March 30, 1998.

During the preparation of this paper, the first author was supported in part by NSF Grants

DMS-9103553 and DMS-9500892. The third author was supported in part by NSF Grants DMS-

9106938, DMS-9208282, DMS-9305857, and DMS-9403887.

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