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
for each place v of K. The
sectional capacity S1{£) is an asymptotic measure of the volume of the set of adelic
sections of
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
= {/ e AK ®K T(£®")) : ||/„||„ 1 for all v} .
where A^ is the adele ring of K. Fix a Haar measure vol on
and let
be the volume of a fundamental domain for
Then S7(C) is defined by:
(£)) = 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
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)\
over Ev, where I I
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 ] : ^ ( £
, 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.
Previous Page Next Page