Capacity, Green's functions, and Intersection Theory
Dr. Lau
Chi Fong
Abstract
Chinburg originally introduced
sectional capacity as an arithmetic
measure for sets on an algebraic variety over a global field. The sets in
question are adelic. There is a component set for
each place of the ground field. An ample divisor produces the size measure. The
Gillet-Soulé Arithmetic
Amplitude Theorem motivated sectional capacity. This gives asymptotics
for measuring the space of sections with sup norm at most 1.
We
discuss work of Chinburg, Rumely,
and the speaker relating sectional capacity and arithmetic intersection theory.
The main theorem asserts that the sectional capacity is a limit of top
self-intersection numbers of metrized line bundles.
This limit is relative to a canonical sequence of models determined by the nonarchimedean part of the set. The metrics come from smoothings of plurisubharmonic extremal functions attached to archimedean
components of the set. This is the strongest result of its type with a
formulation within Gillet-Soulé theory.
Conjecturally, however, the sectional capacity should be a top intersection
number of an adelic line bundle with singular metrics. The talk will discuss
evidence for such an intersection theory.
References:
1. T. Chinburg,
Capacity Theory on Varieties, Compositio Math 80
(1991), 71-84.
2. H. Gillet and C. Soulé, Amplitude Arithmétique, CRAS Paris
307 (1988), Série I (Math), 887-890.
3. R. Rumely,
C.F. Lau and R. Varley, Existence of the Sectional Capacity, Memoires
of the AMS 690 (2000), AMS,
4. T. Chinburg,
C.F. Lau and R. Rumely, Capacity Theory and Arithmetic Intersection Theory, Duke
Mathematical Journal
117, no 2 (2003),
229-285.
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Room 517, Meng Wah Complex |
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All are welcome |
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