# An arithmetic capacity on Grassmannian varieties

#### Abstract

Capacity theory on the complex plane \$\doubc\$ is a non-linear measure theory for the size of subsets of \$\doubc\$. Classical theorems of Carlson, Fekete, Polya and Szego show that this measure has arithmetic implications. In the 1980's, Cantor and Rumely developed adelic capacity theory on the projective line \$\IP\sp1\$ and algebraic curves respectively. Recently Chinburg defined a measure of the size of an adelic set on an algebraic variety relative to an ample and effective divisor, which he called sectional capacity, as the limit of the volume of the multiples of the sections of the line bundle associated with the divisor with sup-norm not greater than 1 on the adelic set. One hopes that this will enable us to study sets of Galois conjugates of algebraic points on algebraic varieties. The existence of the limit defining sectional capacity is not known to exist in general. It was shown by Rumely that this limit exists on algebraic curves and is closely related to the Cantor-Rumely theory. In this thesis, we show that sectional capacity exists for quite general adelic sets on the Grassmannian varieties.

Mathematics

#### Recommended Citation

Lau, Chi Fong, "An arithmetic capacity on Grassmannian varieties" (1993). Dissertations available from ProQuest. AAI9321427.
https://repository.upenn.edu/dissertations/AAI9321427

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