Adelic solenoid I: Structure and topology

TL;DR

This paper explores the topology of the adelic solenoid via its relation to the adelic Riemann sphere, proving a Birkhoff-Grothendieck type theorem for holomorphic vector bundles with rational Chern characters.

Contribution

It establishes a Birkhoff-Grothendieck theorem for adelic structures, showing vector bundles split into line bundles with rational Chern characters, and describes the Picard group and K-ring structure.

Findings

Holomorphic vector bundles split into line bundles with rational Chern characters.

The Picard group is isomorphic to the additive group of rational numbers.

The K-ring contains new elements factoring the tautological class.

Abstract

Topologically the adelic Riemann sphere is the suspension of the adelic solenoid and because of this relation, here we study the adelic solenoid by studying the adelic Riemann sphere topology. The main result is the Birkhoff-Grothendieck Theorem: A holomorphic vector bundle splits as a sum of holomorphic line bundles whose Chern character is now a rational number. As a consequence, the Picard group is isomorphic to the additive group of rational numbers and the $K$--ring has new elements that factor the tautological class.