Adams operations on the virtual K-theory of P(1,n)

TL;DR

This paper studies the virtual K-theory ring of the orbifold P(1,n), identifying its structure, Adams operations, and relations to crepant resolutions, extending previous results for specific cases.

Contribution

It provides a new presentation of the virtual K-theory ring of P(1,n) and establishes a connection to the K-theory of crepant resolutions, generalizing prior work.

Findings

Identified the group of virtual line elements.

Established a surjective homomorphism to the K-theory of a crepant resolution.

Found a subring isomorphic to the K-theory of the resolution.

Abstract

We analyze the structure of the virtual (orbifold) K-theory ring of the complex orbifold P(1,n) and its virtual Adams (or power) operations, by using the non-Abelian localization theorem of Edidin-Graham. In particular, we identify the group of virtual line elements and obtain a natural presentation for the virtual K-theory ring in terms of these virtual line elements. This yields a surjective homomorphism from the virtual K-theory ring of P(1,n) to the ordinary K-theory ring of a crepant resolution of the cotangent bundle of P(1,n) which respects the Adams operations. Furthermore, there is a natural subring of the virtual K-theory ring of P(1,n) which is isomorphic to the ordinary K-theory ring of the resolution. This generalizes the results of Edidin-Jarvis-Kimura who proved the latter for n=2,3.