## 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 [D. Edidin and W. Graham, Nonabelian localization in equivariant K-theory and Riemann-Roch for quotients, Adv. Math. 198(2) (2005) 547-582]. 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 [D. Edidin, T. J. Jarvis and T. Kimura, Chern classes and compatible power operation in inertial K-theory, Ann. K-Theory (2016)], who proved the latter for n = 2, 3.

Original language | English (US) |
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Article number | 1750149 |

Journal | Journal of Algebra and Its Applications |

Volume | 16 |

Issue number | 8 |

DOIs | |

State | Published - Aug 1 2017 |

## Keywords

- Adams operation
- crepant resolution conjecture
- Equivalent K -theory
- Gromov-Witten theory
- orbifold K -theory

## ASJC Scopus subject areas

- Algebra and Number Theory
- Applied Mathematics