2013/02/14 by Takashi Kimura, Kimura, Takashi, Ross Sweet +1
Mathematics · #14H10 (Secondary) #14N35 #19L47 #53D45 (Primary) 19L10 #55N15 #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1302.3524
openalex publication_date 2013/02/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.