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

Abstract

We analyze the structure of the virtual (orbifold) [Formula: see text]-theory ring of the complex orbifold [Formula: see text] 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 [Formula: see text]-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 [Formula: see text]-theory ring in terms of these virtual line elements. This yields a surjective homomorphism from the virtual [Formula: see text]-theory ring of [Formula: see text] to the ordinary [Formula: see text]-theory ring of a crepant resolution of the cotangent bundle of [Formula: see text] which respects the Adams operations. Furthermore, there is a natural subring of the virtual K-theory ring of [Formula: see text] 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 [Formula: see text]-theory, Ann. K-Theory (2016)], who proved the latter for [Formula: see text].