New Products, Chern Classes, and Power Operations in Orbifold K-theory

Details New Products, Chern Classes, and Power Operations in Orbifold K-theory New Products, Chern Classes, and Power Operations in Orbifold K-theory

2012-02-03

arxiv.org/abs/1202.0603v1

Mathematics

Algebraic Geometry

39 pp

Scientific paper

We develop a theory of inertial pairs on smooth, separated Deligne-Mumford quotient stacks. An inertial pair determines inertial products and an inertial Chern character. Every vector bundle V on such a stack defines two new inertial pairs and we recover, as special cases, both the orbifold product and the virtual product of [GLSUX07]. We show that for strongly Gorenstein inertial pairs there is also a theory of Chern classes and compatible power operations. An important application is to show that there is a theory of Chern classes and compatible power operations for the virtual product. We also show that when the stack is a quotient [X/G], with G diagonalizable, inertial K-theory has a lambda-ring structure. This implies that for toric Deligne-Mumford stacks there is a corresponding lambda-ring structure associated to virtual K-theory. As an example we compute the semi-group of lambda-positive elements in the virtual lambda-ring of the weighted projective stack P(1,2). Using the virtual orbifold line elements in this semi-group, we obtain a simple presentation of the K-theory ring with the virtual product and a simple description of the virtual first Chern classes. This allows us to prove that the completion of this ring with respect to the augmentation ideal is isomorphic to the usual K-theory of the resolution of singularities of the cotangent bundle T*P(1,2). We interpret this as a manifestation of mirror symmetry, in the spirit of the Hyper-Kaehler Resolution Conjecture.

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