1202.0603 (Dan Edidin et al.)

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

Dan Edidin, Tyler J. Jarvis, Takashi Kimura

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.

View original: http://arxiv.org/abs/1202.0603