Takashi Kimura

For a smooth Deligne-Mumford stack X we describe a large number of inertial products on K(IX) and A*(IX) and corresponding inertial Chern characters. We do this by developing a theory of inertial pairs. Each inertial pair determines an inertial product on K(IX) and an inertial product on A*(IX) and Chern character ring homomorphisms between them. We show that there are many inertial pairs; indeed, every vector bundle V on X defines two new inertial pairs. We recover, as special cases, the orbifold products of Chen-Run, Abramovich-Graber-Vistoli, Jarvis-Kaufmann-Kimura, and Edidin-Jarvis-Kimura and the virtual product of Gonzalez-Lupercio-Segovia-Uribe-Xicotencatl. We also introduce an entirely new product we call the localized orbifold product, which is defined on the complexification of K(IX). The inertial products developed in this paper are used in a subsequent paper to describe a theory of inertial Chern classes and power operations in inertial K-theory. These constructions provide new manifestations of mirror symmetry, in the spirit of the Hyper-Kaehler Resolution Conjecture.Comment: 20 pages. Several minor errors corrected. To appear in Annals of K-Theor

In this paper, we give some new genus-3 universal equations for Gromov-Witten invariants of compact symplectic manifolds. These equations were obtained by studying new relations in the tautological ring of the moduli space of 2-pointed genus-3 stable curves. A byproduct of our search for genus-3 equations is a new genus-2 universal equation for Gromov-Witten invariants.

For a smooth Deligne-Mumford stack X we describe a large number of inertial products on K(IX) and A*(IX) and corresponding inertial Chern characters. We do this by developing a theory of inertial pairs. Each inertial pair determines an inertial product on K(IX) and an inertial product on A*(IX) and Chern character ring homomorphisms between them. We show that there are many inertial pairs; indeed, every vector bundle V on X defines two new inertial pairs. We recover, as special cases, the orbifold products of Chen-Run, Abramovich-Graber-Vistoli, Jarvis-Kaufmann-Kimura, and Edidin-Jarvis-Kimura and the virtual product of Gonzalez-Lupercio-Segovia-Uribe-Xicotencatl. We also introduce an entirely new product we call the localized orbifold product, which is defined on the complexification of K(IX). The inertial products developed in this paper are used in a subsequent paper to describe a theory of inertial Chern classes and power operations in inertial K-theory. These constructions provide new manifestations of mirror symmetry, in the spirit of the Hyper-Kaehler Resolution Conjecture.

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.

In their 2007 paper, Jarvis, Kaufmann, and Kimura defined the full orbifoldK-theory of an orbifold , analogous to the Chen-Ruan orbifold cohomology of in that it uses the obstruction bundle as a quantum correction to the multiplicative structure. We give an explicit algorithm for the computation of this orbifold invariant in the case when arises as an abelian symplectic quotient. To this end, we introduce the inertial K-theory associated to a T -action on a stably complex manifold M, where T is a compact abelian Lie group. Our methods are integral K-theoretic analogues of those used in the orbifold cohomology case by Goldin, Holm, and Knutson in 2005. We rely on the K-theoretic Kirwan surjectivity methods developed by Harada and Landweber. As a worked class of examples, we compute the full orbifold K-theory of weighted projective spaces that occur as a symplectic quotient of a complex affine space by a circle. Our computations hold over the integers, and in the particular case of these weighted projective spaces, we show that the associated invariant is torsion-free.

We give a purely equivariant construction of orbifold products for quotient Deligne-Mumford stacks [X/G] where G is an arbitrary linear algebraic group (not necessarily finite). The key to our construction is the definition of the "logarithmic trace" of an equivariant vector bundle. We also prove that there is an orbifold Chern character homomorphism which induces an isomorphism of a canonical summand in the orbifold Grothendieck ring with the orbifold Chow ring. As an application we obtain an associative orbifold product on the Grothendieck ring of [X/G] (as opposed to its inerita stack) taken with complex coefficients.