Delocalized Chern character for stringy orbifold K-theory

In this paper, we define a stringy product on K^*_{orb}(\XX) \otimes \C , the orbifold K-theory of any almost complex presentable orbifold \XX. We establish that under this stringy product, the de-locaized Chern character ch_{deloc} : K^*_{orb}(\XX) \otimes \C \longrightarrow H^*_{CR}(\XX), after a canonical modification, is a ring isomorphism. Here H^*_{CR}(\XX) is the Chen-Ruan cohomology of \XX. The proof relies on an intrinsic description of the obstruction bundles in the construction of Chen-Ruan product. As an application, we investigate this stringy product on the equivariant K-theory $K^*_G(G)$ of a finite group $G$ with the conjugation action. It turns out that the stringy product is different from the Pontryajin product (the latter is also called the fusion product in string theory).Comment: 34 pages. Final version to appear in Trans. of AMS. Improve the expositions and Change of the title thanks the referee

Local Gromov-Witten invariants of Blowups of Fano surfaces

In this paper, using the degeneration formula we obtain a blowup formulae of local Gromov-Witten invariants of Fano surfaces. This formula makes it possible to compute the local Gromov-Witten invariants of non-toric Fano surfaces form toric Fano surface, such as del Pezzo surfaces. This formula also verifed an expectation of Chiang-Klemm-Yau-Zaslow.Comment: 13 page