Relative Gromov-Witten invariants of projective completions of vector bundles

It was proved by Fan and Lee(2016) and Fan(2017) that the absolute Gromov-Witten invariants of two projective bundles P(Vi)→ X are identified canonically when the total Chern classes c(V1) and c(V2) satisfy c(V1)= c(V2) for two bundles V1 and V2 over a smooth projective variety X. In this paper, we show that the relative Gromov-Witten invariants of(P(Vi ⊕ O), P(Vi)), i = 1, 2 are identified canonically when c(V1)= c(V2),where P(Vi ⊕ O) are the projective completions of the bundles Vi → X, and the projective bundles P(Vi) are the exceptional divisors in P(Vi ⊕ O).

Chen-Ruan Cohomology and Stringy Orbifold K-Theory for Stable Almost Complex Orbifolds

Comparing to the construction of stringy cohomology ring of equivariant sta-ble almost complex manifolds and its relation with the Chen-Ruan cohomology ring of the quotient almost complex orbifolds,the authors construct in this note a Chen-Ruan cohomology ring for a stable almost complex orbifold.The authors show that for a finite group G and a G-equivariant stable almost complex manifold X,the G-invariant part of the stringy cohomology ring of(X,G)is isomorphic to the Chen-Ruan cohomology ring of the global quotient stable almost complex orbifold[X/G].Similar result holds when G is a torus and the action is locally free.Moreover,for a compact presentable stable almost complex orbifold,they study the stringy orbifold K-theory and its relation with Chen-Ruan cohomology ring.