For ordinary flops, the correspondence defined by the graph closure is shown to give equivalence of Chow motives and to preserve the Poincaré pairing. In the case of simple ordinary flops, this correspondence preserves the big quantum cohomology ring after an analytic continuation over the extended Kähler moduli space.For Mukai flops, it is shown that the birational map for the local models is deformation equivalent to isomorphisms. This implies that the birational map induces isomorphisms on the full quantum rings and all the quantum corrections attached to the extremal ray vanish.(1) Is there a canonical correspondence between the cohomology groups of K-equivalent smooth varieties?(2) Is there a modified ring structure which is invariant under the K-equivalence relation?The following conjecture was advanced by Y. Ruan [24] and the third author [26] in response to these questions.Conjecture 0.4. K-equivalent smooth varieties have canonically isomorphic quantum cohomology rings over the extended Kähler moduli spaces.

Abstract.A new tautological relation of M 3,1 in codimension 3 is derived and proved, using an invariance constraint from our previous work.

The purpose of this short article is to prove a product formula relating the log Gromov-Witten invariants of V × W with those of V and W in the case the log structure on V is trivial. Introduction.Product formulas in the literature of Gromov-Witten (GW) theory started with [15] for the genus zero Gromov-Witten invariants. It was soon generalized in [8] to (absolute) GW invariants in any genus. There is also an orbifold version in [7]. In this paper, we extend the product formula to the setting of relative GW theory or more generally log GW theory.

In this paper we prove the invariance of quantum rings for general ordinary flops, whose local models are certain non-split toric bundles over arbitrary smooth bases. An essential ingredient in the proof is a quantum splitting principle which reduces a statement in Gromov-Witten theory on non-split bundles to the case of split bundles.

No abstract