Invariance of quantum rings under ordinary flops III: A quantum splitting principle

Abstract

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.

A quantum splitting principle and an application

“…Remark 0.6. -The reduction technique in [LLQW14] by blow-ups does not seem to work in our situation, as the Brauer group is a birational invariant.…”

“…It's worth noting that splitting principle has already inspired some of the techniques in [LLQW14]. Roughly speaking, they apply base-changes along blow-ups to turn a vector bundle into a split one, and relate back via degeneration formula.…”

On Gromov–Witten Theory of Projective Bundles

“…In particular, it's important in Gromov-Witten theory as well. For example, Gromov-Witten theory of projective bundles has appeared as a central subject in crepant transformation conjecture for ordinary flops (e.g., [LLW11,LLW13,LLQW14]), and the later proposals of the functoriality of Gromov-Witten theory ( [LLW16,LLW15]) also emphasize the role of projective bundles. Besides, projective bundle appears naturally in the degeneration to the normal cone which is an important construction in some Gromov-Witten results (e.g., [MP06,HLR08]).…”

“…When the projective bundle is not split, such a fiberwise torus action is absent. A degeneration argument is applied in [LLQW14] and has had some success in a limited form of quantum splitting principle.…”

Birational Calabi-Yau manifolds have the same small quantum products

“…It was shown in [LLW11], [LLW13] and [LLQW14] that certain birational isomorphisms called ordinary flops induce isomorphisms between big quantum cohomology rings up to analytic continuation. In [Wan02, Conjecture IV], it was conjectured that a small perturbation of a birational isomorphism between Calabi-Yau manifolds is a sequence of ordinary flops.…”