Invariance of quantum rings under ordinary flops II: A quantum Leray–Hirsch theorem
This is the second of a sequence of papers proving the quantum invariance for ordinary flops over an arbitrary smooth base. In this paper, we complete the proof of the invariance of the big quantum rings under ordinary flops of split type. To achieve that, several new ingredients are introduced. One is a quantum Leray-Hirsch theorem for the local model (a certain toric bundle) which extends the quantum D-module of the Dubrovin connection on the base by a Picard-Fuchs system of the toric fibers.
... f the toric fibers. Non-split flops as well as further applications of the quantum Leray-Hirsch theorem will be discussed in subsequent papers. In particular, a quantum splitting principle is developed in part III (Lee, Lin, Qu and Wang, "Invariance of quantum rings under ordinary flops III", Cambridge Journal of Mathematics, 2016), which reduces the general ordinary flops to the split case solved here. This is, as far as we know, the first result on the quantum invariance under the K-equivalence (crepant transformation) [Wan04, Wan03] where the local structure of the exceptional loci cannot be deformed to any explicit (for example, toric) geometry and the analytic continuation is nontrivial. This is also the first result for which the analytic continuation is established with nontrivial Birkhoff factorization. Several new ingredients are introduced in the course of the proof. One main technical ingredient is the quantum Leray-Hirsch theorem for the local model, which is related to the canonical lift of the quantum D-module from the base to the total space of a (toric) bundle. The techniques developed in this paper are applicable to more general cases and will be discussed in subsequent papers. Conventions. This paper is strongly correlated with [LLW16], which will be referred to as Part I throughout the paper. All conventions and the notation introduced there carry over to this paper.