Invariance of Quantum Rings under Ordinary Flops II: A quantum Leray--Hirsch theorem

Abstract

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 splitting 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 Dubrovin connection on the base by a Picard--Fuchs system of the toric fibers. Nonsplit 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 which reduces the general ordinary flops to the split case solved here.