Invariance of Quantum Rings under Ordinary Flops I: Quantum corrections and reduction to local models

TL;DR

This paper establishes the invariance of quantum cohomology rings under ordinary flops by analyzing quantum corrections, reducing to local models, and developing new theorems to handle general cases.

Contribution

It introduces methods to prove quantum invariance under flops over arbitrary bases, including quantum corrections, reductions to local models, and a quantum splitting principle.

Findings

Quantum corrections fix the defect in the cup product.

Reduction to local models simplifies the invariance proof.

Complete invariance established for all ordinary flops.

Abstract

This is the first of a sequence of papers proving the quantum invariance under ordinary flops over an arbitrary smooth base. In this first part, we determine the defect of the cup product under the canonical correspondence and show that it is corrected by the small quantum product attached to the extremal ray. We then perform various reductions to reduce the problem to the local models. In Part II, we develop a quantum Leray--Hirsch theorem and use it to show that the big quantum cohomology ring is invariant under analytic continuations in the K\"ahler moduli space for ordinary flops of splitting type. In Part III, together with F. Qu, we remove the splitting condition by developing a quantum splitting principle, and hence solve the problem completely.