Birational Calabi-Yau manifolds have the same small quantum products

TL;DR

This paper proves that birational projective Calabi-Yau manifolds share the same small quantum cohomology algebra after a change of Novikov rings, using symplectic cohomology and Hamiltonian Floer cohomology as key tools.

Contribution

It establishes an isomorphism between small quantum cohomology algebras of birational Calabi-Yau manifolds, revealing a deep invariance under birational transformations.

Findings

Birational Calabi-Yau manifolds have identical small quantum cohomology after a change of Novikov rings.

Symplectic cohomology can be used to compare quantum products of different manifolds.

Subvarieties of positive codimension are shown to be stably displaceable.

Abstract

We show that any two birational projective Calabi-Yau manifolds have isomorphic small quantum cohomology algebras after a certain change of Novikov rings. The key tool used is a version of an algebra called symplectic cohomology, which is constructed using Hamiltonian Floer cohomology. Morally, the idea of the proof is to show that both small quantum products are identical deformations of symplectic cohomology of some common open affine subspace. Part of the proof uses the fact that subvarieties of positive codimension are stably displaceable.