Quantum flips I: local model

TL;DR

This paper investigates how quantum cohomology behaves under simple flips, constructing embeddings that reveal functoriality beyond K-equivalence, especially in local models.

Contribution

It introduces a deformation of the inverse Chow motive correspondence that induces a non-linear embedding of quantum cohomology, demonstrating functoriality beyond K-equivalent transformations.

Findings

Constructs a deformation of the Chow motive correspondence.

Establishes a non-linear embedding of quantum cohomology.

Provides examples of functoriality beyond K-equivalence.

Abstract

We study analytic continuations of quantum cohomology under simple flips $f: X \dashrightarrow X'$ along the extremal ray quantum variable $q^\ell$. The inverse correspondence $\Psi = [\Gamma_f]^*$ by the graph closure gives an embedding of Chow motives $[\hat{X}'] \hookrightarrow [\hat{X}]$ which preserves the Poincar\'e pairing. We construct a deformation $\widehat{\Psi}$ of $\Psi = [\Gamma_f]^*$ which induces a non-linear embedding $$QH(X') \hookrightarrow QH(X)$$ in the category of $F$-manifolds into the regular integrable loci of $QH(X)$ near $q^\ell = \infty$. This provides examples of functoriality of quantum cohomology beyond $K$-equivalent transformations. In this paper, we focus on the case when $X$ and $X'$ are (projective) local models.