Quantum flips I: local model

Abstract

We study analytic continuations of quantum cohomology under simple flips $f:X⇢X^{\prime }$ along the extremal ray quantum variable $q^{\ell }$. The inverse correspondence $\Psi =[{\Gamma }_f{]}^{\ast }$ by the graph closure gives an embedding of Chow motives $[{\hat{X}}^{\prime }]\hookrightarrow [\hat{X}]$ which preserves the Poincaré pairing. We construct a deformation $\hat{\Psi }$ of $\Psi =[{\Gamma }_f{]}^{\ast }$ which induces a non-linear embedding $$QH(X^{\prime })\hookrightarrow QH(X)$$ 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 $X^{\prime }$ are (projective) local models.

50 pages; v2: typos fixed

References in corpus (1)

• Global Mirrors and Discrepant Transformations for Toric Deligne-Mumford Stacks