Extremal transitions via quantum Serre duality

TL;DR

This paper demonstrates that the genus-zero Gromov--Witten theories of toric hypersurfaces connected by extremal transitions are related through analytic continuation and parameter restriction of their quantum D-modules, providing geometric insight.

Contribution

It establishes a precise relationship between the quantum D-modules of toric hypersurfaces involved in extremal transitions, revealing the geometric origin of analytic continuation and parameter restriction.

Findings

Quantum D-module of $ ilde Z$ recovers that of $Z$ after analytic continuation.

Provides geometric explanation for analytic continuation and restriction parameters.

Shows the relation holds for toric hypersurfaces related by extremal transitions.

Abstract

Two varieties $Z$ and $\widetilde Z$ are said to be related by extremal transition if there exists a degeneration from $Z$ to a singular variety $\overline Z$ and a crepant resolution $\widetilde Z \to \overline Z$. In this paper we compare the genus-zero Gromov--Witten theory of toric hypersurfaces related by extremal transitions arising from toric blow-up. We show that the quantum $D$-module of $\widetilde Z$, after analytic continuation and restriction of a parameter, recovers the quantum $D$-module of $Z$. The proof provides a geometric explanation for both the analytic continuation and restriction parameter appearing in the theorem.