Toric Landau-Ginzburg models in threefold divisorial contractions

TL;DR

This paper studies the relationship between quantum periods and toric Landau-Ginzburg models in divisorial contractions of Fano threefolds, providing a mirror approach to Sarkisov links.

Contribution

It proves a regularized period identity for divisorial contractions in Fano threefolds, advancing mirror symmetry techniques in this context.

Findings

Established the limit identity for regularized quantum periods.

Provided a mirror symmetry framework for Sarkisov links.

Enhanced understanding of Landau-Ginzburg models in threefold contractions.

Abstract

We investigate quantum periods and toric Landau-Ginzburg models under divisorial contractions of terminal Fano threefolds. Let $g:Y \rightarrow X$ be a divisorial contraction between $\mathbb{Q}$-factorial Fano threefolds with ordinary terminal singularities and $E$ be the exceptional divisor. Assuming that the center of the contraction is either a smooth point, a terminal quotient point, a point of type cA/n, or a smooth curve with singularities of type cA or cA/n, we prove the regularized period identity $$ \lim_{r\to+\infty}\hat{G}_{Y,rE}(t)=\hat{G}_X(t) $$ where $\hat{G}_{Y,rE}(t)$ and $\hat{G}_X(t)$ are the regularized quantum periods of $(Y,rE)$ and $X$ respectively. This gives a mirror approach to the computation of the Sarkisov links and higher syzygies of central models of dimension 3.