Overview
This study investigates quantum periods and toric Landau-Ginzburg (LG) models in the context of divisorial contractions involving terminal Fano threefolds. The core of the research establishes a regularized period identity under specific conditions for the center of contraction.
Research Context
The research focuses on transformations between $\mathbb{Q}$-factorial Fano threefolds, denoted as $g:Y \rightarrow X$, where $g$ represents a divisorial contraction. Both $Y$ and $X$ are characterized by ordinary terminal singularities. A key element of this contraction is the exceptional divisor, $E$. The investigation is conditioned on the nature of the center of contraction, which is assumed to be one of the following:
- A smooth point.
- A terminal quotient point.
- A point of type cA/n.
- A smooth curve with singularities of type cA or cA/n.
Approach
The approach involves analyzing and proving a specific regularized period identity. This identity relates the regularized quantum periods of two different geometric configurations. Specifically, the study aims to demonstrate that:
$$ \lim_{r\to+\infty}\hat{G}_{Y,rE}(t)=\hat{G}_X(t) $$Here, $\hat{G}_{Y,rE}(t)$ represents the regularized quantum period associated with the pair $(Y,rE)$, and $\hat{G}_X(t)$ denotes the regularized quantum period of $X$. The limit as $r$ approaches positive infinity is central to this identity.
Findings
The primary finding is the proof of the regularized period identity:
$$ \lim_{r\to+\infty}\hat{G}_{Y,rE}(t)=\hat{G}_X(t) $$This identity holds when the center of the divisorial contraction $g:Y \rightarrow X$ meets the specified conditions: being 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.
Why This Matters
The established regularized period identity provides a mirror approach. This approach is instrumental for the computation of Sarkisov links and higher syzygies, specifically within central models of dimension 3.