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Fibers of Landau-Ginzburg models and rationality

Sukjoo Lee, Victor Przyjalkowski

TL;DR

The paper investigates how the rationality of a Fano threefold $X$ is encoded in its mirror Landau–Ginzburg models. The main result establishes that $X$ is rational if and only if the monodromy around every reducible fiber of a generic mirror LG model is unipotent, with a generalization to higher Picard ranks via parameterized LG models. The authors prove this by a meticulous, case-by-case analysis across deformation families, leveraging Calabi–Yau compactifications, isomonodromic deformations, and the Griffiths–Landman–Grothendieck criterion to connect fiber reducibility and monodromy. The work provides a concrete bridge between rationality questions in algebraic geometry and mirror-symmetric structures, offering a robust framework for detecting rationality through the topology of mirror families and a comprehensive data table for the 105 Fano threefold families.

Abstract

In this article, we study how the rationality of a Fano threefold is reflected in its standard mirror Landau-Ginzburg model and its deformations. The main result is that a Fano threefold is rational if and only if the monodromy around every reducible fiber of its generic mirror Landau-Ginzburg model is unipotent.

Fibers of Landau-Ginzburg models and rationality

TL;DR

The paper investigates how the rationality of a Fano threefold is encoded in its mirror Landau–Ginzburg models. The main result establishes that is rational if and only if the monodromy around every reducible fiber of a generic mirror LG model is unipotent, with a generalization to higher Picard ranks via parameterized LG models. The authors prove this by a meticulous, case-by-case analysis across deformation families, leveraging Calabi–Yau compactifications, isomonodromic deformations, and the Griffiths–Landman–Grothendieck criterion to connect fiber reducibility and monodromy. The work provides a concrete bridge between rationality questions in algebraic geometry and mirror-symmetric structures, offering a robust framework for detecting rationality through the topology of mirror families and a comprehensive data table for the 105 Fano threefold families.

Abstract

In this article, we study how the rationality of a Fano threefold is reflected in its standard mirror Landau-Ginzburg model and its deformations. The main result is that a Fano threefold is rational if and only if the monodromy around every reducible fiber of its generic mirror Landau-Ginzburg model is unipotent.
Paper Structure (8 sections, 18 theorems, 54 equations, 1 figure)

This paper contains 8 sections, 18 theorems, 54 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Mirror Landau--Ginzburg models for Fano 3-folds
  3. Toric Landau--Ginzburg models and Calabi--Yau compactifications
  4. Parameterized Landau--Ginzburg models
  5. Proof of Theorem \ref{['theorem:any rank rationality']}
  6. The case $X$ is rational
  7. The case $X$ is not rational
  8. The table

Key Result

Theorem 1.1

Let $X$ be a smooth Fano variety of dimension $n$. Let $w\colon Y\to {\mathbb C}$ be a Calabi--Yau compactification of a standard toric Landau--Ginzburg model for a Fano threefold $X$ or a Landau--Ginzburg model of Givental's type for a Fano complete intersection $X$. Then where $\rho_{\lambda}$ is the number of irreducible components of $w^{-1}(\lambda)$.

Figures (1)

  • Figure 2: Open Book Figure

Theorems & Definitions (37)

  • Theorem 1.1: Prz13, ChP18, PS15
  • Theorem 1.2: KP09
  • Example 1.3: see Example \ref{['eg:2-12']} for details
  • Theorem 1.4
  • Remark 1.5
  • Example 1.6
  • Example 1.7
  • Definition 2.1.1
  • Theorem 2.1.2: see Prz16
  • Remark 2.1.3: cf. ChP20 and KasPr22
  • ...and 27 more