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On finiteness of fiber space structures of klt Calabi-Yau pairs in dimension 3

Fulin Xu

TL;DR

The paper addresses finiteness of fiber space structures for a fixed 3‑dimensional klt Calabi–Yau pair $(X,\Delta)$ by embedding the problem in the cone conjecture framework and boundedness theory. It develops a comprehensive machinery: index‑1 covers, polyhedral cone criteria, VHS/period maps, and boundedness results for curve fibrations, all culminating in an extraction theorem that enables product splittings over a finite base change. The main contributions are (i) proving the cone conjecture in several 3‑fold CY‑klt cases, (ii) establishing boundedness and marked‑curve fibrations for strict terminal $\mathbb{Q}$‑factorial CY 3‑folds, and (iii) deducing finiteness of all fiber space structures up to $Aut(X,\Delta)$, including handling the non‑strict case via quotient and covering arguments. Together, these results extend the cone conjecture toolkit to singular CY varieties in dimension 3 and provide a robust pathway to finiteness statements with potential implications for mirror symmetry and birational geometry.

Abstract

We prove that for a fixed klt Calabi-Yau pair $(X,Δ)$ of dimension $3$, the set of fiber space structures of $X$ is finite up to $Aut(X,Δ)$.

On finiteness of fiber space structures of klt Calabi-Yau pairs in dimension 3

TL;DR

The paper addresses finiteness of fiber space structures for a fixed 3‑dimensional klt Calabi–Yau pair by embedding the problem in the cone conjecture framework and boundedness theory. It develops a comprehensive machinery: index‑1 covers, polyhedral cone criteria, VHS/period maps, and boundedness results for curve fibrations, all culminating in an extraction theorem that enables product splittings over a finite base change. The main contributions are (i) proving the cone conjecture in several 3‑fold CY‑klt cases, (ii) establishing boundedness and marked‑curve fibrations for strict terminal ‑factorial CY 3‑folds, and (iii) deducing finiteness of all fiber space structures up to , including handling the non‑strict case via quotient and covering arguments. Together, these results extend the cone conjecture toolkit to singular CY varieties in dimension 3 and provide a robust pathway to finiteness statements with potential implications for mirror symmetry and birational geometry.

Abstract

We prove that for a fixed klt Calabi-Yau pair of dimension , the set of fiber space structures of is finite up to .
Paper Structure (24 sections, 41 theorems, 8 equations)

This paper contains 24 sections, 41 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Types of divisors
  4. Various cones
  5. Pairs
  6. Index 1 covers
  7. Polyhedral type cones
  8. A variant of the cone conjecture
  9. Variation of Hodge structures
  10. Period domains and Period maps
  11. Boundedness results for fibered Calabi-Yau pairs in dimension $3$
  12. Strict klt Calabi-Yau pairs
  13. Boundedness of bases
  14. Boundedness of rational multi-sections
  15. Boundedness of marked curve fibered Calabi-Yau pairs
  16. ...and 9 more sections

Key Result

Theorem 1.2

Let $(X,\Delta)$ be a projective $\mathbb{Q}$-factorial klt Calabi-Yau pair in dimension $3$, then (1) If $\kappa(-K_X)>0$, then the cone conjecture holds for $(X,\Delta)$. (2) If $\kappa(-K_X)=0$, we may run $K_X+(1+\varepsilon)\Delta$-MMP for a small positive rational number $\varepsilon$ and term

Theorems & Definitions (103)

  • Conjecture 1.1
  • Remark
  • Theorem 1.2
  • Remark
  • Corollary 1.3
  • Lemma 2.1
  • Proof
  • Lemma 2.2
  • Proof
  • Proposition 2.3
  • ...and 93 more