[Paper Review] Boundedness of polarized log Calabi-Yau fibrations

Junpeng Jiao|arXiv (Cornell University)|Feb 15, 2022

Geometry and complex manifolds4 citations

TL;DR

This paper establishes the boundedness of polarized Calabi-Yau fibrations modulo crepant birational equivalence under natural geometric constraints: when the base is bounded and the general fibers form a bounded family of log Calabi-Yau pairs. The key result proves that good minimal models with intermediate Kodaira dimension and such fibrations are bounded modulo crepant birational equivalence, extending boundedness results to Calabi-Yau fibrations beyond Fano or canonical types.

ABSTRACT

In this paper, we investigate the boundedness of log pairs with log Calabi--Yau fibration structures. We prove that total spaces of log Calabi--Yau fibrations are bounded modulo crepant birational equivalence when the Iitaka volumes of log canonical divisors are bounded and general fibers are in a bounded family of polarized log Calabi--Yau pairs.

Motivation & Objective

• To establish boundedness of good minimal models with intermediate Kodaira dimension via their Iitaka fibrations.
• To investigate the birational boundedness of log Calabi-Yau fibrations where general fibers are polarized Calabi-Yau pairs.
• To extend boundedness results from Fano and canonical types to Calabi-Yau fibrations in the minimal model program.
• To provide a framework for boundedness under Iitaka volume and volume constraints on fibers.

Proposed method

• Introduces the Iitaka volume as a measure of positivity for $Χ$-divisors, generalizing the standard volume for linear systems.
• Defines the family $\mathcal{G}_{klt}(d,\mathcal{I},v,u)$ of klt pairs with bounded Iitaka volume and fiber volume, and proves boundedness modulo crepant birational equivalence.
• Applies the MMP and relative MMP techniques to reduce the problem to lower-dimensional cases, using the existence of good minimal models and relative good minimal models.
• Employs the theory of generalized log Calabi-Yau fibrations and moduli b-divisors to control singularities and ensure boundedness under flops.
• Uses boundedness of moduli families and volume bounds to control the geometry of fibers and total spaces.
• Applies results from [HX13] on good minimal models and crepant birational equivalence to conclude boundedness up to flops.

Experimental results

Research questions

• RQ1 Are good minimal models with intermediate Kodaira dimension and Calabi-Yau fibrations bounded modulo crepant birational equivalence?
• RQ2 Under what conditions is the family of log Calabi-Yau fibrations bounded when the base and fiber volumes are fixed?
• RQ3 Can boundedness be established for Calabi-Yau fibrations with a rational section or multi-section, even when fibers are not Fano?
• RQ4 How does the Iitaka volume control the boundedness of fibrations in the minimal model program?

Key findings

• The family $\mathcal{G}_{klt}(d,\mathcal{I},v,u)$ of $d$-dimensional klt pairs with coefficients in a DCC set $\mathcal{I}$, Iitaka volume $v$, and fiber volume $u$ is bounded modulo crepant birational equivalence.
• Smooth projective Calabi-Yau varieties of dimension $d$ with an elliptic fibration and a rational section are bounded modulo flop when the fiber volume is fixed.
• The boundedness result extends to cases where the general fiber is of Fano type or admits a rational multi-section of bounded degree.
• The existence of a divisor $A$ with fixed volume on general fibers ensures boundedness modulo crepant birational equivalence, even without assuming $-K_{X_g}$ ampleness.
• The proof relies on the boundedness of moduli families and the termination of the MMP for generalized pairs with $K_{Y}+\Delta+\mathbf{M}\sim_{\mathbb{Q}}0$.
• The result implies that Calabi-Yau fibrations with bounded base and bounded fiber geometry are birationally bounded, generalizing prior results on elliptic and Fano-type fibrations.