[Paper Review] Log Calabi-Yau fibrations

Caucher Birkar|arXiv (Cornell University)|Nov 26, 2018

Algebraic Geometry and Number Theory9 references21 citations

TL;DR

This paper establishes boundedness results for log Calabi-Yau fibrations, particularly those of Fano type, by proving that such fibrations form bounded families under natural numerical and singularity conditions. It introduces a framework using generalized pairs, complements, and inductive techniques on Mori fiber spaces to show that coefficients, nef parts, and singularities are uniformly controlled, leading to boundedness of the total space and base under Fano-type conditions.

ABSTRACT

In this paper we study boundedness properties and singularities of log Calabi-Yau fibrations, particularly those admitting Fano type structures. A log Calabi-Yau fibration roughly consists of a pair $(X,B)$ with good singularities and a projective morphism $X o Z$ such that $K_X+B$ is numerically trivial over $Z$. This class includes many central ingredients of birational geometry such as Calabi-Yau and Fano varieties and also fibre spaces of such varieties, flipping and divisorial contractions, crepant models, germs of singularities, etc.

Motivation & Objective

• To establish boundedness of log Calabi-Yau fibrations of Fano type under numerical and singularity constraints.
• To investigate the behavior of singularities on the total space and base of such fibrations.
• To prove the existence of bounded complements and uniform lower bounds on lc thresholds for these fibrations.
• To extend boundedness results to towers of Mori fiber spaces and generalized log Calabi-Yau fibrations.
• To provide a foundation for moduli theory and inductive arguments in birational geometry by unifying key objects like Fano, Calabi-Yau, and crepant models.

Proposed method

• Uses the framework of generalized pairs with nef parts to control singularities and boundedness via inductive arguments on Mori fiber spaces.
• Applies boundedness of relative complements and rational approximation of boundary divisors to control coefficients and Cartier indices.
• Employs induction on the number of steps in a tower of fibrations, reducing to lower-dimensional cases via general fiber analysis.
• Applies the DCC property of discriminant b-divisors and boundedness of log pairs to control singularities across the fibration.
• Uses the fact that $K_X + B \sim_{\mathbb{Q}} 0$ and $-K_X$ big over $Z$ to ensure existence of uniform lower bounds on lc thresholds.
• Relies on the boundedness of Néron-Severi groups and very ample divisors to conclude log boundedness of the total space.

Experimental results

Research questions

• RQ1 Under what conditions do log Calabi-Yau fibrations of Fano type form bounded families?
• RQ2 How do singularities behave on the total space and base of such fibrations?
• RQ3 Do bounded (klt or lc) complements exist for log Calabi-Yau fibrations with Fano-type structures?
• RQ4 Can uniform lower bounds on lc thresholds be established for these fibrations?
• RQ5 What is the boundedness behavior of towers of Mori fiber spaces and generalized log Calabi-Yau fibrations?

Key findings

• The pairs $(X,B)$ in a $(d,r,\epsilon)$-Fano type fibration are log bounded when $K_X + B \sim_{\mathbb{R}} 0$ and coefficients of $B$ lie in a DCC set.
• There exists a DCC set $\Psi$, a natural number $p$, and a positive real number $\delta$ depending only on $d, l, \Phi, \epsilon$ such that coefficients of $B_i$ are in $\Psi$, $pM_i'$ is Cartier, and $(X_i, B_i + M_i)$ is generalised $\delta$-lc.
• The total space $X_{l-1}$ in a tower of fibrations is bounded when $K_{X_{l-1}} + B_{l-1} + \Delta_{l-1} + M_{l-1} \sim_{\mathbb{Q}} h^*(L + A)$ and $-(K_{X_{l-1}} + \Delta_{l-1})$ is ample over $X_l$.
• When $K_X + B \sim_{\mathbb{Q}} 0$, the generalized pair $(X_i, B_i + M_i)$ is generalised $\delta$-lc for a fixed $\delta > 0$, independent of $i$, due to boundedness of coefficients and Cartier index.
• The volume of $H$ is bounded from above, and $H^{d-1} \cdot B$ is bounded, implying log boundedness of $(X,B)$ under Fano-type conditions.
• The base $Z$ and the total space $X$ are bounded when the fibration is of Fano type and the general fiber is log bounded, via induction and boundedness of relative complements.