[Paper Review] On the moduli b-divisors of lc-trivial fibrations

Osamu Fujino, Yoshinori Gongyo|arXiv (Cornell University)|Oct 18, 2012

Algebraic Geometry and Number Theory9 references4 citations

TL;DR

This paper establishes that the moduli b-divisor of an lc-trivial fibration is b-nef and abundant by reducing the problem to a klt-trivial fibration via a generically finite base change and applying the semi-stable minimal model program. The key result confirms the b-nefness and abundance of the moduli part, supporting the broader conjecture on b-semi-ampleness in the minimal model program.

ABSTRACT

Roughly speaking, by using the semi-stable minimal model program, we prove that the moduli part of an lc-trivial fibration coincides with that of a klt-trivial fibration induced by adjunction after taking a suitable generically finite cover. As an application, we obtain that the moduli part of an lc-trivial fibration is b-nef and abundant by Ambro's result on klt-trivial fibrations.

Motivation & Objective

• To prove that the moduli b-divisor of an lc-trivial fibration is b-nef and abundant.
• To reduce the study of lc-trivial fibrations to klt-trivial fibrations via a generically finite base change.
• To establish a link between the moduli part of lc-trivial fibrations and that of klt-trivial fibrations induced by adjunction.
• To provide a new proof of b-nefness and abundance without relying on period maps or Torelli theorems.
• To partially address open problems on b-semi-ampleness in the context of canonical bundle formulae.

Proposed method

• Use the semi-stable minimal model program to construct a generically finite base change that transforms an lc-trivial fibration into a klt-trivial fibration.
• Apply adjunction to obtain a klt-trivial fibration from a log canonical center of minimal dimension over the base.
• Construct a common resolution between the original and transformed fibrations to compare their moduli b-divisors.
• Show that the moduli b-divisor of the original fibration coincides with that of the klt-trivial fibration after base change.
• Leverage Ambro’s result on klt-trivial fibrations to deduce b-nefness and abundance for the original moduli b-divisor.
• Use the fact that the moduli b-divisor pulls back via a generically finite morphism to preserve its b-nef and abundant properties.

Experimental results

Research questions

• RQ1 Does the moduli b-divisor of an lc-trivial fibration remain b-nef and abundant?
• RQ2 Can the moduli b-divisor of an lc-trivial fibration be related to that of a klt-trivial fibration via base change?
• RQ3 Is it possible to prove b-nefness and abundance without using period maps or infinitesimal Torelli theorems?
• RQ4 To what extent does the semi-stable minimal model program allow reduction from lc to klt fibrations in the context of moduli b-divisors?
• RQ5 What conditions ensure the b-semi-ampleness of the moduli b-divisor in lc-trivial fibrations?

Key findings

• The moduli b-divisor of an lc-trivial fibration is b-nef and abundant, as established via reduction to klt-trivial fibrations.
• After a suitable generically finite base change, the moduli b-divisor of an lc-trivial fibration coincides with that of the induced klt-trivial fibration.
• The proof avoids reliance on the mixed period map or infinitesimal mixed Torelli theorem, relying instead on the semi-stable minimal model program.
• The moduli b-divisor remains b-nef and abundant after pullback via a generically finite morphism, preserving its geometric properties.
• The result supports the conjecture that the moduli b-divisor is b-semi-ample in many cases, particularly when the dimension of log canonical centers is small.
• When the dimension of log canonical centers dominant over the base is zero, the moduli b-divisor is numerically trivial.