A Remark on Fujino's work on the canonical bundle formula via period maps

TL;DR

This paper extends Fujino's results on the semi-ampleness of the moduli part in the canonical bundle formula to primitive symplectic varieties with mild singularities, connecting it to the structure of K-trivial varieties.

Contribution

It generalizes Fujino's semi-ampleness result to a broader class of varieties and links the problem to the structure theory of K-trivial varieties.

Findings

Semi-ampleness established for primitive symplectic varieties with mild singularities.

Reduction of semi-ampleness questions to Calabi-Yau conditions for K-trivial varieties.

Answers a question posed by Fujino regarding the general fibers.

Abstract

Fujino gave a proof in [Fuj03] for the semi-ampleness of the moduli part in the canonical bundle formula in the case when the general fibers are K3 surfaces or Abelian varieties. We show a similar statement when the general fibers are primitive symplectic varieties with mild singularities. This answers a question of Fujino raised in the same article. Moreover, using the structure theory of varieties with trivial first Chern class, we reduce the question of semi-ampleness in the case of families of K-trivial varieties to a question when the general fibers satisfy a slightly weaker Calabi-Yau condition.