The Euler Characteristic Formula for Logarithmic Minimal Degenerations of Surfaces with Kodaira Dimension Zero and its application to Calabi-Yau Threefolds with a pencil

TL;DR

This paper establishes an Euler characteristic formula for certain degenerations of surfaces with Kodaira dimension zero and applies it to bound the number of singular fibers in abelian fibred Calabi-Yau threefolds, advancing understanding of their degenerations.

Contribution

It proves a new Euler characteristic formula for logarithmic minimal degenerations of surfaces with Kodaira dimension zero and applies it to analyze singularities and bound singular fibers in Calabi-Yau threefolds.

Findings

Euler characteristic formula for degenerations proved

Singularities of degenerations characterized in specific cases

Upper bound established for singular fibers in Calabi-Yau threefolds

Abstract

In this paper, the Euler characteristic formula for projective logarithmic minimal degenerations of surfaces with Kodaira dimension zero over a 1-dimensional complex disk is proved under a reasonable assumption and as its application, the singularity of logarithmic minimal degenerations are determined in the abelian or hyperelliptic case. By globalizing this local analysis of singular fibres via generalized canonical bundle formulae due to Fujino-Mori, we bound the number of singular fibres of abelian fibred Calabi-Yau threefolds from above,which was previously done by Oguiso in the potentially good reduction case.