Remarks on BCOV invariants and degenerations of Calabi-Yau manifolds

TL;DR

This paper relates BCOV invariants to singularities in Calabi-Yau degenerations and establishes inequalities and bounds for Calabi-Yau families over Riemann surfaces using complex geometric methods.

Contribution

It demonstrates that total singularities can be expressed via BCOV invariants and proves Arakelov type inequalities and Euler number bounds for Calabi-Yau families.

Findings

Total singularities expressed by BCOV invariants

Established Arakelov inequalities for Calabi-Yau families

Derived Euler number bounds for Calabi-Yau degenerations

Abstract

For a one parameter family of Calabi-Yau threefolds, Green, Griffiths and Kerr have expressed the total singularities in terms of the degrees of Hodge bundles and Euler number of the general fiber. In this paper, we show that the total singularities can be expressed by the sum of asymptotic values of BCOV invariants, studied by Fang, Lu and Yoshikawa. On the other hand, by using Yau's Schwarz lemma, we prove Arakelov type inequalities and Euler number bound for Calabi-Yau family over a compact Riemann surface.