Degenerations of Calabi-Yau threefolds and BCOV invariants

TL;DR

This paper investigates the asymptotic behavior of BCOV invariants in degenerating Calabi-Yau threefolds, establishing the rationality of divergence coefficients and linking them to geometric data through semi-stable reduction.

Contribution

It provides the first analysis of BCOV invariant asymptotics for degenerating Calabi-Yau threefolds, including a proof of rationality and geometric interpretation of divergence coefficients.

Findings

Rationality of the logarithmic divergence coefficient

Geometric expression of the divergence coefficient

Application of semi-stable reduction techniques

Abstract

In their papers published in 1993 and 1994, by expressing certain physical quantity in two distinct ways, Bershadsky-Cecotti-Ooguri-Vafa discovered a remarkable equivalence between Ray-Singer analytic torsion and elliptic instanton numbers for Calabi-Yau threefolds. After their discovery, in a paper published in 2008, a holomorphic torsion invariant for Calabi-Yau threefolds corresponding to the physical quantity was constructed and is called BCOV invariant. In this article, we study the asymptotic behavior of BCOV invariants for algebraic one-parameter degenerations of Calabi-Yau threefolds. We prove the rationality of the coefficient of logarithmic divergence and give its geometric expression by using a semi-stable reduction of the given family.