An extension of BCOV invariant

Abstract

Bershadsky, Cecotti, Ooguri and Vafa constructed a real valued invariant for Calabi-Yau manifolds, which is called the BCOV invariant. In this paper, we consider a pair $(X,Y)$, where $X$ is a compact Kaehler manifold and $Y\in \left|K_X^m\right|$ with $m\in \mathbb{Z}\backslash \{0,-1\}$. We extend the BCOV invariant to such pairs. If $m=-2$ and $X$ is a rigid del Pezzo surface, the extended BCOV invariant is equivalent to Yoshikawa's equivariant BCOV invariant. If $m=1$, the extended BCOV invariant is well-behaved under blow-up. It was conjectured that birational Calabi-Yau threefolds have the same BCOV invariant. As an application of our extended BCOV invariant, we show that this conjecture holds for Atiyah flops.

33 pages, this version combines the previous version (arXiv:1905.03964v1) and arXiv:1902.08062, accepted by IMRN