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 Kähler manifold and $Y\in \big |K_X^m\big |$ 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 blowup. It was conjectured that birational Calabi–Yau three-folds have the same BCOV invariant. As an application of our extended BCOV invariant, we show that this conjecture holds for Atiyah flops.
Manin's conjecture on a nonsingular quartic del Pezzo surface