An extension of BCOV invariant

TL;DR

This paper extends the BCOV invariant to pairs of a compact Kähler manifold and a divisor, demonstrating its behavior under specific conditions and confirming a conjecture for Atiyah flops.

Contribution

The paper introduces an extension of the BCOV invariant to pairs (X,Y) with new properties and verifies the birational invariance conjecture for Atiyah flops.

Findings

Extended BCOV invariant for pairs (X,Y)

Equivalence to Yoshikawa's invariant for specific cases

Birational invariance confirmed for Atiyah flops

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 $Y\in\big|K_X^m\big|$ 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 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.