| 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 Kahler manifold and Y is an element of vertical bar K-X(m)vertical bar with m is an element of Z\{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. |
