BCOV invariant for Calabi-Yau pairs

Abstract

Bershadsky, Cecotti, Ooguri and Vafa constructed a real valued invariant for Calabi-Yau manifolds, which is now called BCOV invariant.We consider a pair (X, Y ), where X is a Kähler manifold and Y ⊆ X is a pluricanonical divisor. We will call such (X, Y ) a Calabi-Yau pair. In this paper, we extend the BCOV invariant to Calabi-Yau pairs. We show that the extended BCOV invariant for rigid del Pezzo surfaces is equivalent to Yoshikawa's equivariant BCOV invariant. We also study the behavior of the extended BCOV invariant under blowing up.We expect that these considerations may eventually lead to a positive answer to Yoshikawa's conjecture that the BCOV invariant for Calabi-Yau threefold is a birational invariant.