KASS seminar

Abstract: The BCOV invariant is a real valued invariant of Calabi-Yau manifolds (CY's), constructed by Fang-Lu-Yoshikawa in dimension 3 and Eriksson-F.-Mourougane in general. It builds on a combination of holomorphic analytic torsions of sheaves of holomorphic differentials, first studied by string theorists Bershadsky-Cecotti-Ooguri-Vafa. The BCOV invariant induces a smooth function on the moduli space of polarized CY's, and it satisfies a differential equation relating it to the Weil-Petersson form and the curvatures of the Griffiths bundles. In previous works with D. Eriksson and C. Mourougane, we studied the singularities of the BCOV invariant for one-parameter degenerations of CY’s. This can be interpreted as the boundary conditions of the differential equation. Motivated by mirror symmetry, we asked ourselves about the positivity properties of these singularities, and whether the BCOV invariant provides a psh exhaustion function on the moduli space. In this talk I will report on work in progress with D. Eriksson on these questions.