Kähler Forms for Families of Calabi–Yau Manifolds

Abstract

Kähler–Einstein metrics for polarized families $(f:\mathcal{X}\to S,{\lambda }_{\mathcal{X}/S})$ of Calabi–Yau manifolds define a natural hermitian metric on the relative canonical bundle $K_{\mathcal{X}/S}$. The computation of the curvature form being equal to the pullback of the Weil–Petersson form up to a numerical constant is used for the construction of a Kähler–Einstein form on $\mathcal{X}$, whose restriction to the fibers is Ricci flat.