Positivity of direct images of fiberwise Ricci-flat metrics on Calabi-Yau fibrations

Abstract

Let X X be a Kähler manifold which is fibered over a complex manifold Y Y such that every fiber is a Calabi-Yau manifold. Let ω \omega be a fixed Kähler form on X X . By Yau’s theorem, there exists a unique Ricci-flat Kähler form ω K E , y \omega _{KE,y} on each fiber X y X_y for y ∈ Y y\in Y which is cohomologous to ω | X y \omega \vert _{X_y} . This family of Ricci-flat Kähler forms ω K E , y \omega _{KE,y} induces a smooth ( 1 , 1 ) (1,1) -form ρ \rho on X X under a normalization condition. In this paper, we prove that the direct image of ρ n + 1 \rho ^{n+1} is positive on the base Y Y . We also discuss several byproducts including the local triviality of families of Calabi-Yau manifolds.