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

Let X X be a Kähler manifold which is fibered over complex Y"> Y encoding="application/x-tex">Y such that every fiber Calabi-Yau manifold. alttext="omega"> ? encoding="application/x-tex">\omega fixed form on . By Yau’s theorem, there exists unique Ricci-flat alttext="omega Subscript upper K E comma y"> K E , y encoding="application/x-tex">\omega _{KE,y} each X encoding="application/x-tex">X_y for alttext="y element-of ? encoding="application/x-tex">y\in Y cohomologous to vertical-bar Sub y Baseline"> | \vert _{X_y} This family of forms induces smooth alttext="left-parenthesis 1 right-parenthesis"> stretchy="false">( 1 stretchy="false">) encoding="application/x-tex">(1,1) -form alttext="rho"> ?<!-- ? encoding="application/x-tex">\rho under normalization condition. In this paper, we prove the direct image alttext="rho Superscript n plus 1"> n + encoding="application/x-tex">\rho ^{n+1} positive base We also discuss several byproducts including local triviality families manifolds.