Young-Jun Choi

Let $p:X\rightarrow Y$ be a surjective holomorphic submersion between complex manifolds such that every fiber $X_y:=p^{-1}(y)$ for $y\in Y$,  is a Calabi-Yau manifold, i.e., a compact K\"ahler manifold with trivial canonical line bundle. This is called a \emph{family of Calabi-Yau manifolds} or a \emph{Calabi-Yau fibration}. If $(X,\omega)$ is a K\"ahler manifold, then every fiber $X_y$ equpped has a unique Ricci-flat K\"ahler metric whose associated K\"ahler form belongs to the fixed K\"ahler class $[\omega\vert_{X_y}]$ by Calabi-Yau theorem. This family of Ricci-flat metrics induces the fiberwise Ricci-flat metric on a Calabi-Yau fibration.
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In this talk, we discuss the semi-positivity (on the total space $X$) of the fiberwise Ricci-flat metrics
on Calabi-Yau fibrations.