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 K\"ahler manifold. This is called a \emph{(smooth) family of K\"ahler manifolds}. If every fiber admits a K\"ahler-Einstein metric, which is a  K\"ahler metric whose Ricci curvature tensor is proportional to the K\"ahler metric, then this family of K\"ahler-Einstein metrics induces the \emph{fiberwise K\"ahler-Einstein metric} (by a suitable normalization condition). Schumacher have shown that the fiberwise K\"ahler-Einstein metric on a family of canonically polarized compact K\"ahler manifolds is semi-positive. Moreover, we also proved that if the family is not locally trivial, then the fiberwise K\"ahler-Einstein metric is strictly positive. As a consequence he obtained several applications to the geometry of moduli spaces.
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In the first talk, we discuss the positivity of the fiberwise K\"ahler-Einstein metric on a family of bounded strongly pseudoconvex domains. And in the second talk, we discuss the semi-positivity of the fiberwise Ricci-flat metric on a family of Calabi-Yau manifolds.

Attention: Séance spéciale du séminaire.