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  1. ; ; (, Journal für die reine und angewandte Mathematik (Crelles Journal))
    null (Ed.)
    Abstract Given a Kähler fiber space p : X → Y {p:X\to Y} whose generic fiber is of general type, we prove that the fiberwise singular Kähler–Einstein metric inducesa semipositively curved metric on the relative canonical bundle K X / Y {K_{X/Y}} of p . We also propose a conjectural generalization of this result for relativetwisted Kähler–Einstein metrics. Then we show that our conjecture holds trueif the Lelong numbers of the twisting current are zero. Finally, we explain the relevance of our conjecture for the study of fiberwise Song–Tian metrics (which represent the analogue of KE metrics for fiber spaces whose generic fiber has positive but not necessarily maximal Kodaira dimension). 
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  2. ; (, Pure and Applied Mathematics Quarterly)
    null (Ed.)
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  3. ; ; ; ; (, Journal of Singularities)
    null (Ed.)
    Full Text Available 
  4. ; ; (, Journal of the Institute of Mathematics of Jussieu)
    null (Ed.)
    Let $$p:X\rightarrow Y$$ be an algebraic fiber space, and let $$L$$ be a line bundle on $$X$$ . In this article, we obtain a curvature formula for the higher direct images of $$\unicode[STIX]{x1D6FA}_{X/Y}^{i}\otimes L$$ restricted to a suitable Zariski open subset of $$X$$ . Our results are particularly meaningful if $$L$$ is semi-negatively curved on $$X$$ and strictly negative or trivial on smooth fibers of $$p$$ . Several applications are obtained, including a new proof of a result by Viehweg–Zuo in the context of a canonically polarized family of maximal variation and its version for Calabi–Yau families. The main feature of our approach is that the general curvature formulas we obtain allow us to bypass the use of ramified covers – and the complications that are induced by them. 
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  5. ; (, Journal of convex analysis)
    null (Ed.)
    Suppose $$f(x,y) + \frac{\kappa}{2} \|x\|^2 - \frac{\sigma}{2}\|y\|^2$$ is convex where $$\kappa\ge 0, \sigma>0$$, and the argmin function $$\gamma(x) = \{ \gamma : \inf_y f(x,y) = f(x,\gamma)\}$$ exists and is single valued. We will prove $$\gamma$$ is differentiable almost everywhere. As an application we deduce a minimum principle for certain semiconcave subsolutions. 
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