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Given an effectively parameterized family $f:X\to S$ of canonically polarizedmanifolds, the K\"ahler-Einstein metrics on the fibers induce a hermitianmetric on the relative canonical bundle $K_{X/S}$. We use a global ellipticequation to show that this metric is strictly positive everywhere and giveestimates.  The direct images $R^{n-p}f_*\Omega^p_{X/S}(K_{X/S}^{\otimes m})$, $m > 0$,carry induced natural hermitian metrics. We prove an explicit formula for thecurvature tensor of these direct images with estimates, which impliesNakano-positivity for $p=n$.  The Kodaira-Spencer map induces morphisms $S^p\mathcal T_S \to R^pf_*\Lambda^p T_{X/S}$. For a suitable value of $p$ (and $\dim S=1$) a naturalhermitian metric on $T_S$ of negative curvature is induced. A differentialgeometric proof for hyperbolicity properties of the moduli space follows.

Curvature of higher direct images and applications (Curvature of $R^{n-p}f_*\Omega^p_{X/S}(K_{X/S}^{\otimes m})$ and applications)