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

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

2010-02-25

arxiv.org/abs/1002.4858v2

Mathematics

Complex Variables

Minor typos and such corrected

Scientific paper

Given an effectively parameterized family $f:X\to S$ of canonically polarized manifolds, the K\"ahler-Einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle $K_{X/S}$. We use a global elliptic equation to show that this metric is strictly positive everywhere and give estimates. 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 the curvature tensor of these direct images with estimates, which implies Nakano-positivity for $p=n$. The Kodaira-Spencer map induces morphisms $S^p\mathcal T_S \to R^p f_*\Lambda^p T_{X/S}$. For a suitable value of $p$ (and $\dim S=1$) a natural hermitian metric on $T_S$ of negative curvature is induced. A differential geometric proof for hyperbolicity properties of the moduli space follows.

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