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

TL;DR

This paper investigates the curvature properties of higher direct image sheaves associated with families of canonically polarized manifolds, establishing positivity results and applications to moduli space hyperbolicity.

Contribution

It provides explicit curvature formulas and positivity results for direct images, and applies these to derive hyperbolicity properties of moduli spaces.

Findings

Proves strict positivity of the induced hermitian metric on the relative canonical bundle.

Establishes Nakano-positivity for certain direct image sheaves.

Derives hyperbolicity properties of moduli spaces using differential geometric methods.

Abstract

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.