Algebraic Fiber Spaces and Curvature of Higher Direct Images

Abstract

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.

Curvature of higher direct images

“…We consider the case where the hermitian bundle (L, h) is relatively positive, which means that ω Xs := ω X | Xs are Kähler forms on the n-dimensional fibers X s . Then one has the notion of the horizontal lift v s of a tangent vector ∂ s on the base S (see Section 4.1 for (1) The same curvature formula was also proved in [4] using a different method of computation.…”

On the Kodaira dimension of base spaces of families of manifolds

“…Viehweg's conjecture was finally solved in complete generality by the fundamental work of Campana and Pȃun [CP15] and more recently by Popa and Schnell [PS17]. For the more analytic counterparts of these results please see [VZ03], [Sch12], [TY15], [BPW17], [TY16], [PTW18] and [Den18].…”

Hyperbolicity for log smooth families with maximal variation

“…Using it, the author proved Brody hyperbolicity of moduli stacks of polarized monifolds with semi-ample canonical bundles. To and Yeung [TY15] proved that the base spaces of effective parametrized (a more restrictive condition replacing maximal variation) families of canonically polarized manifolds are Kobayashi hyperbolic, from a differential geometric point of view; see also Schumacher [Sch17] and Berndtsson, Pȃun and Wang [BPW17]. Deng [Den18a] generalizes the Kobayashi hyperbolicity result to smooth families of minimal varieties of general type, where the author combined the Hodge theoretical method in [VZ03] and its refinement in [PTW18], together with differential geometric methods.…”