Finsler metrics and Kobayashi hyperbolicity of the moduli spaces of canonically polarized manifolds. (English) Zbl 1319.32024

The main result of the paper is the following. Let \(\pi: \mathcal{X}\rightarrow S\) be an effectively parametrized holomorphic family of compact canonically polarized complex manifolds over a complex manifold \(S\). Then \(S\) admits a \(C^\infty\) \(\mathrm{Aut}(\pi)\)-invariant Finsler metric whose holomorphic sectional curvature is bounded above by a negative constant. In particular, \(S\) is Kobayashi hyperbolic.


32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
32Q20 Kähler-Einstein manifolds
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
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