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.

MSC:

 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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