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On the Brody hyperbolicity of moduli spaces for canonically polarized manifolds. (English) Zbl 1042.14010

Given a polynomial \(h\), let \({\mathcal M}_h\) be the moduli stack of canonically polarized complex manifolds with Hilbert polynomial \(h\). In a previous work, the first author introduced a coarse quasi-projective moduli scheme \(M_h\) for \({\mathcal M}_h\) [see E. Viehweg, Quasi-projective moduli for polarized manifolds, Springer-Verlag, Berlin (1995; Zbl 0844.14004)]. In the paper under review, the authors prove the following.
Theorem. Assume that for some quasi-projective variety \(U\) there exists a family \(f:V\to U\in {\mathcal M}_h\) for which the induced morphism \(\varphi:U\to M_h\) is quasi-finite over its image. Then, \(U\) is Brody hyperbolic.
In general, it is a hard question to decide whether a space is Brody-hyperbolic or not.
We recall that a complex space \(U\) is said to be Brody hyperbolic if every holomorphic map from the complex plane to \(U\) is constant. (In the case where \(U\) is a compact manifold, this is equivalent to Kobayashi hyperbolicity).
An algebraic version of this theorem has been obtained by S. Kovács [J. Algebr. Geom. 9, 165–174 (2000; Zbl 0970.14008)].

MSC:

14J15 Moduli, classification: analytic theory; relations with modular forms
14J10 Families, moduli, classification: algebraic theory
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds

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