Brody hyperbolicity of base spaces of certain families of varieties

Abstract

We prove that quasi-projective base spaces of smooth families of minimal varieties of general type with maximal variation do not admit Zariski dense entire curves. We deduce the fact that moduli stacks of polarized varieties of this sort are Brody hyperbolic, answering a special case of a question of Viehweg and Zuo. For two-dimensional bases, we show analogous results in the more general case of families of varieties admitting a good minimal model.

Algebraic fiber spaces and curvature of higher direct images

“…We note that Theorem 1.2 (and a few other results we prove in this article) have also been announced by S.-K. Yeung and W. To in their joint project [46]. For questions related to Brody hyperbolicity we refer to the very recent preprint [35] by M. Popa, B. Taji and L. Wu (and the references therein).…”

Higgs Bundles in Geometry and Arithmetic

“…The negative holomorphic sectional curvature derived as above plays the same role as the negative holomorphic sectional curvature associated to horizontal period maps in Hodge theory. It implies Brody and Kobayashi hyperbolities of U in [VZ03], [PTW19] and [Den18a]. Also, the big Picard theorem was recently proven in [DLSZ19].…”

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].…”