Semi-positivity of logarithmic cotangent bundle and Shafarevich-Viehweg's conjecture, after Campana, Paun, Taji...

Abstract

Proven by A. Parshin and S. Arakelov in the early 70's, Shafaverich hyperbolicity conjecture states that a family of curves of genus $g\ge2$ parametrized by a non hyperbolic curve (\emph{i.e.} isomorphic to $\mathbb{P}^1$, $\mathbb{C}$, $\mathbb{C}^*$ or an elliptic curve) has to be isotrivial : the moduli of smooth fibres are constant. In higher dimensions, Viehweg's works on moduli of canonically polarized manifolds led him to generalize this statement in the following way: if a family of canonically polarized manifolds (parametrized by a quasi-projective base) has maximal variation, the base is then of log-general type. It can be thought as an algebraic hyperbolicity property which is expected to hold for the moduli space.Adapting results due to Y. Miyoaka on generic semi-positivity of cotangent bundle to the framework of pairs, F. Campana and M. P\u{a}un recently obtained a positive answer to Viehweg's conjecture. We will also take the opportunity of this talk to present the classification of orbifolds as developed in Campana's works. This setting is moreover the right one to state the optimal version of Viehweg's conjecture proven by B. Taji