Picard hyperbolicity of manifolds admitting nilpotent harmonic bundles

Abstract

BENOÎT CADOREL AND YA DENG A. For a quasi-compact Kähler manifold endowed with a nilpotent harmonic bundle whose Higgs field is injective at one point, we prove that is pseudo-algebraically hyperbolic, pseudo-Picard hyperbolic, and is of log general type. Moreover, we prove that there is a finite unramified cover ˜ of from a quasi-projective manifold ˜ so that any projective compactification of ˜ is pseudo-algebraically hyperbolic, pseudo-Picard hyperbolic and is of general type. As a byproduct, we establish some criterion of pseudo-Picard hyperbolicity and pseudo-algebraic hyperbolicity for quasi-compact Kähler manifolds.

Big Picard theorems and algebraic hyperbolicity for varieties admitting a variation of Hodge structures

“…In [CD21] Cadorel and the author generalized Theorems A and B and Corollary C in this paper to quasi-compact Kähler manifolds admitting nilpotent Higgs bundles. More recently, in [CDY22, Theorem 0.1] Cadorel, Yamanoi and the author proved that for any complex quasi-projective normal variety X, if there is a big representation : π 1 (X) → GL N (C) such that the Zariski closure of (π 1 (X)) is a semisimple algebraic group, then there is a proper Zariski closed subset Z X such that • any closed subvariety of X not contained in X is log general type; • X is Picard hyperbolic modulo Z.…”