Families of varieties of general type over compact bases

Abstract

Let f : X → Y be a smooth family of canonically polarized complex varieties over a smooth base. Generalizing the classical Shafarevich hyperbolicity conjecture, Viehweg conjectured that Y is necessarily of log general type if the family has maximal variation. A somewhat stronger and more precise version of Viehweg's conjecture was shown by the authors in [S. Kebekus, S.J. Kovács, Families of canonically polarized varieties over surfaces, preprint math.AG/0511378; Invent. Math. (2008), doi: 10.1007/s00222-008-0128-8; S. Kebekus, S.J. Kovács, The structure of surfaces mapping to the moduli stack of canonically polarized varieties, arXiv: 0707.2054v1 [math.AG], 2007] in the case where Y is a quasi-projective surface.Assuming that the minimal model program holds, this very short paper proves the same result for projective base manifolds Y of arbitrary dimension.

Brody hyperbolicity of base spaces of certain families of varieties

“…Roughly speaking, Viehweg predicted that for families with maximal variation, a log smooth compactification (Y, D) of V is of log general type. The proof of the original statement of the conjecture, in the canonically polarized case, was established in important special cases in [VZ02], [KK08], [KK08b], [KK10], [Pat12], and was recently completed by Campana and Pȃun [CP15, Thm. 8.1]; for a more detailed overview of this body of work and for further references, please see [PS17, §1.2].…”

Viehwegʼs hyperbolicity conjecture is true over compact bases

“…We also show that the conjecture holds when the compactification of U is not uniruled. The paper, at least in spirit, is the continuation of the very short paper [KK08]. That paper proves Viehweg's hyperbolicity conjecture over compact bases assuming the full Minimal Model Program and the Abundance conjecture.…”

Positive sheaves of differentials coming from coarse moduli spaces

“…The "Viehweg-Zuo" sheaf A was crucial in the study of hyperbolicity properties of manifolds that appear as bases of families of maximal variation and has been used to show that any minimal model program of the pair (Y, D) factors the moduli map, [KK08a,KK08b,KK08c], see also the survey [KS06]. In spite of its importance, little is known about further properties of the sheaf A .…”