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.