2009/05/31 by Kelly Jabbusch, Stefan Kebekus
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Base (topology) #Boundary (topology) #Combinatorics #Conjecture #Filtration (mathematics) #Geometry #Geometry and complex manifolds #Manifold (fluid mechanics) #Mathematical analysis #Mathematics #Pure mathematics #Surface (topology) #math.AG #math.CV #math.DG #msc:14D06 #msc:14D20
paper · pdf · doi:10.1007/s00209-010-0758-6
published as Mathematische Zeitschrift, Volume 269, Issue 3 (2011), Page 847-878 · Final version, to appear in Math. Zeitschrift
arxiv created 2010/07/14 · openalex publication_date 2010/08/05 · arxiv updated 2011/11/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Consider a smooth, projective family of canonically polarized varieties over a smooth, quasi-projective base manifold Y, all defined over the complex numbers. It has been conjectured that the family is necessarily isotrivial if Y is special in the sense of Campana. We prove the conjecture when Y is a surface or threefold. The proof uses sheaves of symmetric differentials associated to fractional boundary divisors on log canonical spaces, as introduced by Campana in his theory of Orbifoldes Geometriques. We discuss a weak variant of the Harder-Narasimhan Filtration and prove a version of the Bogomolov-Sommese Vanishing Theorem that take the additional fractional positivity along the boundary into account. A brief, but self-contained introduction to Campana's theory is included for the reader's convenience.