Families of Canonically Polarized Varieties
نویسنده
چکیده
Shafarevich's hyperbolicity conjecture asserts that a family of curves over a quasi-projective 1-dimensional base is isotrivial unless the logarithmic Kodaira dimension of the base is positive. More generally it has been conjectured by Viehweg that the base of a smooth family of canonically polarized varieties is of log general type if the family is of maximal variation. In this paper, we relate the variation of a family to the logarithmic Kodaira dimension of the base and give an affirmative answer to Viehweg's conjecture for families parametrized by surfaces.
منابع مشابه
Nonexistence of Asymptotic Git Compactification
We provide examples of families of (log) smooth canonically polarized varieties, including smooth weighted pointed curves and smooth hypersurfaces in P with large degree such that the Chow semistable limits under distinct pluricanonical embeddings do not stabilize.
متن کاملPositivity of Relative Canonical Bundles for Families of Canonically Polarized Manifolds
Given an effectively parameterized family of canonically polarized manifolds the Kähler-Einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle. We use a global elliptic equation to show that this metric is strictly positive. For degenerating families we obtain a singular hermitian metric. Applications concern the curvature of the classical and generalized Weil...
متن کاملPositivity of Relative Canonical Bundles of Families of Canonically Polarized Manifolds
The Kähler-Einstein metrics on the fibers of an effectively parameterized family of canonically polarized manifolds induce a hermitian metric on the relative canonical bundle. We use a global elliptic equation to show that this metric is strictly positive. Applications concern the curvature of the classical and generalized Weil-Petersson metrics and hyperbolicity of moduli spaces.
متن کاملRigidity for Families of Polarized Calabi-yau Varieties
In this paper, we study the analogue of the Shafarevich conjecture for polarized Calabi-Yau varieties. We use variations of Hodge structures and Higgs bundles to establish a criterion for the rigidity of families. We then apply the criterion to obtain that some important and typical families of Calabi-Yau varieties are rigid, for examples., Lefschetz pencils of Calabi-Yau varieties, strongly de...
متن کاملThe Topology of Conjugate Varieties
Serre [Se 64] and Abelson [Ab 74] have produced examples of conjugate algebraic varieties which are not homeomorphic. We show that if the field of definition of a polarized projective variety coincides with its field of moduli then all of its conjugates have the same topological type. This immediately extends the class of varietie s known to possess invariant topological type to all canonically...
متن کامل