0000000000563765

AUTHOR

Eckart Viehweg

showing 3 related works from this author

Special Families of Curves, of Abelian Varieties, and of Certain Minimal Manifolds over Curves

2006

This survey article discusses some results on the structure of families f:V-->U of n-dimensional manifolds over quasi-projective curves U, with semistable reduction over a compactification Y of U. We improve the Arakelov inequality for the direct images of powers of the dualizing sheaf. For families of Abelian varieties we recall the characterization of Shimura curves by Arakelov equalities. For families of curves we recall the characterization of Teichmueller curves in terms of the existence of certain sub variation of Hodge structures. We sketch the proof that the moduli scheme of curves of genus g>1 can not contain compact Shimura curves, and that it only contains a non-compact Shimura c…

AlgebraAbelian varietyShimura varietyPure mathematicsMathematics::Algebraic GeometryModuli schemeMathematics::Number TheorySheafCompactification (mathematics)Abelian groupHodge structureHiggs bundleMathematics
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Numerical bounds for semi-stable families of curves or of certain higher-dimensional manifolds

2005

Given an open subset U U of a projective curve Y Y and a smooth family f : V → U f:V\to U of curves, with semi-stable reduction over Y Y , we show that for a subvariation V \mathbb {V} of Hodge structures of R 1 f ∗ C V R^1f_*\mathbb {C}_V with rank ( V ) > 2 \textrm {rank} (\mathbb {V})>2 the Arakelov inequality must be strict. For families of n n -folds we prove a similar result under the assumption that the ( n , 0 ) (n,0) component of the Higgs bundle of V \mathbb {V} defines a birational map.

CombinatoricsProjective curveAlgebra and Number TheoryReduction (recursion theory)Hodge bundleComponent (group theory)Geometry and TopologyRank (differential topology)MathematicsHiggs bundleJournal of Algebraic Geometry
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Complex multiplication, Griffiths-Yukawa couplings, and rigidity for families of hypersurfaces

2003

Let M(d,n) be the moduli stack of hypersurfaces of degree d > n in the complex projective n-space, and let M(d,n;1) be the sub-stack, parameterizing hypersurfaces obtained as a d fold cyclic covering of the projective n-1 space, ramified over a hypersurface of degree d. Iterating this construction, one obtains M(d,n;r). We show that M(d,n;1) is rigid in M(d,n), although the Griffiths-Yukawa coupling degenerates for d<2n. On the other hand, for all d>n the sub-stack M(d,n;2) deforms. We calculate the exact length of the Griffiths-Yukawa coupling over M(d,n;r), and we construct a 4-dimensional family of quintic hypersurfaces, and a dense set of points in the base, where the fibres ha…

Algebra and Number TheoryDegree (graph theory)Mathematics - Complex Variables14D0514J3214D07Complex multiplicationYukawa potentialRigidity (psychology)14J70ModuliCombinatoricsAlgebraMathematics - Algebraic Geometry14J70; 14D05; 14D07; 14J32HypersurfaceMathematics::Algebraic GeometryMathematikFOS: MathematicsGeometry and TopologyComplex Variables (math.CV)Algebraic Geometry (math.AG)Stack (mathematics)Mathematics
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