Symmetry and Variation of Hodge Structures

Details Symmetry and Variation of Hodge Structures Symmetry and Variation of Hodge Structures

2003-10-10

arxiv.org/abs/math/0310153v2

Mathematics

Algebraic Geometry

38 pages, to appear in Asian J. Math., Volume in honour of Y.T. Siu's 60-th birthday. Revision, we correct the main theorem re

Scientific paper

The main problem addressed in the paper is the Torelli problem for n-dimensional varieties of general type, more specifically for varieties with ample canonical bundle. It asks under which geometrical condition for a variety the period map for the Hodge structure of weight n is a local embedding. We define a line bundle to be almost very ample iff the associated linear system is base point free and yields an injective morphism. We define instead a line bundle to be quasi very ample if it yields a birational morphism which is a local embedding on the complement of a finite set. Our main result is the existence of infinitely many families of surfaces of general type, with quasi very ample canonical bundle, each yielding an irreducible connected component of the moduli space, such that the period map has everywhere positive dimensional fibres. These surfaces are surfaces isogenous to a product, i.e., quotients of a product of curves by the free action of a finite group G. In the paper we also give some sufficient conditions in order that global double Torelli holds for these surfaces, i.e., the isomorphism type of the surface is reconstructed from the fundamental group plus the Hodge structure on the cohomology algebra. We do this via some useful lemmas on the action of an abelian group on the cohomology of an algebraic curve. We also establish a birational description of the moduli space of curves of genus 3 with a non trivial 3-torsion divisor.

Affiliated with

Also associated with

No associations

Rating

Symmetry and Variation of Hodge Structures does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Symmetry and Variation of Hodge Structures, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Symmetry and Variation of Hodge Structures will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-217070