2003/10/10 by Ingrid Bauer, Ingrid C. Bauer, Fabrizio Catanese +3
Mathematics · #14C30 #14C34 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Geometry and complex manifolds #math.AG #math.CV #msc:14C30 #msc:14C34
paper · pdf · doi:10.48550/arxiv.math/0310153
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 replacing almost very ample by quasi very ample, which is in one way better and in one way worse
openalex publication_date 2003/10/10 · arxiv created 2004/02/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.