2003/07/04 by Fabrizio Catanese, Catanese, Fabrizio · 2 citations
Mathematics · #14J15 #32G05 #32G08 #32G13 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14J15 #msc:32G05 #msc:32G08 #msc:32G13
paper · pdf · doi:10.48550/arxiv.math/0307067
29 pages, to appear in Annali di Mat. pura e appl., volume in memory of Fabio Bardelli
arxiv created 2003/07/04 · arxiv updated 2009/11/30
Main topic of the paper is the determination, for a compact complex manifold M, of the class of manifolds X which are deformation equivalent to it. If M is a complex torus, then also X is so. After describing the structure of principal holomorphic torus bundles over curves, a similar result (stability by deformations in the large) is obtained also for the latter class of manifolds. A section of the paper is devoted to the structure of principal holomorphic torus bundles over tori, establishing the Riemann bilinear relations, and exhibiting a so called Appell Humbert family, which could conjecturally give all small deformations. Finally, a general definition is given of so called Blanchard -Calabi manifolds, which, using an old construction of Sommese, show that the family of complex structures on the differentiable manifold underlying the product of a curve of genus g≥ 2 with a 2-dimensional complex torus admits several distinct deformation types.