Deformations in the large of some complex manifolds, I

Details Deformations in the large of some complex manifolds, I Deformations in the large of some complex manifolds, I

2003-07-04

arxiv.org/abs/math/0307067v1

Mathematics

Algebraic Geometry

29 pages, to appear in Annali di Mat. pura e appl., volume in memory of Fabio Bardelli

Scientific paper

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 $g\geq 2$ with a 2-dimensional complex torus admits several distinct deformation types.

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