Lectures on deformations of complex manifolds (Deformations from differential graded viewpoint)

By Marco Manetti. 2005

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Preface This paper is based on a course given at the " Dottorato di Ricerca in Matemat-ica " of the University of Rome " La Sapienza " in the Academic year 2000/2001. The intended aim of the course was to rapidly introduce, although not in an exhaustive way, the non-expert PhD student to deformations of compact complex manifolds, from the very beginning to some recent (i.e. at that time not yet published) results. With the term " deformation theory " , the mathematicians usually intend a set of deformation theories, each one of which studies small parametric variation of a specific mathematical structure, for example: deformation theory of complex manifolds, deformation theory of associative algebras, deformations of schemes, deformations of representations and much more. Every mathematician which tries to explain and investigate deformation theory has to deal with two opposite features: order and chaos. Chaos: the various deformation theories often rely on theorems which are proved using very different tools, from families of elliptic differential operators of Kodaira and Spencer [41] to ringed toposes of Illusie [32]. Order: all the deformation theories have lots of common features; for instance they have a vector space of first order deformations (usually the H 1 of some complex) and they have an obstruction space (usually an H 2). Another unifying aspect of all deformation theories is summarized in the slogan " In characteristic 0 every deformation problem is governed by a differential graded Lie algebra " , which underlie some ideas given, mostly in private communications, by Quillen, Deligne, Drinfeld and other about 20 years ago. More recently (especially in [42] and [44]) these ideas have shown a great utility and possibility of development. Nowadays this approach to deformation theory is a very active area of research which is usually called deformation theory via DGLA or extended deformation theory. The goal of these notes is to give a soft introduction to extended deformation theory. In view of the aim (and the hope) of keeping this paper selfcontained, user friendly and with a tolerating number of pages, we consider only deformations of compact complex manifolds. Anyhow, most part of the formalism and of the results that we prove here will apply to many other deformation problems. The first part of the paper (Chapters I, II and III) is a classical introduction to deformations of compact complex manifolds; the beginners can find here the main …

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9a7b77daf534e7d543f86d1ae8f969765289ce3d

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