Existence and geometry of Legendre moduli spaces
type: Item
instance of: scholarly article
title: Existence and geometry of Legendre moduli spaces
zbMATH Open document ID: 0904.53047
DOI: 10.1007/PL00004337
author: Sergei Merkulov
published in: Mathematische Zeitschrift
publication date: 1997-12-10T00:00:00Z
review text: One of the most useful characteristics of an affine connection on a manifold \(M\) is its (restricted) holonomy group which is defined, up to conjugation, as a subgroup of \(GL (T_tM)\) consisting of all automorphisms of the tangent space \(T_tM\) at a point \(t\in M\) induced by parallel translations along the \(t\)-based contractible loops in \(M\). Which groups can occur as holonomies of affine connections? It is well-known that any closed subgroup of a general linear group can be realised as holonomy of some affine connection (which in general has a non-vanishing torsion tensor). The same question, when posed in the class of torsion-free affine connections only, is not yet answered. Long ago, Berger presented a very restricted list of possible irreducibly acting holonomies of torsion-free affine connections. His list was complete in the part of metric connections (and later much work has been done to refine this ``metric'' part of his list), while the situation with holonomies of non-metric torsion-free affine connections was and remains very unclear. As usual in representation theory, in order to get a deeper understanding of all irreducible real holonomies it is worthwhile trying to address a complex version of the problem. The main result of this paper says that any holomorphic irreducible 1-flat \(G\)-structure as well as any holomorphic torsion-free affine connection with irreducibly acting holonomy group can, in principle, be constructed by twistor methods. The first part of the paper solves a moduli problem which is of interest on its own, without references to differential geometry or twistor theory, and which is a generalization of the moduli problem solved by Kodaira in 1962. Kodaira's initial data is a pair, \(X\hookrightarrow Y\), consisting of a compact complex submanifold \(X\) of a complex manifold \(Y\). The object of study is the set \(M\) of complex submanifolds \(X_t\hookrightarrow Y\) obtained from \(X\) by all possible holomorphic deformatio…
Mathematics Subject Classification ID: 53C56
Mathematics Subject Classification ID: 32G10
Mathematics Subject Classification ID: 53B05
Mathematics Subject Classification ID: 53C10
Mathematics Subject Classification ID: 32M10
Mathematics Subject Classification ID: 32L25
zbMATH DE Number: 1096018
zbMATH Keywords: twistor theory
zbMATH Keywords: deformation theory
zbMATH Keywords: contact manifolds
zbMATH Keywords: complex homogeneous manifolds
zbMATH Keywords: holonomy of torsion-free affine connections
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1096018