The deformation theory of representations of fundamental groups of compact Kähler manifolds. (English) Zbl 0678.53059

Let \(\Gamma\) be the fundamental group of a compact Kähler manifold \(M\) and \(G\) a real algebraic Lie group. The set \(\mathcal R = \mathcal R(\Gamma,G)\) of the representations \(\Gamma\to G\) is naturally an affine variety. In the present paper the authors study the local structure of \(\mathcal R\) at a point \(\rho\) under various conditions. It is proved that \(\mathcal R\) is locally isomorphic to a cone defined by a finite number of quadratic equations in \(Z^1(\Gamma,\mathfrak g_{\operatorname{ad} \rho})\) of the space of Lie algebra valued 1-cocycles. These equations are determined by the cup product induced by the Lie product (the obstruction map). This includes the following cases: (i) \(G\) is compact, (ii) \(\rho\) is the monodromy of a variation of Hodge structure on \(M\), and (iii) \(G\) is the automorphism group of a Hermitian symmetric space with certain condition.
This deformation problem of \(\rho\) is equivalent as groupoids, to the deformation problem of flat connections of an associated principal bundle \(P\). This latter in turn is equivalent to that of \(\operatorname{ad} P\) valued exterior differential forms on \(M\) associated with the connection on \(P\).


53C55 Global differential geometry of Hermitian and Kählerian manifolds
14D15 Formal methods and deformations in algebraic geometry
58H15 Deformations of general structures on manifolds
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