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Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization. (English) Zbl 0669.58008
The aim of this very thorough and exciting paper is to present a method of constructing representations of fundamental groups in complex geometry, using techniques of partial differential equations. Among others, the author solves the Yang-Mills equations on holomorphic vector bundles with interaction terms, over compact and some noncompact Kähler manifolds, yielding flat connections when certain Chern numbers vanish. As an application, a criterion is given in the compact case for a variety to be uniformized by any particular bounded symmetric domain.
Reviewer: J.Szilasi

##### MSC:
 58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 53C05 Connections (general theory) 32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
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