Simpson, Carlos T. Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization. (English) Zbl 0669.58008 J. Am. Math. Soc. 1, No. 4, 867-918 (1988). 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 Cited in 21 ReviewsCited in 299 Documents 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) Keywords:Kähler manifold; Hodge structure; Yang-Mills equations; holomorphic vector bundles; Chern numbers × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Lars V. Ahlfors, An extension of Schwarz’s lemma, Trans. Amer. Math. Soc. 43 (1938), no. 3, 359 – 364. · Zbl 0018.41002 [2] Thierry Aubin, Sur la fonction exponentielle, C. R. Acad. Sci. Paris Sér. A-B 270 (1970), A1514 – A1516 (French). · Zbl 0197.47802 [3] Kevin Corlette, Flat \?-bundles with canonical metrics, J. 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