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Mixed Hodge structures on homotopy groups. (English) Zbl 0597.55009

P. Deligne [Publ. Math., Inst. Hautes Étud. Sci. 40, 5-57 (1972; Zbl 0219.14007); ibid. 44, 5-77 (1975; Zbl 0237.14003)] defined mixed Hodge structures and showed that the cohomology of every algebraic variety over \({\mathbb{C}}\) has a natural mixed Hodge structure. J. W. Morgan [ibid. 48, 137-204 (1978; Zbl 0401.14003)], using Sullivan’s minimal models, showed that the rational homotopy Lie algebra and rational homotopy type of every smooth variety have natural mixed Hodge structures. In this note we announce an extension of mixed Hodge theory to arbitrary varieties and homotopy fibers of morphisms between varieties. The latter is a major step in extending asymptotic Hodge theory to homotopy groups and periods of iterated integrals. The bar construction and Kuo-Tsai Chen’s iterated integrals provide the link between Hodge theory and homotopy groups.

MSC:

55P62 Rational homotopy theory
55Q35 Operations in homotopy groups
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
57T30 Bar and cobar constructions
14F35 Homotopy theory and fundamental groups in algebraic geometry
55P35 Loop spaces
Full Text: DOI

References:

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