Hain, Richard M. Mixed Hodge structures on homotopy groups. (English) Zbl 0597.55009 Bull. Am. Math. Soc., New Ser. 14, 111-114 (1986). 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. Cited in 2 ReviewsCited in 6 Documents 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 Keywords:smooth variety; varieties; homotopy fibers of morphisms between varieties; asymptotic Hodge theory; homotopy groups; periods of iterated integrals; bar construction Citations:Zbl 0219.14007; Zbl 0237.14003; Zbl 0401.14003 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Kuo Tsai Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977), no. 5, 831 – 879. · Zbl 0389.58001 [2] Kuo Tsai Chen, Reduced bar constructions on de Rham complexes, Algebra, topology, and category theory (a collection of papers in honor of Samuel Eilenberg), Academic Press, New York, 1976, pp. 19 – 32. [3] Pierre Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5 – 57 (French). Pierre Deligne, Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5 – 77 (French). [4] Pierre Deligne, Phillip Griffiths, John Morgan, and Dennis Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), no. 3, 245 – 274. · Zbl 0312.55011 · doi:10.1007/BF01389853 [5] Alan H. Durfee and Richard M. Hain, Mixed Hodge structures on the homotopy of links, Math. Ann. 280 (1988), no. 1, 69 – 83. · Zbl 0617.14012 · doi:10.1007/BF01474182 [6] Richard M. Hain, Iterated integrals and homotopy periods, Mem. Amer. Math. Soc. 47 (1984), no. 291, iv+98. · Zbl 0539.55002 · doi:10.1090/memo/0291 [7] Richard M. Hain, The de Rham homotopy theory of complex algebraic varieties. I, \?-Theory 1 (1987), no. 3, 271 – 324. · Zbl 0637.55006 · doi:10.1007/BF00533825 [8] Richard M. Hain and Steven Zucker, Unipotent variations of mixed Hodge structure, Invent. Math. 88 (1987), no. 1, 83 – 124. · Zbl 0622.14007 · doi:10.1007/BF01405093 [9] Stephen Halperin, Finiteness in the minimal models of Sullivan, Trans. Amer. Math. Soc. 230 (1977), 173 – 199. · Zbl 0364.55014 [10] S. Halperin, Lectures on minimal models, Mém. Soc. Math. France (N.S.) 9-10 (1983), 261. · Zbl 0536.55003 [11] Bruno Harris, Harmonic volumes, Acta Math. 150 (1983), no. 1-2, 91 – 123. · Zbl 0527.30032 · doi:10.1007/BF02392968 [12] John W. Morgan, The algebraic topology of smooth algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 48 (1978), 137 – 204. · Zbl 0401.14003 [13] M. Pulte, Thesis, University of Utah, 1985. [14] Joseph Steenbrink, Limits of Hodge structures, Invent. Math. 31 (1975/76), no. 3, 229 – 257. · Zbl 0303.14002 · doi:10.1007/BF01403146 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.