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The de Rham homotopy theory of complex algebraic varieties. I. (English) Zbl 0637.55006

Generalizing work by P. Deligne [Publ. Math., Inst. Hautes Etud. Sci. 40, 5-57 (1971; Zbl 0219.14007); 44, 5-77 (1975; Zbl 0237.14003)] and J. W. Morgan [ibid. 48, 137-204 (1978; Zbl 0401.14003); 64, 185 (1986; Zbl 0617.14013)] the aim of this paper is to construct natural mixed Hodge structures (MHSs) on invariants of the algebraic topology of an arbitrary complex algebraic variety V, which are accessible to rational homotopy theory, e.g., the rational homotopy Lie algebra \(\pi_*(\Omega V)\otimes Q\), if V is nilpotent. The author’s approach is in two steps: he first constructs a natural geometric commutative Q-dga for V, together with a natural enrichment of structure up to a mixed Hodge diagram - in short a de Rham mixed Hodge complex (MHC) for V - and second he uses the de Rham MHC to propagate MHSs on the Q-homotopy invariants. As is well known, one may reduce in step 1 to the case of the geometric realization of a smooth simplicial variety V.; here the author uses the trick of a de Rham-Sullivan type theorem with differential local coefficients; (see, e.g. S. Halperin [Mem. Soc. Math. Fr., Nouv. Ser. 9-10 (1983; Zbl 0536.55003)]); thus obtaining the MHC for IV.I by assembling the various MHCs for \(V_ n\), which were known to exist in the smooth case. In step 2, the idea is to obtain various MHSs from the de Rham MHC not by passing through the Sullivan minimal model as in Morgan’s construction, but rather directly, by the use of two sided (reduced) bar constructions on dga’s and MHCs.
Reviewer: St.Papadima

MSC:

55P62 Rational homotopy theory
14F35 Homotopy theory and fundamental groups in algebraic geometry
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
57T30 Bar and cobar constructions
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