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Mixed Hodge structures and Sullivan’s minimal models of Sasakian manifolds. (Structures de Hodge mixtes et modèles minimaux de Sullivan des variétés sasakiennes.) (English. French summary) Zbl 1403.53040
Let \(M\) be a compact \((2n+1)\)-dimensional Sasakian manifold with \(n\geq 2\). In the present work, the author proves that the Malčev Lie algebra of the fundamental group \(\pi_1(M)\) admits a quadratic presentation, i.e., is a quotient of a free Lie algebra by an ideal generated in degree two. The proof of this result is mainly based on a clever usage of Morgan’s techniques of mixed Hodge diagrams. With the same ingredient, it is proved that a compact \((2n+1)\)-dimensional nilmanifold admits a Sasakian structure if and only if it is a Heisenberg nilmanifold \(H_{2n+1}/\Gamma\).

MSC:
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
55P62 Rational homotopy theory
58A14 Hodge theory in global analysis
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