In this note, we survey various facets of formality, with emphasis on the geometric and algebraic implications of -formality, and its relations to the cohomology jump loci and the Bieri-Neumann-Strebel invariant. We also produce examples of -manifolds such that, for every compact Kähler manifold , the product has the rational homotopy type of a Kähler manifold, yet admits no Kähler metric.
Key Words: Formality, fundamental group, cohomology jumping loci, holonomy Lie algebra, Bieri-Neumann-Strebel invariant, Malcev completion, lower central series, Kähler manifold, quasi-Kähler manifold, Milnor fiber, hyperplane arrangement, Artin group, Bestvina-Brady group, pencil, fibration, monodromy
2000 Mathematics Subject Classification: Primary: 55P62, 57M07,
Secondary: 14F35, 20J05, 55N25.
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