E1–formality of complex algebraic varieties

Cirici, Joana¹ ; Guillén, Francisco²

¹Fachbereich Mathematik und Informatik, Freie Universität Berlin, Arnimallee 3, D-14195 Berlin, Germany
²Departament d’Àlgebra i Geometria, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain

Algebraic and Geometric Topology, Tome 14 (2014) no. 5, pp. 3049-3079

Résumé

Let X be a smooth complex algebraic variety. Morgan showed that the rational homotopy type of X is a formal consequence of the differential graded algebra defined by the first term E1(X,W) of its weight spectral sequence. In the present work, we generalize this result to arbitrary nilpotent complex algebraic varieties (possibly singular and/or non-compact) and to algebraic morphisms between them. In particular, our results generalize the formality theorem of Deligne, Griffiths, Morgan and Sullivan for morphisms of compact Kähler varieties, filling a gap in Morgan’s theory concerning functoriality over the rationals. As an application, we study the Hopf invariant of certain algebraic morphisms using intersection theory.

Détail

DOI : 10.2140/agt.2014.14.3049

Classification : 32S35, 55P62
Keywords: rational homotopy, mixed Hodge theory, formality, minimal models, weight filtration, cohomological descent, Hopf invariant

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