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$E_1$–formality of complex algebraic varieties

Joana Cirici and Francisco Guillén

Algebraic & Geometric Topology 14 (2014) 3049–3079

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.

rational homotopy, mixed Hodge theory, formality, minimal models, weight filtration, cohomological descent, Hopf invariant
Mathematical Subject Classification 2010
Primary: 32S35, 55P62
Received: 29 October 2013
Revised: 29 January 2014
Accepted: 4 February 2014
Published: 5 November 2014
Joana Cirici
Fachbereich Mathematik und Informatik
Freie Universität Berlin
Arnimallee 3
D-14195 Berlin
Francisco Guillén
Departament d’Àlgebra i Geometria
Universitat de Barcelona
Gran Via 585
08007 Barcelona