E1–formality of complex algebraic varieties

Cirici, Joana; Guillén, Francisco

doi:10.2140/agt.2014.14.3049

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$E_1$–formality of complex algebraic varieties

Joana Cirici and Francisco Guillén

Algebraic & Geometric Topology 14 (2014) 3049–3079

DOI: 10.2140/agt.2014.14.3049

Abstract

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 $E_{1} (X, W)$ 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.

Keywords

rational homotopy, mixed Hodge theory, formality, minimal models, weight filtration, cohomological descent, Hopf invariant

Mathematical Subject Classification 2010

Primary: 32S35, 55P62

References

Publication

Received: 29 October 2013

Revised: 29 January 2014

Accepted: 4 February 2014

Published: 5 November 2014

Authors

Joana Cirici
Fachbereich Mathematik und Informatik Freie Universität Berlin Arnimallee 3 D-14195 Berlin Germany

Francisco Guillén
Departament d’Àlgebra i Geometria Universitat de Barcelona Gran Via 585 08007 Barcelona Spain

Volume 14, issue 5 (2014)