#### Volume 14, issue 5 (2014)

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1472-2739 ISSN (print): 1472-2747
$E_1$–formality of complex algebraic varieties

### Joana Cirici and Francisco Guillén

Algebraic & Geometric Topology 14 (2014) 3049–3079
##### 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 ${E}_{1}\left(X,W\right)$ 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