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$E_{1}$-Formality of complex algebraic varieties
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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.
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CIRICI, Joana and GUILLÉN SANTOS, Francisco. $E_{1}$-Formality of complex algebraic varieties. Algebraic and Geometric Topology. 2014. Vol. 14, num. 3049-3079. ISSN 1472-2747. [consulted: 2 of July of 2026]. Available at: https://hdl.handle.net/2445/62303