E_1-Formality of Complex Algebraic Varieties

TL;DR

This paper extends Morgan's formality results from smooth complex algebraic varieties to more general nilpotent varieties and morphisms, enhancing understanding of their rational homotopy types and functorial properties.

Contribution

It generalizes the formality theorem to arbitrary nilpotent complex algebraic varieties and morphisms, including singular and non-compact cases, filling a gap in Morgan's theory.

Findings

Generalization of formality to singular and non-compact varieties

Extension of functoriality of algebraic morphisms in rational homotopy theory

Application to studying Hopf invariants via intersection theory

Abstract

Let X be a smooth complex algebraic variety. Morgan [Mor78] showed that the rational homotopy type of X is a formal consequence of the differential graded algebra defined by the first term 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. The result for algebraic morphisms generalizes the Formality Theorem of [DGMS75] for compact K\"ahler varieties, filling a gap in Morgan's theory concerning functoriality over the rational numbers. As an application, we study the Hopf invariant of certain algebraic morphisms using intersection theory.