E₁–formality of complex algebraic varieties

Abstract

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 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. The result for algebraic morphisms generalizes the Formality Theorem of [DGMS75] for compact Kähler 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.

Naturality properties and comparison results for topological and infinitesimal embedded jump loci

“…This transversality property is a substantial non-abelian extension of the corresponding rank 1 result, proved in [10] in the case when ι is the standard isomorphism Cˆ» Ý Ñ GL 1 pCq. To state the quasi-projective analogue of the formality property, we need to recall from [23,5] some relevant facts. Every quasi-projective manifold M is of the form MzD, where M is a smooth, projective variety and D is a normal-crossing divisor in M. A regular map between two such manifolds, f : M Ñ M 1 , is induced by a regular mapf : M Ñ M 1 with the property thatf´1pD 1 q Ď D. The manifold M admits as a finite cdga model over C…”

“…We want to prove a quasi-projective analogue of Proposition 3.4. For that, we will need the theory of relative minimal models for mixed Hodge diagrams (MHDs, for short), developed by Cirici and Guillén in [5]. We start by recalling some pertinent definitions and results from [5].…”

Weight decompositions of Thom spaces of vector bundles in rational homotopy

“…For the smooth case, the above result is due to Morgan [Mor78]. The definition of mixed Hodge cdga of [CG14] differs by a décalage from the one introduced here. This does not affect the above result.…”

Derived A-infinity algebras and their homotopies

“…The r-translation and the r-cone. The r-path construction is related to a translation functor depending on r (see [CG16] and [CG14] for similar constructions in the categories of filtered complexes and filtered commutative dgas respectively). Furthermore, the cone obtained via this translation allows one to detect E r -quasi-isomorphisms, as we shall see next.…”