2001/07/31 by Ludmil Katzarkov, L. Katzarkov, Tony Pantev +3 · 22 citations
Mathematics · #Algebra over a field #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Cofibration #Cohomology #Functor #Fundamental group #Hodge conjecture #Hodge dual #Hodge theory #Homotopy #Homotopy and Cohomology in Algebraic Topology #Homotopy group #Mathematics #Pure mathematics #Regular homotopy #math.AG #math.AT
paper · pdf · doi:10.1112/s0010437x07003351
published in Compositio Mathematica 144(3), 582-632 (Cambridge University Press) · 57 pages. This new version has been globally reorganized and includes additional results and applications. Minor corrections
arxiv created 2005/06/16 · openalex publication_date 2008/05/01 · arxiv updated 2014/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Abstract We use Hodge theoretic methods to study homotopy types of complex projective manifolds with arbitrary fundamental groups. The main tool we use is the schematization functor X ↦ (X⊗ \mathbb C)^\mathrm sch , introduced by the third author as a substitute for the rationalization functor in homotopy theory in the case of non-simply connected spaces. Our main result is the construction of a Hodge decomposition on (X⊗ \mathbb C)^\mathrm sch . This Hodge decomposition is encoded in an action of the discrete group \mathbb C× δ on the object (X⊗ \mathbb C)^\mathrm sch and is shown to recover the usual Hodge decomposition on cohomology, the Hodge filtration on the pro-algebraic fundamental group, and, in the simply connected case, the Hodge decomposition on the complexified homotopy groups. We show that our Hodge decomposition satisfies a purity property with respect to a weight filtration, generalizing the fact that the higher homotopy groups of a simply connected projective manifold have natural mixed Hodge structures. As applications we construct new examples of homotopy types which are not realizable as complex projective manifolds and we prove a formality theorem for the schematization of a complex projective manifold.