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Cofibrant models of diagrams: Mixed Hodge structures in rational homotopy

Author: Joana Cirici
Journal: Trans. Amer. Math. Soc. 367 (2015), 5935-5970
MSC (2010): Primary 18G55, 32S35
Published electronically: October 3, 2014
MathSciNet review: 3347193
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Abstract: We study the homotopy theory of a certain type of diagram category whose vertices are in variable categories with a functorial path, leading to a good calculation of the homotopy category in terms of cofibrant objects. The theory is applied to the category of mixed Hodge diagrams of differential graded algebras. Using Sullivan's minimal models, we prove a multiplicative version of Beilinson's Theorem on mixed Hodge complexes. As a consequence, we obtain functoriality for the mixed Hodge structures on the rational homotopy type of complex algebraic varieties. In this context, the mixed Hodge structures on homotopy groups obtained by Morgan's theory follow from the derived functor of the indecomposables of mixed Hodge diagrams.

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Joana Cirici
Affiliation: Fachbereich Mathematik und Informatik, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany

Received by editor(s): August 21, 2013
Received by editor(s) in revised form: January 20, 2014
Published electronically: October 3, 2014
Additional Notes: This research was financially supported by the Marie Curie Action through PCOFUND-GA-2010-267228, and partially supported by the Spanish Ministry of Economy and Competitiveness MTM 2009-09557 and the DFG under project SFB 647.
Article copyright: © Copyright 2014 American Mathematical Society