Homotopy theory of mixed Hodge complexes. (English) Zbl 1359.55013

Mixed Hodge complexes were introduced by P. Deligne [Publ. Math., Inst. Hautes Étud. Sci. 44, 5–77 (1974; Zbl 0237.14003)] in order to extend his theory of mixed Hodge structures on the cohomology of algebraic varieties to the singular case, via simplicial resolution.
The authors study the homotopy theory of mixed Hodge complexes within the homotopical framework of Cartan-Eilenberg categories [F. Guillén et al., J. Pure Appl. Algebra 214, No. 2, 140–164 (2010; Zbl 1198.18006)]. Using Deligne’s décalage [loc. cit.], the authors show that the homotopy categories associated with the two notions of mixed Hodge complex introduced by Deligne [loc. cit.] and A. A. Beilinson [Contemp. Math. 55, 35–68 (1986; Zbl 0621.14011)], respectively, are equivalent. Then, Beilinson’s [loc. cit.] and J. A. Carlson’s results [Journées de géométrie algébrique, Angers/France 1979, 107–127 (1980; Zbl 0471.14003)] on mixed Hodge complexes and extensions of mixed Hodge structures follow.


55U35 Abstract and axiomatic homotopy theory in algebraic topology
32S35 Mixed Hodge theory of singular varieties (complex-analytic aspects)
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