Tohoku Mathematical Journal

Homotopy theory of mixed Hodge complexes

Joana Cirici and Francisco Guillén

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Abstract

We show that the category of mixed Hodge complexes admits a Cartan-Eilenberg structure, a notion introduced by Guillén-Navarro-Pascual-Roig leading to a good calculation of the homotopy category in terms of (co)fibrant objects. Using Deligne's décalage, we show that the homotopy categories associated with the two notions of mixed Hodge complex introduced by Deligne and Beilinson respectively, are equivalent. The results provide a conceptual framework from which Beilinson's and Carlson's results on mixed Hodge complexes and extensions of mixed Hodge structures follow easily.

Article information

Source
Tohoku Math. J. (2) Volume 68, Number 3 (2016), 349-375.

Dates
Received: 17 February 2014
Revised: 25 November 2014
First available in Project Euclid: 23 September 2016

Permanent link to this document
http://projecteuclid.org/euclid.tmj/1474652264

Digital Object Identifier
doi:10.2748/tmj/1474652264

Mathematical Reviews number (MathSciNet)
MR3550924

Zentralblatt MATH identifier
06662223

Subjects
Primary: 55U35: Abstract and axiomatic homotopy theory
Secondary: 32S35: Mixed Hodge theory of singular varieties [See also 14C30, 14D07]

Keywords
Mixed Hodge theory homotopical algebra mixed Hodge complex filtered derived category weight filtration absolute filtration diagram category Cartan-Eilenberg category décalage

Citation

Cirici, Joana; Guillén, Francisco. Homotopy theory of mixed Hodge complexes. Tohoku Math. J. (2) 68 (2016), no. 3, 349--375. doi:10.2748/tmj/1474652264. http://projecteuclid.org/euclid.tmj/1474652264.


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References

  • A. A. Beǐlinson, Notes on absolute Hodge cohomology, Applications of algebraic $K$-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983), 35–68, Contemp. Math. 55, Amer. Math. Soc., Providence, RI, 1986.
  • K. S. Brown, Abstract homotopy theory and generalized sheaf cohomology, Trans. Amer. Math. Soc. 186 (1973), 419–458.
  • J. A. Carlson, Extensions of mixed Hodge structures, Journées de Géometrie Algébrique d'Angers Juillet 1979/Algebraic Geometry, Angers, 1979, pp. 107–127, Sijthoff & Noordhoff, Alphen aan den Rijn–Germantown, Md., 1980.
  • H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton, N. J., 1956.
  • J. Cirici, Cofibrant models of diagrams: mixed Hodge structures in rational homotopy, Trans. Amer. Math. Soc. 367 (2015), 5935–5970.
  • J. Cirici and F. Guillén, ${E}_1$-formality of complex algebraic varieties, Algebr. Geom. Topol. 14 (2014), 3049–3079.
  • D.-C. Cisinski, Catégories dérivables, Bull. Soc. Math. France 138 (2010), 317–393.
  • P. Deligne, Théorie de Hodge, II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5–57.
  • P. Deligne, Théorie de Hodge, III, Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5–77.
  • F. El Zein, Mixed Hodge structures, Trans. Amer. Math. Soc. 275 (1983), 71–106.
  • S. Gelfand and Y. Manin, Methods of homological algebra, Springer Monographs in Mathematics, Springer-Verlag, Berlin, second ed., 2003.
  • P. Griffiths and W. Schmid, Recent developments in Hodge theory: a discussion of techniques and results, Discrete subgroups of Lie groups and applicatons to moduli, 31–127, Oxford Univ. Press, Bombay, 1975.
  • F. Guillén, V. Navarro, P. Pascual and A. Roig, A Cartan-Eilenberg approach to homotopical algebra, J. Pure Appl. Algebra 214 (2010), 140–164.
  • R. M. Hain, The de Rham homotopy theory of complex algebraic varieties II, $K$-Theory 1 (1987), 481–497.
  • S. Halperin and D. Tanré, Homotopie filtrée et fibrés $C^\infty$, Illinois J. Math. 34 (1990), 284–324.
  • P. S. Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs 99, American Mathematical Society, Providence, RI, 2003.
  • A. Huber, Mixed motives and their realization in derived categories, Lecture Notes in Mathematics 1604, Springer-Verlag, Berlin, 1995.
  • L. Illusie, Complexe cotangent et déformations, I, Lecture Notes in Mathematics 239, Springer-Verlag, Berlin, 1971.
  • B. Keller, Chain complexes and stable categories, Manuscripta Math. 67 (1990), 379–417.
  • G. Laumon, Sur la catégorie dérivée des $\mathcal{D}$-modules filtrés, Algebraic geometry (Tokyo/Kyoto, 1982), Lecture Notes in Mathematics 1016, 151–237, Springer, Berlin, 1983.
  • M. Levine, Mixed motives, Handbook of $K$-theory, Vol. 1, 2, 429–521, Springer, Berlin, 2005.
  • J. W. Morgan, The algebraic topology of smooth algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 48 (1978), 137–204.
  • V. Navarro, Sur la théorie de Hodge-Deligne, Invent. Math. 90 (1987), 11–76.
  • K. H. Paranjape, Some spectral sequences for filtered complexes and applications, J. Algebra 186 (1996), 793–806.
  • P. Pascual, Some remarks on Cartan-Eilenberg categories, Collect. Math. 63 (2012), 203–216.
  • C. Peters and J. Steenbrink, Mixed Hodge structures, A Series of Modern Surveys in Mathematics 52, Springer-Verlag, Berlin, 2008.
  • D. Quillen, Homotopical algebra, Lecture Notes in Mathematics 43, Springer-Verlag, Berlin, 1967.
  • M. Saito, Mixed Hodge modules, Publ. Res. Inst. Math. Sci. 26 (1990), 221–333.
  • M. Saito, Mixed Hodge complexes on algebraic varieties, Math. Ann. 316 (2000), 283–331.
  • R. W. Thomason, Homotopy colimits in the category of small categories, Math. Proc. Cambridge Philos. Soc. 85 (1979), 91–109.
  • J.-L. Verdier, Des catégories dérivées des catégories abéliennes, Astérisque 239 (1997), xii+253.