Cofibrant models of diagrams: Mixed Hodge structures in rational homotopy
HTML articles powered by AMS MathViewer
- by Joana Cirici PDF
- Trans. Amer. Math. Soc. 367 (2015), 5935-5970 Request permission
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.References
- Hans J. Baues, Obstruction theory on homotopy classification of maps, Lecture Notes in Mathematics, Vol. 628, Springer-Verlag, Berlin-New York, 1977. MR 0467748
- Hans Joachim Baues, Algebraic homotopy, Cambridge Studies in Advanced Mathematics, vol. 15, Cambridge University Press, Cambridge, 1989. MR 985099, DOI 10.1017/CBO9780511662522
- A. A. Beĭlinson, Notes on absolute Hodge cohomology, Applications of algebraic $K$-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983) Contemp. Math., vol. 55, Amer. Math. Soc., Providence, RI, 1986, pp. 35–68. MR 862628, DOI 10.1090/conm/055.1/862628
- A. K. Bousfield and V. K. A. M. Gugenheim, On $\textrm {PL}$ de Rham theory and rational homotopy type, Mem. Amer. Math. Soc. 8 (1976), no. 179, ix+94. MR 425956, DOI 10.1090/memo/0179
- Kenneth S. Brown, Abstract homotopy theory and generalized sheaf cohomology, Trans. Amer. Math. Soc. 186 (1973), 419–458. MR 341469, DOI 10.1090/S0002-9947-1973-0341469-9
- J. Cirici and F. Guillén, ${E}_1$-formality of complex algebraic varieties, Algebr. Geom. Topol., to appear.
- —, Homotopy theory of mixed Hodge complexes, Preprint, arXiv:math/1304.6236 [math.AG] (2013).
- Denis-Charles Cisinski, Catégories dérivables, Bull. Soc. Math. France 138 (2010), no. 3, 317–393 (French, with English and French summaries). MR 2729017, DOI 10.24033/bsmf.2592
- Pierre Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5–57 (French). MR 498551
- Pierre Deligne, Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5–77 (French). MR 498552
- Pierre Deligne, Phillip Griffiths, John Morgan, and Dennis Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), no. 3, 245–274. MR 382702, DOI 10.1007/BF01389853
- W. G. Dwyer and J. Spaliński, Homotopy theories and model categories, Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 73–126. MR 1361887, DOI 10.1016/B978-044481779-2/50003-1
- Yves Félix, Stephen Halperin, and Jean-Claude Thomas, Rational homotopy theory, Graduate Texts in Mathematics, vol. 205, Springer-Verlag, New York, 2001. MR 1802847, DOI 10.1007/978-1-4613-0105-9
- Phillip Griffiths and John Morgan, Rational homotopy theory and differential forms, 2nd ed., Progress in Mathematics, vol. 16, Springer, New York, 2013. MR 3136262, DOI 10.1007/978-1-4614-8468-4
- Francisco Guillén and Vicente Navarro Aznar, Un critère d’extension des foncteurs définis sur les schémas lisses, Publ. Math. Inst. Hautes Études Sci. 95 (2002), 1–91 (French). MR 1953190, DOI 10.1007/s102400200003
- F. Guillén, V. Navarro, P. Pascual, and Agustí Roig, A Cartan-Eilenberg approach to homotopical algebra, J. Pure Appl. Algebra 214 (2010), no. 2, 140–164. MR 2559687, DOI 10.1016/j.jpaa.2009.04.009
- Richard M. Hain, The de Rham homotopy theory of complex algebraic varieties. II, $K$-Theory 1 (1987), no. 5, 481–497. MR 934453, DOI 10.1007/BF00536980
- Stephen Halperin and Daniel Tanré, Homotopie filtré e et fibrés $C^\infty$, Illinois J. Math. 34 (1990), no. 2, 284–324 (French). MR 1046566
- Mark Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999. MR 1650134
- K. H. Kamps and T. Porter, Abstract homotopy and simple homotopy theory, World Scientific Publishing Co., Inc., River Edge, NJ, 1997. MR 1464944, DOI 10.1142/9789812831989
- John W. Morgan, The algebraic topology of smooth algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 48 (1978), 137–204. MR 516917
- V. Navarro Aznar, Sur la théorie de Hodge-Deligne, Invent. Math. 90 (1987), no. 1, 11–76 (French). MR 906579, DOI 10.1007/BF01389031
- Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR 0223432
- A. Radulescu-Banu, Cofibrations in homotopy theory, Preprint, arXiv:math/0610009v4 [math.AT] (2009).
- Beatriz Rodríguez González, Simplicial descent categories, J. Pure Appl. Algebra 216 (2012), no. 4, 775–788. MR 2864852, DOI 10.1016/j.jpaa.2011.10.003
- Dennis Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269–331 (1978). MR 646078
- R. W. Thomason, Homotopy colimits in the category of small categories, Math. Proc. Cambridge Philos. Soc. 85 (1979), no. 1, 91–109. MR 510404, DOI 10.1017/S0305004100055535
- B. Totaro, Topology of singular algebraic varieties, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 533–541. MR 1957063
Additional Information
- Joana Cirici
- Affiliation: Fachbereich Mathematik und Informatik, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany
- Email: jcirici@math.fu-berlin.de
- 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.
- © Copyright 2014 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 367 (2015), 5935-5970
- MSC (2010): Primary 18G55, 32S35
- DOI: https://doi.org/10.1090/S0002-9947-2014-06405-2
- MathSciNet review: 3347193