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Sur la théorie de Hodge-Deligne. (On the Hodge-Deligne theory). (French) Zbl 0639.14002
Let X be a complex algebraic variety. By the work of Deligne and Hodge, the singular cohomology groups H n(X,\({\mathbb{Q}})\), \(n\geq 0\), carry canonical mixed Hodge structures, which moreover are functorial in X. The paper under review exploits some ideas of Deligne and others in order to prove that in certain cases, not only the global cohomology, but also all (local) cohomology groups as well as the rational homotopy carry canonical and functorial mixed Hodge structures. Specifically, let \(f: X\to {\mathbb{C}}\) be an analytic function, and let V be an analytic subspace of \(f^{-1}(0)\). If \({\mathbb{R}}\psi ({\mathbb{Q}})\) is the complex of “cycles proches” and \({\mathbb{R}}\Phi ({\mathbb{Q}})\) is the complex of vanishing cycles, the author is able to prove that, if V is a compact algebraic variety, then the groups H n(V,\({\mathbb{R}}\psi ({\mathbb{Q}}))\) and H n(V,\({\mathbb{R}}\Phi ({\mathbb{Q}}))\), \(n\geq 0\), carry canonical and functorial mixed Hodge structures such that the logarithm of the unipotent part of the monodromy is a morphism of mixed Hodge structures of type (-1,-1). This result extends previous results of Clemens and Steenbrink.
On the other hand, using some ideas of Deligne and Morgan, the author also proves that such mixed Hodge structures can be also encountered at the rational homotopy level. As an example, one has the following result: if \(F_ x\) is the Milnor fibre of \(f: X\to {\mathbb{C}},\) \(x\in X\), the Sullivan’s minimal model associated to \(F_ x\) has a mixed Hodge structure; in particular, if \(F_ x\) is simply-connected, then the rational homotopy groups of \(F_ x\) have mixed Hodge structures. The paper is technically very elaborated, the author dealing first with the details of the general theory, and then applying them to situations like those sketched above.
Reviewer: L.Bădescu

14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14F35 Homotopy theory and fundamental groups in algebraic geometry
14F20 Étale and other Grothendieck topologies and (co)homologies
Full Text: DOI EuDML
[1] Barlet, D.: Contribution du cup-produit de la fibre de Milnor aux pôles de |f|2. Ann. Inst. Fourier, Grenoble,34, (4) 75-107 (1984) · Zbl 0525.32007
[2] Bousfield, A.K., Gugenheim, V.K.A.M.: On PL De Rham theory and rational homotopy type. Mem. Am. Math. Soc.179, 1-94 1976 · Zbl 0338.55008
[3] Clemens, C.H.: Degenerations of Kähler manifolds. Duke Math. J.44, 215-290 (1977) · Zbl 0353.14005
[4] Deligne, P.: Comparaison avec la théorie transcendente, Exp. XIV, dans SGA 7 II. Lect. Notes Math., Vol. 340. Berlin-Heidelberg-New York: Springer 1973
[5] Deligne, P.: Poids dans la Cohomologie des Variétés Algébriques, Actes du Congrès International des Mathématiciens, Vancouver, 1974, pp. 79-85
[6] Deligne, P.: Théorie de Hodge II. Publ. Math., Inst. Hautes Etud. Sci.40, 5-57 (1972) · Zbl 0219.14007
[7] Deligne, P.: Théorie de Hodge III. Publ. Math., Inst. Hautes Etud. Sci.44, 5-77 (1975) · Zbl 0292.14005
[8] Deligne, P.: La conjecture de Weil, II. Publ. Math., Inst. Hautes Etud. Sci.52, 137-252 (1980) · Zbl 0456.14014
[9] Dupont, J.L.: Curvature and characteristic classes. Lect. Notes Math., Vol. 640. Berlin Heidelberg New York: Springer 1978 · Zbl 0373.57009
[10] Fujiki, A.: Duality of mixed Hodge structures of algebraic varieties. Publ. Res. Inst. Math. Sci.16, 635-667 (1980) · Zbl 0475.14006
[11] Godement, R.: Topologie algébrique et théorie des faisceaux. Paris: Hermann 1958 · Zbl 0080.16201
[12] Griffiths, P.A., Schmid, W.: Recent developments in Hodge theory: A discussion of techniques and results, Proceedings of the International Colloquium on Discrete Subgroups of Lie Groups (Bombay, 1973). Oxford Univ. Press, 1975 · Zbl 0355.14003
[13] Grothendieck, A.: On the De Rham cohomology of algebraic varieties. Publ. Math. Inst. Hautes Etud. Sci.29, 96-103 (1966) · Zbl 0145.17602
[14] Guillén, F.: Une relation entre la filtration de Zeeman et la filtration par le poids de Deligne, Compos. Math.61, 201-228 (1987) · Zbl 0617.14011
[15] Guillén, F., Navarro Aznar, V., Puerta, F.: Théorie de Hodge via schemas cubiques, notes polycopiées, Universitat Politècnica de Catalunya, 1982
[16] Hain, R.M.: The de Rham homotopy theory of complex algebraic varieties. Preprint, University of Utah, 1984 · Zbl 0637.55006
[17] Lê, D.T.: Remarks on relative monodromy, Real and Complex Singularities (Oslo, 1976). Alphen van den Rijn: Sijthoff-Noordhoff, 1977
[18] MacLane, S.: Categories for the working mathematician. New York: Springer 1971 · Zbl 0705.18001
[19] Morgan, J.W.: The algebraic topology of smooth algebraic varieties. Publ. Math., Inst. Hautes Etud. Sci.48, 137-204 (1978) · Zbl 0401.14003
[20] Pascual Gainza, P.: Contribucions als espais algebraics, tesi doctoral, Universitat Autònoma de Barcelona, Barcelona, 1983
[21] Pascual Gainza, P.: On the simple object of a diagram in a closed model category. Math. Proc. Camb. Philos. Soc.100, 459-474 (1986) · Zbl 0619.18004
[22] Quillen, D.G.: Rational homotopy theory. Ann. Math.90, 205-295 (1969) · Zbl 0191.53702
[23] Segal, G.: Classifying spaces and spectral secuences. Publ. Math., Inst. Hautes Etud. Sci.34, 105-112 (1968) · Zbl 0199.26404
[24] Steenbrink, J.H.: Limits of Hodge structures. Invent. Math.31, 229-257 (1976) · Zbl 0312.14007
[25] Steenbrink, J.H.M.: Mixed Hodge structure on the vanishing cohomology. Real and Complex Singularities (Oslo, 1976). Alphen van den Rijn: Sijthoff-Noordhoff, 1977 · Zbl 0373.14007
[26] Steenbrink, J.H.M.: Mixed Hodge structures associated with isolated singularities. Proc. Symp. Pure Math.40, (part 2) 513-536 (1983) · Zbl 0515.14003
[27] Steenbrink, J.H.M.: Semicontinuity of the singularity spectrum. Invent. Math.79, 557-565 (1985) · Zbl 0568.14021
[28] Steenbrink, J.H., Zucker, S.: Variation of mixed Hodge structure. I. Invent. Math.80, 489-542 (1985) · Zbl 0626.14007
[29] Sullivan, D.: Infinitesimal computations in topology. Publ. Math., Inst. Hautes Etud. Sci.47, 269-331 (1977) · Zbl 0374.57002
[30] Swan, R.: Thom’s theory of differential forms on simplicial sets. Topology44, 271-273 (1975) · Zbl 0319.58004
[31] Thom, R.: Opérations en cohomologie réelle. Exp. 17. Sem. H. Cartan, E.N.S. Paris, 1954-1955
[32] Varchenko, A.N.: Asymptotic Hodge structure in the vanishing cohomology. Math. USSR, Izv.18, 469-512 (1982) · Zbl 0489.14003
[33] Whitney, H.: Geometric integration theory. Princenton, Univ. Press, 1957 · Zbl 0083.28204
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