Skip to main content
SpringerLink
Log in
Book cover

Complex Manifolds pp 39–142Cite as

An Overview of Recent Advances in Hodge Theory

  • Chapter

Abstract

Our aim, in writing this article, is to give a survey of the progress in Hodge theory over the past 15 years. Of course, it was first necessary to decide what we mean by “Hodge theory”! This is by no means an easy question, and our answer to it shaped the contents of the article. Indeed, the nature of the subject has evolved over the years.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Adams, J.: On the cobar construction. In: Colloque de topologie algébrique (Louvain, 1956). Masson 1957, pp. 81–87

    Google Scholar 

  2. Anderson, G.: Cyclotomy and an extension of the Taniyama group. Comp. Math. 57 (1986) 153–217

    MATH  Google Scholar 

  3. Andreotti, A., Vesentini, E.: Carleman estimates for the Laplace-Beltrami equation on complex manifolds. Publ. Math. I.H.E.S. 25 (1966) 81–130

    Google Scholar 

  4. Arapura, D.: Hodge theory with local coefficients and fundamental groups of varieties. Bull. Am. Math. Soc. 20 (1989) 169–172

    MATH  MathSciNet  Google Scholar 

  5. Barlet, D.: Filtration de Hodge asymptotique et pôles de ∫\f\ 2λ. Prépubl. Inst. Henri Cartan 88/7

    Google Scholar 

  6. Barlet, D.: Symétrie de Hodge pour le polynôme de Bernstein-Sato. (Lecture Notes in Mathematics, vol. 1295). Springer 1987, pp. 1–10

    Google Scholar 

  7. Beilinson, A.: Higher regulators and values of L-functions. J. Soviet Math. 30 (1985) 2036–2070

    MATH  Google Scholar 

  8. Beilinson, A.: Notes on absolute Hodge cohomology. In: Applications of Algebraic X-Theory to Algebraic Geometry and Number Theory. (Contemporary Mathematics, vol. 55, part I). Amer. Math. Soc. 1986, pp. 35–68

    Google Scholar 

  9. Beilinson, A.: Height pairing between algebraic cycles. In: K. Ribet (ed.) Current trends in arithmetical algebraic geometry (Contemporary Mathematics, vol. 67). American Mathematical Society 1987, pp. 1–24

    Google Scholar 

  10. Beilinson, A.: On the derived category of perverse sheaves. In: K-Theory, Arithmetic and Geometry, Yu. Manin, ed. (Lecture Notes in Mathematics, vol. 1289) Springer, pp. 27–41

    Google Scholar 

  11. Beilinson A, Bernstein, I.N., Deligne, P.: Faisceaux Pervers. (Astérisque, vol. 100). Soc. Math. Fr., 1982

    Google Scholar 

  12. Beilinson, A., Ginsburg, V.: Mixed categories, Ext-duality and representations (results and conjectures). Preprint 1988

    Google Scholar 

  13. Bernstein, I.N., Gel’fand, I.M., Gel’fand, S.I.: Differential operators on the base affine space and a study of g-modules. In: Lie Groups and their Representation. Halsted 1975, pp. 21–64

    Google Scholar 

  14. Beilinson, A., MacPherson, R., Schechtman, V.: Notes on motivic cohomology. Duke Math. J. 54 (1987) 679–710

    MATH  MathSciNet  Google Scholar 

  15. Bloch, S.: Deligne groups. Manuscript (unpubl.) c. 1972

    Google Scholar 

  16. Bloch, S.: Algebraic K-theory and zeta-functions of elliptic curves. In: Proc. Intern. Congress Helsinki, 1978, pp. 511–515

    Google Scholar 

  17. Bloch, S.: Applications of the dilogarithm function in algebraic K-theory and algebraic geometry. In: Proceedings of the International Symposium on Algebraic Geometry 1977. Kinokuniya Book Store, 1978, pp. 103–114

    Google Scholar 

  18. Bloch, S.: The dilogarithm and extensions of Lie algebras. (Lecture Notes in Mathematics, vol. 854). Springer, 1981, pp. 1–23

    Google Scholar 

  19. Bloch, S.: Height pairing between algebraic cycles.

    Google Scholar 

  20. Bloch, S, Ogus, A.: Gersten’s conjecture and the homology of schemes. Ann. Sci. Ec. Norm. Sup. 7 (1974) 181–202

    MATH  MathSciNet  Google Scholar 

  21. Bloom, T., Herrera, M.: The de Rham cohomology of an analytic space. Invent. math. 7 (1969) 275–296

    MATH  MathSciNet  Google Scholar 

  22. Borel, A.: Cohomologie de SL n et valeurs de fonctions zêta. Ann. Sci. Ec. Norm. Sup. 7 (1974) 613–636

    MathSciNet  Google Scholar 

  23. Borel, A., et al.: Intersection Cohomology. (Progress in Mathematics, vol. 50). Birkhäuser, 1984

    MATH  Google Scholar 

  24. Borel, A., et al.: Algebraic D-modules. (Perspective in Mathematics, vol. 2). Academic Press, 1987

    Google Scholar 

  25. Bousfield, A., Kan, D.: Homotopy limits, completions and localizations. (Lectures Notes in Mathematics, vol. 304). Springer, 1972

    MATH  Google Scholar 

  26. Brieskorn, E.: Die Monodromie der isolierten Singularitäten von Hyperflächen. Manuscr. Math. 2 (1970) 103–161

    MATH  MathSciNet  Google Scholar 

  27. Brylinski, J.-L.: Cohomologie d’intersection et faisceaux pervers. (Séminaire Bourbaki Exposé 585, February 1982, in: Astérisque, vol.92–93). Soc. Math. Fr, 1982, pp. 129–157

    Google Scholar 

  28. Brylinski, J.-L.: Modules holonomes à singularités régulières et filtration de Hodge. In: Algebraic Geometry. Proceedings, la Rabida 1981. (Lecture Notes in Math, vol.961). Springer, 1982, pp. 1–21. II (Astérisque, vol. 101–102). Soc. Math. Fr, 1983, pp. 75–117

    Google Scholar 

  29. Brylinski, J.-L.: Transformations canoniques, dualité projective, théorie de Lefschetz, transformation de Fourier et sommes trigonométriques. In: Géométrie et Analyse Microlocales. (Astérisque, vol. 140–141). Soc. Math. Fr. 1986, pp. 3–134

    Google Scholar 

  30. Carlson, J.: Extensions of mixed Hodge structures. In: Journées de Géométrie Algébrique d’Angers 1979. Sijthoff and Nordhoff, 1980, pp. 107–127

    Google Scholar 

  31. Carlson, J.: The geometry of the extension class of a mixed Hodge structure. In: Algebraic Geometry, Bowdoin 1985, Proc. Symp. Pure Math. 46, Part 2. Amer. Math. Soc., pp. 199–222

    Google Scholar 

  32. Carlson, J, Clemens, H, Morgan, J.:. On the mixed Hodge structure associated to π3 of a simply-connected projective manifold. Ann. Sci. Ec. Norm. Sup. 14 (1981) 323–338

    MATH  MathSciNet  Google Scholar 

  33. Carlson, J., Hain, R: Extensions of variations of mixed Hodge structure. In: Proc. of the Luminy Conference on Hodge Theory, 1987, Astérisque. Soc. Math. Fr. (to appear)

    Google Scholar 

  34. Cattani, E., Kaplan, A.: On the SL(2)-orbits in Hodge theory (with an appendix by P. Deligne). I.H.E.S, preprint M/82/58 (1982)

    Google Scholar 

  35. Cattani, E., Kaplan, A.: Polarized mixed Hodge structures and the monodromy of a variation of Hodge structure. Invent, math. 67 (1982) 101–115

    MATH  MathSciNet  Google Scholar 

  36. Cattani, E., Kaplan, A.: Degenerating variations of Hodge structures. In: Proc. of the Luminy Conference on Hodge Theory, 1987. Astérisque, vol. 179–180 (1989) 67–96

    Google Scholar 

  37. Cattani, E., Kaplan, A., Schmid, W.: Degeneration of Hodge structures. Ann. Math. 123 (1986) 457–535

    MATH  MathSciNet  Google Scholar 

  38. Cattani, E., Kaplan, A., Schmid, W.: L2 and intersection cohomologies for a polarizable variation of Hodge structure. Invent. math. 87 (1987) 217–252

    MATH  MathSciNet  Google Scholar 

  39. Cheeger, J.: On the Hodge theory of Riemannian pseudomanifolds. In: Geometry of the Laplace operator. Proc. Symp. Pure Math., vol.36, 1980, pp.91–146

    Google Scholar 

  40. Cheeger, J.: Hodge theory of complex cones. In: Analyse et Topologie sur les Espaces Singuliers. (Astérisque, vol. 101–102). Soc. Math. Fr., 1983, pp. 118–134

    Google Scholar 

  41. Cheeger, J., Goresky, M., MacPherson, R.: L2-cohomology and intersection homology of singular algebraic varieties. In: Ann. Math. Studies, vol.102. Princeton Univ. Press, 1982, pp. 303–340

    Google Scholar 

  42. Cheeger, J., Simons, J.: Differential characters and geometric invariants. In: Geometry and Topology (Lecture Notes in Mathematics, vol.1167). Springer, 1985, pp. 50–80

    Google Scholar 

  43. Chen, K.-T.: Iterated path integrals. Bull. Am. Math. Soc. 83 (1977) 831–879

    MATH  Google Scholar 

  44. Chen, K.-T.: Reduced bar constructions on de Rham complexes. In: Algebra, Topology and Category Theory (ed. A. Heller and M. Tierney). Academic Press, 1976, pp. 19–32

    Google Scholar 

  45. Chen, K.-T.: Circular bar construction. J. Alg. 57 (1979) 466–483

    MATH  Google Scholar 

  46. Clemens, C.H.: Picard-Lefschetz theorem for families of nonsingular algebraic varieties acquiring ordinary singularities. Trans. Am. Soc. 136 (1969) 93–108

    MATH  MathSciNet  Google Scholar 

  47. Clemens, C.H.: Degeneration of Kahler manifolds. Duke Math. J. 44 (1977) 215–290

    MATH  MathSciNet  Google Scholar 

  48. Corlette, K: Flat G-bundles with canonical metrics. J. Diff. Geom. 28 (1988) 361–382

    MATH  MathSciNet  Google Scholar 

  49. Deligne, P.: Théorème de Lefschetz et critères de dégénérescence de suites spectrales. Publ. Math. I.H.E.S. 35 (1968) 107–126

    MATH  Google Scholar 

  50. Deligne, P.: Equations différentielles à points singuliers réguliers. (Lecture Notes in Mathematics, vol. 163). Springer, 1970

    MATH  Google Scholar 

  51. Deligne, P.: Travaux de Griffiths. Séminaire Bourbaki, Exposé 376, June 1970. (Lecture Notes in Mathematics, vol. 180). Springer, 1971, pp. 213–237

    Google Scholar 

  52. Deligne, P.: Théorie de Hodge I. Actes, Congrès Intern. Math. Nice 1970, pp. 425–430. II, Publ. Math. I.H.E.S. 40 (1971) 5–58. Ill, Publ. Math. I.H.E.S. 44 (1974) 5–77

    Google Scholar 

  53. Deligne, P.: La conjecture de Weil I. Publ. Math. I.H.E.S. 43 (1974) 273–308. II, Publ. Math. I.H.E.S. 52 (1980) 137–252

    Google Scholar 

  54. Deligne, P.: Structures de Hodge mixtes réelles. Appendix to: On the SL(2)- orbits in Hodge theory, by E. Cattani and A. Kaplan

    Google Scholar 

  55. Deligne, R: Théorie de Hodge irrégulière. Handwritten notes 1984

    Google Scholar 

  56. Deligne, P.: Un théorème de finitude pour la monodromie. In: Discrete Groups in Geometry and Analysis (Papers in the Honor of G. D. Mostow on His Sixtieth Birthday). Progress in Mathematics, vol. 67. Birkhäuser, 1987, pp. 1–19

    Google Scholar 

  57. Deligne, P.: Valeurs de fonctions L et périodes d’intégrales. In: Automorphic Forms, Representations and L-functions. Proc. Symp. Pure Math, vol. 33, Part 2. Amer. Math. Soc. 1979, pp. 313–346

    Google Scholar 

  58. Deligne, P, Griffiths, P, Morgan, J, Sullivan, D.: Real homotopy theory of Kähler manifolds. Invent. math. 29 (1975) 245–274

    MATH  MathSciNet  Google Scholar 

  59. Deligne, P, Milne, J.-S, Ogus, A, Shih, K.-Y.: Hodge cycles, motives and Shimura varieties. (Lecture Notes in Mathematics, vol. 900). Springer, 1982

    MATH  Google Scholar 

  60. Du Bois, P.: Complexe de de Rham filtré d’une variété singulière. Bull. Soc. Math. Fr. 109 (1981) 41–81

    MATH  Google Scholar 

  61. Du Bois, P.: Structure de Hodge mixte sur la cohomologie évanescente. In: Ann. Inst. Fourier 35 (1985) 191–213

    MATH  Google Scholar 

  62. Durfee, A.: A naive guide to mixed Hodge theory. In: Proc. Symp. Pure Math, vol.40, part I. Amer. Math. Soc. 1983, pp.313–320

    Google Scholar 

  63. Durfee, A, Hain, R: Mixed Hodge structures on the homotopy of links. Math. Ann. 280 (1988) 69–83

    MATH  MathSciNet  Google Scholar 

  64. Durfee, A, Saito, M.: Mixed Hodge structures on intersection cohomology of links. Compos. Math. 76 (1990) 49–67

    MATH  MathSciNet  Google Scholar 

  65. El Zein, F.: Complexe dualisant et applications à la classe fondamentale d’un cycle. Mémoire Bull. Soc. Math. Fr. 58 (1978) 5–66

    Google Scholar 

  66. El Zein, F.: Complexe de Hodge mixte filtré. CRAS Paris 295 (1982) 669–672

    MATH  Google Scholar 

  67. El Zein, F.: Mixed Hodge structures. Proc. Symp. Pure Math., vol.40, Part I, 1983, 345–352

    Google Scholar 

  68. El Zein, F.: Dégénérescence diagonale I. C.R.A.S. Paris 296 (1983) 51–54. II, C.R.A.S. Paris 296 (1983) 199–202

    Google Scholar 

  69. El Zein, F.: Suites spectrales de structures de Hodge mixtes. (Astérisque, vol. 130). Soc. Math. Fr. 1985, pp. 308–329

    Google Scholar 

  70. El Zein, F.: Théorie de Hodge à coefficients: étude locale. C.R.A.S. Paris 307 (1988) 593–598

    MATH  Google Scholar 

  71. El Zein, F.: Théorie de Hodge des cycles évanescents. Ann. Sci. Ec. Norm. Sup. 19 (1986) 107–194

    MATH  Google Scholar 

  72. El Zein, F.: Mixed Hodge structures. Trans. Amer. Math. Soc. 275 (1983) 71–106

    MathSciNet  Google Scholar 

  73. El Zein, F, Zucker, S.: Extendability of normal functions associated to algebraic cycles. In: Topics in Transcendental Algebraic Geometry, P. Griffiths, ed. Princeton University Press, vol. 106, 1986, pp. 269–288

    Google Scholar 

  74. Esnault, H, Viehweg, E.: Deligne-Beilinson cohomology. In: Beilinson’s Conjectures on Special Values of L-functions. (Perspectives in Mathematics, vol.4). Academic Press, pp.43–91

    Google Scholar 

  75. Faltings, G.: Arakelov’s theorem for abelian varieties. Invent. math. 73 (1983) 337–347

    MATH  MathSciNet  Google Scholar 

  76. Faltings, E.: p-adic Hodge theory. J. Amer. Math. Soc. 1 (1988) 255–299

    MATH  MathSciNet  Google Scholar 

  77. Friedman, S.: Global smoothings of varieties with normal crossings. Ann. Math. 118 (1983) 75–114

    MATH  Google Scholar 

  78. Gabber, O.: Pureté de la cohomologie de Goresky-MacPherson. Prépublication I.H.E.S. 1981

    Google Scholar 

  79. Gaffney, M.: A special Stokes’ theorem for complete Riemannian manifolds. Ann. Math. 60 (1954) 140–145

    MATH  MathSciNet  Google Scholar 

  80. Gaffney, M.: Hilbert space methods in the theory of harmonic integrals. Trans. Amer. Math. Soc. 78 (1955) 426–444

    MATH  MathSciNet  Google Scholar 

  81. Gillet, H.: Deligne homology and Abel-Jacobi maps. Bull. Amer. Math. Soc. 10 (1984) 284–288

    MathSciNet  Google Scholar 

  82. Godement, R.: Topologie Algébrique et Théorie des Faisceaux. Hermann, 1958

    MATH  Google Scholar 

  83. Goldman, W., Millson, J.: The deformation theory of representations of fundamental groups of compact Kâhler manifolds. Publ. Math. I.H.E.S. 67 (1988) 43–96

    MATH  MathSciNet  Google Scholar 

  84. Goresky, M., MacPherson, R.: Intersection homology. Topology 19 (1980) 77–129. II, Invent, math. 72 (1983)

    MathSciNet  Google Scholar 

  85. Griffiths, P.: Periods of integrals on algebraic manifolds I. Amer. J. Math. 90 (1968) 568–626. II, Amer. J. Math. 90 (1968) 805–865. III, Publ. Math. H.E.S. (1970) 228–296

    Google Scholar 

  86. Griffiths, P.: Algebraic cycles on algebraic manifolds. In: Algebraic Geometry. (Papers presented at the Bombay Colloquium 1969). Oxford University Press, 1969, pp. 93–191

    Google Scholar 

  87. Griffiths, P.: On the periods of certain rational integrals. Ann. Math. 90 (1969) 460–541

    MATH  Google Scholar 

  88. Griffiths, P.: Periods of integrals on algebraic manifolds: Summary of main results and discussion of open problems. Bull. Amer. Math. Soc. 76 (1970), 228–296

    MATH  MathSciNet  Google Scholar 

  89. Griffiths, S., Schmid, W.: Recent developments in Hodge theory: A discussion of techniques and results. In: Proc. Bombay Colloq. on Discrete Subgroups of Lie Groups (Bombay, 1973). Oxford University Press, 1975, pp. 31–127

    Google Scholar 

  90. Grothendieck, A.: On the de Rham cohomology of algebraic varieties. Publ. Math. I.H.E.S. 29 (1966) 96–103

    Google Scholar 

  91. Guillén, F., Navarro Aznar, V., Puerta, F.: Théorie de Hodge via schémas cubiques. Preprint 1982

    Google Scholar 

  92. Guillén, F., Puerta, F.: Hyperrésolutions cubiques et applications à la théorie de Hodge-Deligne. In: Hodge Theory Proceedings, Sant Cugat, 1985. (Lecture Notes in Mathematics, vol. 1246). Springer, 1987

    Google Scholar 

  93. Guillén, F., Navarro Aznar, V., Pascual-Gainza, P., Puerta, F.: Hyperrésolutions cubiques et descente cohomologique. (Lecture Notes in Mathematics, vol. 1335). Springer, 1988

    MATH  Google Scholar 

  94. Hain, R.: The de Rham homotopy theory of complex algebraic varieties I, II. K-Theory 1 (1987) 271–324, 481–497

    MATH  MathSciNet  Google Scholar 

  95. Hain, R.: The geometry of the mixed Hodge structure on the fundamental group. In: Proc. Symp. Pure Math. Amer. Math. Soc. 1987, pp.247–282

    Google Scholar 

  96. Hain, R., MacPherson, R.: Higher logarithms. I11. J. Math. 34, no. 2 (1990) 392–475

    MATH  MathSciNet  Google Scholar 

  97. Hain, R, Zucker, S.: Unipotent variations of mixed Hodge structure. Invent. math. 88 (1987) 83–124

    MATH  MathSciNet  Google Scholar 

  98. Hain, R, Zucker, S.: A guide to unipotent variations of mixed Hodge structures. In: Hodge Theory Proceedings, Sant Cugat, 1985. (Lecture Notes in Mathematics, vol. 1246). Springer, 1987, pp. 92–106

    Google Scholar 

  99. Harris, B.: Harmonic volumes. Acta Math. 150 (1983) 91–123

    MATH  MathSciNet  Google Scholar 

  100. Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero. Ann. Math. 79 (1964) 109–326

    MATH  MathSciNet  Google Scholar 

  101. Hitchin, N.: The self-duality equation over a Riemann surface. Proc. Lond. Math. Soc. 55 (1987) 59–126

    MATH  MathSciNet  Google Scholar 

  102. Hodge, W.V.D.: The Theory and Applications of Harmonic Integrals. Cambridge University Press, 1941

    Google Scholar 

  103. Hsiang, W.-C, Pati, V.: L2-cohomology of normal algebraic surfaces. Invent. math. 81 (1985) 395–412

    MATH  MathSciNet  Google Scholar 

  104. Janssen, U.: Deligne homology, Hodge-D conjecture and motives. In: Beilin- son’s Conjectures on Special Values of L-functions. (Perspectives in Mathematics, vol.4). Acad. Press, 1988, pp.305–372

    Google Scholar 

  105. Kashiwara, M.: On the maximally overdetermined system of linear differential equations I. Publ. RIMS Kyoto University 10 (1974/75) 563–579 563–579

    MATH  Google Scholar 

  106. Kashiwara, M.: B-functions and holonomic systems, rationality of roots of B-functions. Invent, math. 38 (1976) 33–53

    MATH  MathSciNet  Google Scholar 

  107. Kashiwara, M.: On the holonomic systems of differential equations II. Invent, math. 49 (1978) 121–135

    MATH  MathSciNet  Google Scholar 

  108. Kashiwara, M.: Vanishing cycles and holonomic systems of differential systems of differential equations. In: Algebraic Geometry, M. Raynaud and T. Shioda, eds. (Lecture Notes in Mathematics, vol. 1016). Springer, 1983, pp. 134–142

    Google Scholar 

  109. Kashiwara, M.: The asymptotic behavior of a variation of polarized Hodge structure. Publ. RIMS. Kyoto University 21 (1985) 853–875

    MATH  MathSciNet  Google Scholar 

  110. Kashiwara, M.: A study of a variation of mixed Hodge structure. Publ. RIMS. Kyoto University 22 (1986) 991–1024

    MATH  MathSciNet  Google Scholar 

  111. Kashiwara, M, Kawai, T.: On the holonomic systems of microdifferential equations III. Systems with regular singularities. Publ. R.I.M.S. Kyoto University 17 (1981) 813–979

    MATH  MathSciNet  Google Scholar 

  112. Kashiwara, M, Kawai, T.: The Poincaré lemma for variations of polarized Hodge structures. Publ. R.I.M.S. Kyoto University 23 (1987) 345–407

    MATH  MathSciNet  Google Scholar 

  113. Kashiwara, M, Kawai, T.: Hodge structures and holonomic systems. Proc. Japan Acad. (A) 62 (1986) 1–4

    MATH  MathSciNet  Google Scholar 

  114. Katz, N.: The regularity theorem in algebraic geometry. In: Actes, Congrès Intern. Math. Nice 1970, 1 (1970) 437–443

    Google Scholar 

  115. Landman, A.: On the Picard-Lefschetz transformation for algebraic manifolds acquiring general singularities. Trans. Amer. Math. Soc. 181 (1973) 89–126

    MATH  MathSciNet  Google Scholar 

  116. Laumon, S.: Sur la catégorie dérivée des D-modules filtrés. In: Algebraic Geometry, M. Raynaud and T. Shioda, eds. (Lecture Notes in Mathematics, vol. 1016). Springer, 1983, pp. 151–237

    Google Scholar 

  117. Lê, D.-T.: Sur les noeuds algébriques. Comp. Math. 25 (1972) 281–321

    MATH  Google Scholar 

  118. Lefschetz, S.: L’analysis Situs et la Géométrie Algébrique. Gauthier-Villars, 1924

    MATH  Google Scholar 

  119. Lieberman, D.: Higher Picard varieties. Amer. J. Math. 90 (1968) 1165–1199

    MATH  MathSciNet  Google Scholar 

  120. Loeser, F.: Evaluation d’intégrales et théorie de Hodge. In: Hodge Theory Proceedings, Sant Cugat, 1985. (Lecture Notes in Mathematics, vol. 1246). Springer, 1987, pp. 125–142

    Google Scholar 

  121. Looijenga, E.: L2-cohomology of locally symmetric varieties. Comp. Math. 67 (1988) 3–20

    MATH  MathSciNet  Google Scholar 

  122. Malgrange, B.: Polynôme de Bernstein-Sato et cohomologie évanescente. In: Analyse et Topologie sur les Espaces Singuliers. (Astérisque, vol. 101–102). Soc. Math. Fr. 1983, pp. 243–267

    Google Scholar 

  123. MacPherson, R, Vilonen, K.: Elementary construction of perverse sheaves. Invent, math. 84 (1986) 403–435

    MATH  MathSciNet  Google Scholar 

  124. Mebkhout, Z.: Une équivalence de catégories et une autre équivalence de catégories. Ark. Math. 50 (1984) 51–88

    MathSciNet  Google Scholar 

  125. Mebkhout, Z.: Le formalisme des six opérations de Grothendieck pour les D-modules cohérents. (Travaux en cours, vol. 35). Hermann, 1989

    Google Scholar 

  126. Milne, J.S.: Canonical models of (mixed) Shimura varieties and automorphic vector bundles. In: Automorphic Forms, Shimura Varieties and L-Functions, vol. I. Academic Press, 1990, pp. 283–414

    Google Scholar 

  127. Morgan, J.: The algebraic topology of smooth algebraic varieties. Publ. Math. I.H.E.S. 48 (1978) 137–204. Correction, Publ. Math. I.H.E.S. 64 (1986)

    MATH  Google Scholar 

  128. Nagase, M.: Remarks on the L2-cohomology of singular algebraic surfaces. J. Math. Soc. Japan 41 (1989) 97–116

    MATH  MathSciNet  Google Scholar 

  129. Nagase, M.: Pure Hodge structure on the harmonic L2-forms on singular algebraic surfaces. Publ. RIMS. Kyoto University 24 (1988) 1005–1023

    MATH  MathSciNet  Google Scholar 

  130. Navarro Aznar, V.: Sur la théorie de Hodge-Deligne. Invent, math. 90 (1987) 11–76

    MATH  MathSciNet  Google Scholar 

  131. Navarro Aznar, V.: Sur la théorie de Hodge des variétés algébriques à singularités isoleés. (Astérisque, vol. 130). Soc. Math. Fr. 1985, pp. 272–307

    Google Scholar 

  132. Navarro Aznar, V.: Sur les structures de Hodge mixtes associées aux cycles évanescents. (Lecture Notes in Mathematics, vol. 1246). Springer, 1987, pp. 143–153

    Google Scholar 

  133. Ohsawa, T.: On extension of Hodge theory to Kähler spaces with isolated singularities of restricted type. Publ. RIMS. Kyoto University 24 (1988) 253–263

    MATH  MathSciNet  Google Scholar 

  134. Pham, FStructures de Hodge mixtes associées à un germe de fonction à point singulier isolé. In: Analyse et Topologie sur les Espaces Singuliers. (Astérisque, vol. 101–102). Soc. Math. Fr. 1983, pp. 268–285

    Google Scholar 

  135. Pulte, M.: The fundamental group of a Riemann surface: Mixed Hodge structures and algebraic cycles. Duke J. Math. 57 (1988) 721–760

    MATH  MathSciNet  Google Scholar 

  136. Ramis, J.-P.: Variations sur le thème GAGA. In: P. Lelong, H. Skoda (eds.) Séminaire Pierre Lelong – Henri Skoda (Analyse) Année 1976/77. (Lecture Notes in Mathematics, vol.694). Springer, 1978, pp.228–289

    Google Scholar 

  137. Ramakrishnan, D.: A regulator for curves via the Heisenberg group. Bull. Amer. Math. Soc. 5 (1981) 191–195

    MATH  MathSciNet  Google Scholar 

  138. Ramakrishnan, D.: Regulators, algebraic cycles and values of L-functions. In: Algebraic K-Theory and algebraic number theory. (Contemporary Mathematics, vol.83). Amer. Math. Soc. 1989, pp. 183–310

    Google Scholar 

  139. Ramakrishnan, D.: Valeurs de fonctions L des surfaces d’Hilbert-Blumenthal en s = 1. C.R.A.S. Paris 301 (1985) 809–812

    MATH  MathSciNet  Google Scholar 

  140. Ramakrishnan, D.: Periods of integrals arising from K 1 of Hilbert- Blumenthal surfaces. Preprint 1984

    Google Scholar 

  141. Ramakrishnan, D.: Arithmetic of Hilbert-Blumenthal surfaces. In: Number theory (Montréal 1985) CMS Conf. Proc. Amer. Math. Soc. 1987, pp. 285–370

    Google Scholar 

  142. Ramakrishnan, D.: Analogs of the Bloch-Wigner function for higher polylogarithms. In: Number theory (Montréal 1985) CMS Conf. Proc. Amer. Math. Soc. 1987, pp. 371–376

    Google Scholar 

  143. Rapoport, M., Schappacher, N, Schneider, P.: Beilinson’s Conjectures on Special Values of L-functions. (Perspectives in Math., vol.4). Academic Press, 1988

    MATH  Google Scholar 

  144. Rapoport, M, Zink, T.: Über die lokale Zetafunktionen von Shimuravarietäten. Monodromie filtration und verschwindende Zyklen in ungleicher Charakteristik. Invent, math. 68 (1982) 21–101

    MATH  MathSciNet  Google Scholar 

  145. Saito, K.: Period mapping associated to a primitive form. Publ. RIMS. Kyoto University 19 (1983) 1231–1264

    MATH  Google Scholar 

  146. Saito, MGauss-Manin systems and mixed Hodge structure. Proc. Japan Acad. Sci. (A) 581982) 29–32. Supplement in: Analyse et Topologie sur les Espaces Singuliers. (Astérisque, vol. 101–102). Soc. Math. Fr. 1983, pp. 320–331

    MATH  Google Scholar 

  147. Saito, M.: Hodge filtrations on Gauss-Manin systems. J. Fac. Sci. Univ. Tokyo, Sect. IA (Math.) 30 (1984) 489–498. II, Proc. Japan Acad. (A) 59 (1983)

    MATH  MathSciNet  Google Scholar 

  148. Saito, M.: Modules de Hodge polarisables. Publ. RIMS. Kyoto University 24 (1988) 849–995

    MATH  Google Scholar 

  149. Saito, M.: Hodge structure via filtered D-modules. In: Astérisque, vol. 130, 1985, pp. 342–351

    Google Scholar 

  150. Saito, M.: Mixed Hodge modules. Proc. Japan Acad. (A) 62 (1986) 360–363

    MATH  Google Scholar 

  151. Saito, M.: On the derived category of mixed Hodge modules. Proc. Japan Acad. (A) 62 (1986) 364–366

    MATH  Google Scholar 

  152. Saito, M.: On the structure of Brieskorn lattice. Ann. Inst. Fourier Grenoble 39 (1989) 27–72

    MATH  MathSciNet  Google Scholar 

  153. Saito, M.: Vanishing cycles and mixed Hodge modules. Preprint 1988

    Google Scholar 

  154. Saito, M.: Decomposition theorem for proper Kähler morphisms. Preprint I.H.E.S./M/88/34

    Google Scholar 

  155. Saito, M.: Mixed Hodge modules and admissible variations. CRAS Paris 309 (1989) 351–356

    MATH  Google Scholar 

  156. Saito, M.: Mixed Hodge modules. Publ. RIMS Kyoto University 26 (1990) 221–333

    MATH  Google Scholar 

  157. Saito, M.: Introduction to mixed Hodge modules. RIMS-605, preprint 1987

    Google Scholar 

  158. Saito, M, Zucker, S.: The kernel spectral sequence of vanishing cycles. Duke Math. J. (to appear)

    Google Scholar 

  159. Saper, L.: L2-cohomology and intersection cohomology of certain algebraic varieties with isolated singularities. Invent. math. 82 (1985) 207–255

    MATH  MathSciNet  Google Scholar 

  160. Saper, L, Stern, M.: L2-cohomology of arithmetic varieties. Proc. Nat Acad. Sci. USA 84 (1987) 5516–5519

    MATH  MathSciNet  Google Scholar 

  161. Scherk, J.: A note on two local Hodge filtrations. In: Proc. Symp. Pure Math. 46, Part 2. Amer. Math. Soc. 1983, pp. 473–477

    Google Scholar 

  162. Scherk, J, Steenbrink, J.: On the mixed Hodge structure of the Milnor fiber. Math. Ann. 271 (1985) 641–665

    MATH  MathSciNet  Google Scholar 

  163. Schmid, W.: Variation of Hodge structure: The singularities of the period mapping. Invent. math. 22 (1973) 211–319

    MATH  MathSciNet  Google Scholar 

  164. Schneider, P.: Introduction to the Beilinson conjectures. In: Beilinson’s Conjectures on Special Values of L-functions. (Perspectives in Math, vol. 4). Academic Press, 1988, pp. 1–35

    Google Scholar 

  165. Serre, J.P.: Géométrie algébrique et géométrie analytique. Ann. Inst. Fourier 6 (1955–56) 1–42

    MATH  Google Scholar 

  166. Grothendieck, A.: Séminaire de Géométrie Algébrique du Bois-Marie. Cohomologie Locale des Faisceaux Cohérents et Théorèmes de Lefschetz Locaux et Globaux. North-Holland, 1968

    MATH  Google Scholar 

  167. Artin, M, Grothendieck, A, Verdier, J.-L.: Séminaire de Géométrie Algébrique du Bois-Marie. Théorie des topos et cohomologie étale des schémas (SGA 4). (Lecture Notes in Mathematics, vol. 269). Springer, 1972. II, (Lecture Notes in Mathematics, vol.270). 1972. Ill, (Lecture Notes in Mathematics, vol. 305), 1973

    Google Scholar 

  168. Grothendieck, A.: Séminaire de Géométrie Algébrique du Bois-Marie. Groupes de Monodromie en Géométrie Algébrique (SGA 7 I). (Lecture Notes in Mathematics, vol. 288). Springer, 1972. Groupes de Monodromie en Géométrie Algébrique (SGA 7 II), (Lecture Notes in Mathematics, vol. 340). Springer, 1973

    Google Scholar 

  169. Simpson, C.: Constructing variations of Hodge structures using Yang-Mills theory and application to uniformization. J. Amer. Math. Soc. 1 (1988) 867–918

    MATH  MathSciNet  Google Scholar 

  170. Simpson, C.: Moduli of representations of the fundamental group of a smooth projective variety. Preprint 1989

    Google Scholar 

  171. Simpson, C.: Higgs bundles and local systems. Preprint 1989. Publ. Math., Inst. Hautes Etudes Sci. 75 (1992) 5–95

    MATH  Google Scholar 

  172. Simpson, C.: Harmonic bundles on non-compact curves. Preprint 1989. J. Am. Math. Soc. 3 (1990) 713–770

    MATH  Google Scholar 

  173. Soulé, C.: Régulateurs. Exposé Séminaire Bourbaki 644, in: Séminaire Bourbaki 84/85. (Astérisque, vol. 133–134). Soc. Math. Fr. 1986, pp. 237–253

    Google Scholar 

  174. Soulé, C.: Connexions et classes caractéristiques de Beilinson. In: Algebraic K-Theory and Algebraic Number Theory. (Contemporary Mathematics, vol. 83). Amer. Math. Soc. 1989, pp. 349–376

    Google Scholar 

  175. Steenbrink, J.: Limits of Hodge structures. Invent. math. 31 (1976) 229–257

    MATH  MathSciNet  Google Scholar 

  176. Steenbrink, J.: Mixed Hodge structures on the vanishing cohomology. In: Real and Complex Singularities. Sijthoff and Noordhoff, 1977, pp. 525–563

    Google Scholar 

  177. Steenbrink, J.: Mixed Hodge structures associated with isolated singularities. In: Proc. Symp. Proc. Pure Math. Amer. Math. Soc., vol.40, Part 2, 1983, pp. 513–536

    MathSciNet  Google Scholar 

  178. Steenbrink, J.: The spectrum of hypersurface singularities. In: Proc. of the Luminy Conference on Hodge Theory, 1987. Astérisque. Soc. Math. Fr. (to appear)

    Google Scholar 

  179. Steenbrink, J., van Doom, R.: A supplement to the monodromy theorem. Abh. Math. Sem. Universität Hamburg 59 (1989)

    Google Scholar 

  180. Steenbrink, J., Zucker, S.: Variation of mixed Hodge structure, I. Invent. math. 80 (1985) 489–542

    MATH  MathSciNet  Google Scholar 

  181. Sullivan, D.: Infinitesimal computations in topology. Publ. Math. I.H.E.S. 47 (1977) 269–331

    MATH  Google Scholar 

  182. Tanisaki, T.: Hodge modules, equivariant K-theory and Hecke algebras. Publ. RIMS. Kyoto University 23 (1987) 841–879

    MATH  MathSciNet  Google Scholar 

  183. Tate, J.: Les conjectures de Stark sur les fonctions L d’Artin en s= 0, notes d’un cours à Orsay (rédigées par D. Bernardi et N. Schappacher). (Progress in Mathematics, vol.47). Birkhäuser, 1984

    Google Scholar 

  184. Topics in Transcendental Algebraic Geometry. P. Griffiths, ed. (Ann. of Math. Study, vol. 106). Princeton University Press, 1984

    MATH  Google Scholar 

  185. Varchenko, A.: The asymptotics of holomorphic forms determine a mixed Hodge structure. Soviet Math. Dokl. 22 (1980) 772–775

    MATH  Google Scholar 

  186. Varchenko, A.: Asymptotic Hodge structure in the vanishing cohomology. Math. USSR Izv. 18 (1982) 469–512

    MATH  Google Scholar 

  187. Varchenko, A.: On the monodromy operator in vanishing cohomology and the operator of multiplication by f in the local ring. Soviet Math. Dokl. 24 (1982) 248–252

    Google Scholar 

  188. Varchenko, A.: A lower bound for the codimension of the strata µ = const in terms of the mixed Hodge structure. Moscow Univ. Math. Bull. 37–6 (1982) 30–33

    Google Scholar 

  189. Verdier, J.-L.: Stratifications de Whitney et théorème de Bertini-Sard. Invent. math. 36 (1976) 295–312

    MATH  MathSciNet  Google Scholar 

  190. Verdier, J.-L.: Catégories dérivées, état 0. Séminaire de Géométrie Algébrique du Bois-Marie (SGA 1/2) (Lecture Notes in Mathematics, vol. 569). Springer, 1977, pp.262–311

    Google Scholar 

  191. Verdier, J.-L.: Extension of a perverse sheaf over a closed subspace. (Systèmes différentiels et Singularités. Astérisque, vol. 130). Soc. Math. Fr., 1985, pp. 210–217

    Google Scholar 

  192. Zucker, S.: Hodge theory with degenerating coefficients: L2-cohomology in the Poincaré metric. Ann. Math. 109 (1979) 415–476

    MATH  MathSciNet  Google Scholar 

  193. Zucker, S.: Variation of mixed Hodge structure. II, Invent. math. 80 (1985) 543–565

    MATH  MathSciNet  Google Scholar 

  194. Zucker, S.: L2-cohomology of warped products and arithmetic groups. Invent. math. 70 (1982) 169–218

    MATH  MathSciNet  Google Scholar 

  195. Zucker, S.: Degeneration of mixed Hodge structures. In: Proc. Symp. Pure Math. Amer. Math. Soc, vol.46, 1987, pp.283–293

    MathSciNet  Google Scholar 

  196. Zucker, S.: The Hodge structures on the intersection homology of varieties with isolated singularities. Duke Math. J. 55 (1987) 603–616

    MATH  MathSciNet  Google Scholar 

  197. Zucker, S.: L2-cohomology and intersection homology of locally symmetric varieties III. In: Théorie de Hodge: Luminy, Juin 1987. (Astérisque, vol. 179–180). Soc. Math. Fr. 1989, pp. 245–278

    Google Scholar 

  198. Zucker, S.: Locally homogeneous variations of Hodge structure. L’Enseign. Math. 27 (1981) 243–276

    MATH  MathSciNet  Google Scholar 

  199. Zucker, S.: The Cheeger-Simons invariant as a Chern class. In: Algebraic Analysis, Geometry and Number Theory. (Proc. JAMI Inaugural Conference). JHU Press, 1989, pp. 397–417

    Google Scholar 

Download references

Authors
  1. Jean-Luc Brylinski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Steven Zucker
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Brylinski, JL., Zucker, S. (1998). An Overview of Recent Advances in Hodge Theory. In: Complex Manifolds. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-61299-2_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-61299-2_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-62995-5

  • Online ISBN: 978-3-642-61299-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation