Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Global residues and intersections on a complex manifold


Author: James R. King
Journal: Trans. Amer. Math. Soc. 192 (1974), 163-199
MSC: Primary 32C30
DOI: https://doi.org/10.1090/S0002-9947-1974-0338433-3
MathSciNet review: 0338433
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper is the study of a class of forms $ \eta $ on a complex manifold V which are smooth on $ V - W$ and have poles of kernel type on a complex submanifold W of codimension d; such a form is one whose pull-back to the monoidal transform of V along W has a logarithmic pole. A global existence theorem is proved which asserts that any smooth form $ \varphi $ on W of filtration s (no (p, q) components with $ p < s$) is the residue of a form $ \eta $ of filtration $ s + d$ such that $ d\eta $ is smooth on V. This result is used to construct global kernels for $ \bar \partial $ which establish similar global existence theorems for W with singularities. We then establish formulas connecting intersection and wedge product on the d-cohomology theory of Dolbeault which preserve the Hodge filtration. A number of results are also proved on the integrability of $ {f^\ast}\eta $ where f is a rather general holomorphic map.


References [Enhancements On Off] (What's this?)

  • [1] P. Baum and R. Bott, On the zeroes of meromorphic vector-fields, Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York, 1970, pp. 29-47. MR 41 #6248. MR 0261635 (41:6248)
  • [2] T. Bloom and M. Herrera, De Rham cohomology of an analytic space, Invent. Math. 7 (1969), 275-296. MR 40 #1601. MR 0248349 (40:1601)
  • [3] A. Borel and A. Haefliger, La classe d'homologie fondamentale d'un espace analytique, Bull. Soc. Math. France 89 (1961), 461-513. MR 26 #6990. MR 0149503 (26:6990)
  • [4] G. Bredon, Sheaf theory, McGraw-Hill, New York, 1967. MR 36 #4552. MR 0221500 (36:4552)
  • [5] Michael J. Cowen, Hermitian vector bundles and value distribution for Schubert cycles, Trans. Amer. Math. Soc. 180 (1973), 189-228. MR 0333252 (48:11577)
  • [6] P. Dolbeault, Formes différentielles et cohomologie sur une variété analytique complexe. I, Ann. of Math. (2) 64 (1956), 83-130. MR 18, 670. MR 0083166 (18:670e)
  • [7] R. N. Draper, Intersection theory in analytic geometry, Math. Ann. 180 (1969), 175-204. MR 40 #403. MR 0247134 (40:403)
  • [8] H. Federer, Geometric measure theory, Die Grundlehren der math. Wissenschaften, Band 153, Springer-Verlag, New York, 1969. MR 41 #1976. MR 0257325 (41:1976)
  • [9] R. Godement, Topologie algébrique et théorie des faisceaux, Actualités Sci. Indust., no. 1252, Hermann, Paris, 1958. MR 21 #1583. MR 0102797 (21:1583)
  • [10] P. Griffiths, Some results on algebraic cycles on algebraic manifolds, Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968), Oxford Univ. Press, London, 1969, pp. 93-191. MR 41 #1746. MR 0257092 (41:1746)
  • [11] R. Hartshorne, Local cohomology, A Seminar given by A. Grothendieck, Harvard University, Fall, 1961, Lecture Notes in Math., no. 41, Springer-Verlag, Berlin and New York, 1967. MR 37 #219. MR 0224620 (37:219)
  • [12] R. C. Gunning and H. Rossi, Analytic functions of several complex variables, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Englewood Cliffs, N.J., 1965. MR 31 #4927. MR 0180696 (31:4927)
  • [13] M. Herrera, Résidus multiples sur les espaces complexes, 1972 (preprint). MR 0481091 (58:1237)
  • [14] M. Herrera and D. Lieberman, Residues and principal values on complex spaces, Math. Ann. 194 (1971), 259-294. MR 0296352 (45:5413)
  • [15] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109-326. MR 33 #7333. MR 0199184 (33:7333)
  • [16] F. Hirzebruch, Neue topologische Methoden in der algebraischen Geometrie, Ergebnisse der Mathematik und ihrer Grenzgebiete, Heft 9, Springer-Verlag, Berlin, 1956; English transl., Die Grundlehren der math. Wissenschaften, Band 131, Springer-Verlag, New York, 1966. MR 18, 509; 34 #2573. MR 0082174 (18:509b)
  • [17] L. Kaup, Eine Künnethformel für Fréchetgarben, Math. Z. 97 (1967), 158-168. MR 35 #6868. MR 0216033 (35:6868)
  • [18] J. King, A residue formula for complex subvarieties, Proc. Carolina Conf. on Holomorphic Mappings and Minimal Surfaces (Chapel Hill, N.C., 1970), Department of Mathematics, University of North Carolina, Chapel Hill, N.C., 1970, pp. 43-56. MR 42 #7942. MR 0273061 (42:7942)
  • [19] -, The currents defined by analytic varieties, Acta Math. 127 (1971), 185-220. MR 0393550 (52:14359)
  • [20] -, The Abel-Jacobi map for families of Kähler manifolds (to appear).
  • [21] S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol. II, Wiley, New York, 1969. MR 1393941 (97c:53001b)
  • [22] P. Lelong, Fonctionnelles analytiques et fonctions entières, Université de Montréal, 1967.
  • [23] -, Fonctions plurisousharmoniques et formes différentielles positives, Gordon and Breach, New York; distributed by Dunod, Paris, 1969. MR 39 #4436.
  • [24] -, Séminaire Pierre Lelong, 1970, Springer-Verlag, New York, 1971.
  • [25] R. Narasimhan, Compact analytical varieties, Enseignement Math. (2) 14 (1968), 75-98. MR 39 #3530. MR 0242197 (39:3530)
  • [26] F. Norguet, Introduction à la théorie cohomologique des résidus, Séminaire Pierre Lelong, 1970, Springer-Verlag, New York, 1971. MR 0589934 (58:28648)
  • [27] J.-B. Poly, Sur un théorème de J. Leray en théorie des résidus, C. R. Acad. Sci. Paris Sér. A-B 274 (1972), A171-A174. MR 44 #6994. MR 0289807 (44:6994)
  • [28] G. de Rham, Variétés différentiables. Formes, courants, formes harmoniques, Actualités Sci. Indust., no. 1222, Hermann. Paris, 1955. MR 16, 957.
  • [29] L. Schwartz, Théorie des distributions, Nouvelle édition, entièrement corrigée refondue et augmentée, Hermann, Paris, 1966. MR 0209834 (35:730)
  • [30] W. Stoll, Value distribution of holomorphic maps into compact complex manifolds, Lecture Notes in Math., vol. 135, Springer-Verlag, New York, 1970. MR 42 #2040. MR 0267138 (42:2040)
  • [31] G. Stolzenberg, Volumes, limits, and extensions of analytic varieties, Lecture Notes in Math., no. 19, Springer-Verlag, New York, 1966. MR 34 #6156. MR 0206337 (34:6156)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 32C30

Retrieve articles in all journals with MSC: 32C30


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0338433-3
Keywords: Residue forms, intersection of analytic sets, Dolbeault cohomology, kernels on complex manifolds
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society