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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Zeroes of holomorphic vector fields and Grothendieck duality theory
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by N. R. O’Brian PDF
Trans. Amer. Math. Soc. 229 (1977), 289-306 Request permission

Abstract:

The holomorphic fixed point formula of Atiyah and Bott is discussed in terms of Grothendieck’s theory of duality. The treatment is valid for an endomorphism of a compact complex-analytic manifold with arbitrary isolated fixed points. An expression for the fixed point indices is then derived for the case where the endomorphism belongs to the additive group generated by a holomorphic vector field with isolated zeroes. An application and some examples are given. Two generalisations of these results are also proved. The first deals with holomorphic vector bundles having sufficient homogeneity properties with respect to the action of the additive group on the base manifold, and the second with additive group actions on algebraic varieties.
References
  • Allen Altman and Steven Kleiman, Introduction to Grothendieck duality theory, Lecture Notes in Mathematics, Vol. 146, Springer-Verlag, Berlin-New York, 1970. MR 0274461, DOI 10.1007/BFb0060932
  • M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes. II. Applications, Ann. of Math. (2) 88 (1968), 451–491. MR 232406, DOI 10.2307/1970721
  • Paul F. Baum and Raoul Bott, On the zeros of meromorphic vector-fields, Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York, 1970, pp. 29–47. MR 0261635
  • Raoul Bott, A residue formula for holomorphic vector-fields, J. Differential Geometry 1 (1967), 311–330. MR 232405
    • A. Grothendieck, Théorèmes de dualité pour les faisceaux algébriques cohérents, Séminaire Bourbaki, 9ième année 1956/57, 2ième éd. corrigée, Exposé 149, Secrétariat mathématique, Paris, 1957. —, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux, SGA 2, North-Holland, Amsterdam, 1968.
  • Robin Hartshorne, Local cohomology, Lecture Notes in Mathematics, No. 41, Springer-Verlag, Berlin-New York, 1967. A seminar given by A. Grothendieck, Harvard University, Fall, 1961. MR 0224620, DOI 10.1007/BFb0073971
    • A. Grothendieck, Eléments de géométrie algébrique, Inst. Hautes Études Sci. Publ. Math. Nos. 4, 8, 11, 17, 20, 24, 28, 32 (1960-1967). MR 36 #177a, b, c; 29 #1210; 30 #3885; 33 #7330; 36 #178; 39 #220.
  • Robin Hartshorne, Residues and duality, Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64; With an appendix by P. Deligne. MR 0222093, DOI 10.1007/BFb0080482
  • F. Reese Harvey, Integral formulae connected by Dolbeault’s isomorphism, Rice Univ. Stud. 56 (1970), no. 2, 77–97 (1971). MR 273067
  • F. Hirzebruch, Topological methods in algebraic geometry, Third enlarged edition, Die Grundlehren der mathematischen Wissenschaften, Band 131, Springer-Verlag New York, Inc., New York, 1966. New appendix and translation from the second German edition by R. L. E. Schwarzenberger, with an additional section by A. Borel. MR 0202713
  • Lars Hörmander, An introduction to complex analysis in several variables, Second revised edition, North-Holland Mathematical Library, Vol. 7, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. MR 0344507
  • G. Horrocks, Fixed point schemes of additive group actions, Topology 8 (1969), 233–242. MR 244261, DOI 10.1016/0040-9383(69)90013-5
    • S. Lang, Differentiable manifolds, Addison-Wesley, Reading, Mass., 1972.
  • Gheorghe Lusztig, Remarks on the holomorphic Lefschetz numbers, Analyse globale, (Sém. Math. Supérieures, No. 42, 1969) Presses Univ. Montréal, Montreal, Que., 1971, pp. 193–204. MR 0380821
  • N. R. O’Brian, Zeroes of holomorphic vector fields and the Grothendieck residue, Bull. London Math. Soc. 7 (1975), 33–38. MR 374467, DOI 10.1112/blms/7.1.33
  • Jean-Pierre Ramis and Gabriel Ruget, Résidus et dualité, Invent. Math. 26 (1974), 89–131 (French). MR 352522, DOI 10.1007/BF01435691
  • L. M. Sibner and R. J. Sibner, A note on the Atiyah-Bott fixed point formula, Pacific J. Math. 53 (1974), 605–609. MR 464326, DOI 10.2140/pjm.1974.53.605
  • Yum Tong Siu, Techniques of extension of analytic objects, Lecture Notes in Pure and Applied Mathematics, Vol. 8, Marcel Dekker, Inc., New York, 1974. MR 0361154
  • Domingo Toledo, On the Atiyah-Bott formula for isolated fixed points, J. Differential Geometry 8 (1973), 401–436. MR 336766
  • Yue Lin L. Tong, Integral representation formulae and Grothendieck residue symbol, Amer. J. Math. 95 (1973), 904–917. MR 367255, DOI 10.2307/2373701
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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 229 (1977), 289-306
  • MSC: Primary 58G10; Secondary 32L05
  • DOI: https://doi.org/10.1090/S0002-9947-1977-0445562-5
  • MathSciNet review: 0445562