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Zeroes of holomorphic vector fields and Grothendieck duality theory

Author: N. R. O’Brian
Journal: Trans. Amer. Math. Soc. 229 (1977), 289-306
MSC: Primary 58G10; Secondary 32L05
MathSciNet review: 0445562
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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.

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  • [1] Allen Altman and Steven Kleiman, Introduction to Grothendieck duality theory, Lecture Notes in Mathematics, Vol. 146, Springer-Verlag, Berlin-New York, 1970. MR 0274461
  • [2] 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 0232406,
  • [3] 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
  • [4] Raoul Bott, A residue formula for holomorphic vector-fields, J. Differential Geometry 1 (1967), 311–330. MR 0232405
  • [5] 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.
  • [6] -, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux, SGA 2, North-Holland, Amsterdam, 1968.
  • [7] Robin Hartshorne, Local cohomology, A seminar given by A. Grothendieck, Harvard University, Fall, vol. 1961, Springer-Verlag, Berlin-New York, 1967. MR 0224620
  • [8] 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.
  • [9] Robin Hartshorne, Residues and duality, Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. MR 0222093
  • [10] F. Reese Harvey, Integral formulae connected by Dolbeault’s isomorphism, Rice Univ. Studies 56 (1970), no. 2, 77–97 (1971). MR 0273067
  • [11] F. Hirzebruch, Topological methods in algebraic geometry, Third enlarged edition. New appendix and translation from the second German edition by R. L. E. Schwarzenberger, with an additional section by A. Borel. Die Grundlehren der Mathematischen Wissenschaften, Band 131, Springer-Verlag New York, Inc., New York, 1966. MR 0202713
  • [12] Lars Hörmander, An introduction to complex analysis in several variables, Second revised edition, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. North-Holland Mathematical Library, Vol. 7. MR 0344507
  • [13] G. Horrocks, Fixed point schemes of additive group actions, Topology 8 (1969), 233–242. MR 0244261,
  • [14] S. Lang, Differentiable manifolds, Addison-Wesley, Reading, Mass., 1972.
  • [15] 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
  • [16] N. R. O’Brian, Zeroes of holomorphic vector fields and the Grothendieck residue, Bull. London Math. Soc. 7 (1975), 33–38. MR 0374467,
  • [17] Jean-Pierre Ramis and Gabriel Ruget, Résidus et dualité, Invent. Math. 26 (1974), 89–131 (French). MR 0352522,
  • [18] L. M. Sibner and R. J. Sibner, A note on the Atiyah-Bott fixed point formula, Pacific J. Math. 53 (1974), 605–609. MR 0464326
  • [19] Yum Tong Siu, Techniques of extension of analytic objects, Marcel Dekker, Inc., New York, 1974. Lecture Notes in Pure and Applied Mathematics, Vol. 8. MR 0361154
  • [20] Domingo Toledo, On the Atiyah-Bott formula for isolated fixed points, J. Differential Geometry 8 (1973), 401–436. MR 0336766
  • [21] Yue Lin L. Tong, Integral representation formulae and Grothendieck residue symbol, Amer. J. Math. 95 (1973), 904–917. MR 0367255,

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Keywords: Holomorphic fixed-point formula, Atiyah-Bott formula, holomorphic vector field, isolated degenerate fixed-point, Grothendieck duality theory, Grothendieck residue, local cohomology, Bochner-Martinelli kernel, Cauchy kernel, Todd polynomials
Article copyright: © Copyright 1977 American Mathematical Society