A remark on the Grothendieck residue map

Carrell, James B.

doi:10.1090/S0002-9939-1978-0492408-1

A remark on the Grothendieck residue map
HTML articles powered by AMS MathViewer

by James B. Carrell PDF

Proc. Amer. Math. Soc. 70 (1978), 43-48 Request permission

Abstract:

The purpose of this note is to give a direct proof that a global integral over a compact complex manifold X can be evaluated on the zero set of a meromorphic vector field on X with isolated zeros via a Grothendieck residue morphism. A special case of this evaluation is the meromorphic vector field theorem of Baum and Bott [1]. The present proof suggests some complements of the M.V.F. Theorem which are contained in Theorem 2.

References

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
Paul Baum and Raoul Bott, Singularities of holomorphic foliations, J. Differential Geometry 7 (1972), 279–342. MR 377923
Raoul Bott, Vector fields and characteristic numbers, Michigan Math. J. 14 (1967), 231–244. MR 211416
J. B. Carrell and D. I. Lieberman, Vector fields and Chern numbers, Math. Ann. 225 (1977), no. 3, 263–273. MR 435456, DOI 10.1007/BF01425242
Shiing Shen Chern, Meromorphic vector fields and characteristic numbers, Scripta Math. 29 (1973), no. 3-4, 243–251. MR 412486, DOI 10.1016/0370-1573(77)90076-x
Walter Koppelman, The Cauchy integral for differential forms, Bull. Amer. Math. Soc. 73 (1967), 554–556. MR 216027, DOI 10.1090/S0002-9904-1967-11744-0
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

Integral formulae connected by Dolbeault’s isomorphism

20

B. Malgrange, Ideals of differentiable functions, Tata Institute of Fundamental Research Studies in Mathematics, vol. 3, Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1967. MR 0212575
Fonctions de plusieurs variables complexes, Lecture Notes in Mathematics, Vol. 409, Springer-Verlag, Berlin-New York, 1974 (French). Séminaire François Norguet, octobre 1970–décembre 1973; Edité par François Norguet; À la mémoire d’André Martineau. MR 0346176
Yue Lin L. Tong, Integral representation formulae and Grothendieck residue symbol, Amer. J. Math. 95 (1973), 904–917. MR 367255, DOI 10.2307/2373701
Yue Lin L. Tong, de Rham’s integrals and Lefschetz fixed point formula for $d^{\prime \prime }$cohomology, Bull. Amer. Math. Soc. 78 (1972), 420–422. MR 296355, DOI 10.1090/S0002-9904-1972-12926-4
Jean-Louis Verdier, Base change for twisted inverse image of coherent sheaves, Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968) Oxford Univ. Press, London, 1969, pp. 393–408. MR 0274464