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Transactions of the American Mathematical Society

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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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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