Vol. 53, No. 2, 1974

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A note on the Atiyah-Bott fixed point formula

Lesley Millman Sibner and Robert Jules Sibner

Vol. 53 (1974), No. 2, 605–609

Let f be a holomorphic self map of a compact complex analytic manifold X. The differential of f commutes with and, hence, induces an endomorphism of the -complex of X. If f has isolated simple fixed points, the Lefschetz formula of Atiyah-Bott expresses the Lefschetz number of this endomorphism in terms of local data involving only the map f near the fixed points. For example, if X is a curve, this Lefschetz number is the sum of the residues of (z f(z))1 at the fixed points.

Using a well-known technique of Atiyah-Bott for computing trace formulas, we shall, in this note, give a direct analytic derivation of the Lefschetz number as a residue formula. The formula is valid for holomorphic maps having isolated, but not necessarily simple fixed points.

Mathematical Subject Classification 2000
Primary: 58G05
Secondary: 32L05
Received: 29 May 1973
Published: 1 August 1974
Lesley Millman Sibner
Robert Jules Sibner