A note on the Atiyah-Bott fixed point formula

Lesley Millman Sibner and Robert Jules Sibner

Vol. 53 (1974), No. 2, 605–609

DOI: 10.2140/pjm.1974.53.605

Abstract

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

Milestones

Received: 29 May 1973

Published: 1 August 1974