A note on the Atiyah-Bott fixed point formula

Abstract

Let / be a holomorphic self map of a compact complex analytic manifold X The differential of / commutes with 5 and, hence, induces an endomorphism of the 5-complex of X. If / 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 / near the fixed points. For example, if X is a curve, this Lefschetz number is the sum of the residues of (z — /(z)) 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.