The Atiyah-Bott-Lefschetz formula for elliptic operators on manifolds with conical singularities.

*(English. Russian original)*Zbl 0962.58007
Dokl. Math. 60, No. 1, 54-57 (1999); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 367, No. 3, 303-306 (1999).

From the text: The famous Atiyah-Bott-Lefschetz formula, relates the Lefschetz number of a geometric endomorphism of an elliptic complex on a closed smooth manifold \(M\) to fixed points of the smooth map \(g: M\to M\) that corresponds to this endomorphism. Later, this formula was generaized in various directions. We consider the case of manifolds with conical singularities. The theory of such manifolds and elliptic theory on them are well developed at present. Thus, the problem of finding the Lefschetz number is rather natural in this case. We present the corresponding Atiyah-Bott-Lefschetz formula, restricting ourselves for simplicity to the case of a short complex that consists of a single elliptic operator. The Lefschetz number is then represented as the sum of the standard contribution of “interior” fixed points and the contribution of fixed control points. The latter contribution is expressed in natural terms of analytic families – conormal symbols of the elliptic operator itself and its geometric endomorphism.