On a holomorphic Lefschetz formula in strictly pseudoconvex subdomains of complex manifolds

Abstract

The classical Lefschetz formula expresses the number of fixed points of a continuous map in terms of the transformation induced by on the cohomology of . In 1966, Atiyah and Bott extended this formula to elliptic complexes over a compact closed manifold. In particular, they obtained a holomorphic Lefschetz formula on compact complex manifolds without boundary. Brenner and Shubin (1981, 1991) extended the Atiyah-Bott theory to compact manifolds with boundary. On compact complex manifolds with boundary the Dolbeault complex is not elliptic, therefore the Atiyah-Bott theory is not applicable. Bypassing difficulties related to the boundary behaviour of Dolbeault cohomology, Donnelly and Fefferman (1986) obtained a formula for the number of fixed points in terms of the Bergman metric. The aim of this paper is to obtain a Lefschetz formula on relatively compact strictly pseudoconvex subdomains of complex manifolds with smooth boundary, that is, to find the total Lefschetz number for a holomorphic endomorphism of the Dolbeault complex and to express it in terms of local invariants of the fixed points of .