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

The classical Lefschetz formula expresses the number of fixed points of a continuous map $f\colon M\to M$ in terms of the transformation induced by $f$ on the cohomology of $M$. 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 $X$ with smooth boundary, that is, to find the total Lefschetz number for a holomorphic endomorphism $f^*$ of the Dolbeault complex and to express it in terms of local invariants of the fixed points of $f$.