On a holomorphic Lefschetz formula in strictly pseudoconvex subdomains of complex manifolds
- Krasnoyarsk State University, Krasnoyarsk (Russian Federation)
- Potsdam University, Potsdam (Germany)
The classical Lefschetz formula expresses the number of fixed points of a continuous map f:M{yields}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{sup *} of the Dolbeault complex and to express it in terms of local invariants of the fixed points of f.
- OSTI ID:
- 21260468
- Journal Information:
- Sbornik. Mathematics, Vol. 195, Issue 12; Other Information: DOI: 10.1070/SM2004v195n12ABEH000865; Country of input: International Atomic Energy Agency (IAEA); ISSN 1064-5616
- Country of Publication:
- United States
- Language:
- English
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