Periodic points of holomorphic maps
We give criteria for a holomorphic map to have infinitely many periodic points that are based on the Atiyah-Bott formula for holomorphic Lefschetz numbers. This extends results of Lusztig, Kosniowski, and O'Brian on holomorphic vector fields to the discrete case. The corresponding theory for smooth and continuous maps, using a twisted Lefschetz formula, can be found in our earlier paper [Lect. Notes Math. 1007, 261-293 (1983; Zbl 0524.58037)]. For a holomorphic Morse-Smale diffeomorphism of a compact complex manifold there are two natural filtrations of the De Rham complex, one based on dynamics (due to Smale) and the other based on the complex structure (due to Dolbeault). This gives two natural decompositions of the ordinary Lefschetz number and we show that these coincide. This suggests a strong (and poorly understood) connection between Dolbeault cohomology and handlebody theory. The paper is organized as follows. In Section 1 we study holomorphic maps with finite nondegenerate periodic set using the Atiyah-Bott formula. Section 2 deals with holomorphic Morse-Smale diffeomorphisms. Sections 3 and 4 indicate how to cope with degenerate periodic points. In Section 5 we extend our results to Dolbeault cohomology with twisted coefficients and give a related result for isometries of Riemannian manifolds.
scientific article; zbMATH DE number 4030557