Fixed Points and Torsion on Kahler Manifolds

Abstract

When a 1-parameter group acts by isometries on a Riemannian manifold M, the fixed point set F is nicely behaved. It is known that each component Fo0 of F is a totally geodesic submanifold of M whose dimension has the same parity as the dimension of M (see e.g., S. Kobayashi, Fixed points of isometries, Nagoya Math J., 13 (1958), 63-68). When M is compact Kihler theisometries are holomorphic transformations and the FM are compact Kihler submanifolds (which may reduce to points); in particular, as cycles, these components cannot bound in M. This paper is mainly concerned with the structure of the fixed point set in this Kihlerian case; however, the use of the complex structure is mainly for convenience; our results also hold for the special type of symplectic manifold in which the fundamental exterior 2-form is harmonic. As has been pointed out to us by several people, our situation is equivalent to having a toral group acting complex analytically on a compact Kahler M. Bott [2] has given some important results on the homology of certain homogeneous spaces and the loop space to a group. Our main results, the theorem and corollaries of ? 4 can be considered as direct generalizations of the former (see Corollary 3). Our method yields, at the same time, new proofs of his results. Our proofs are simple applications of another phase of Bott's work, namely his extension of the Morse theory of critical points to functions with non-degenerate critical manifolds [3]. The following simple example illustrates the method. Let S2 be the 2-sphere and let '1t be the 1-parameter group of rotations of S2 about the z axis. The fixed (or stationary) set F of (D, consists of the north and south poles, i.e., the places where the velocity vector X vanishes. Now