An explicit formula for the transversal indices of the lifted Dolbeault operators

Abstract

M. F. Atiyah proved that the index of a transversally elliptic operator relative to a free action can be computed by using indices of elliptic operators on the orbit manifold. In this paper, we derive an explicit formula for the transversal indices on S^1-bundles over complex projective spaces. Using this explicit formula and the index map, we prove Lefschetz formula for the n-dimensional complex projective space with the canonical action of T^{n + 1}.