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

TL;DR

This paper derives an explicit formula for transversal indices of lifted Dolbeault operators on S^1-bundles over complex projective spaces and applies it to prove a Lefschetz formula for complex projective spaces with torus actions.

Contribution

It provides a new explicit formula for transversal indices on specific bundles and uses it to establish a Lefschetz formula for complex projective spaces under torus actions.

Findings

Explicit formula for transversal indices on S^1-bundles over complex projective spaces

Proof of Lefschetz formula for complex projective spaces with T^{n+1} action

Application of index map to derive geometric formulas

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}.