The eta invariant of a circle bundle on a Fano manifold

TL;DR

This paper computes the eta invariant of a spin-c Dirac operator on a circle bundle over a Fano manifold, extending previous results to all adiabatic parameters.

Contribution

It provides a general formula for the eta invariant in terms of Zhang's adiabatic limit, broadening earlier specific cases.

Findings

Explicit formula for eta invariant in terms of Zhang's adiabatic limit

Extension of previous computations to arbitrary adiabatic parameters

Application to circle bundles over Fano manifolds

Abstract

We consider the spin-c Dirac operator on the unit circle bundle of a positive line bundle over a Fano manifold of even complex dimension. We compute the corresponding eta invariant in terms of Zhang's value of its adiabatic limit. This extends the earlier computation of the author from small to arbitrary values of the adiabatic parameter.