The eta invariant in the doubly K\"ahlerian conformally compact Einstein case

On a 3-manifold bounding a compact 4-manifold, let a conformal structure be induced from a complete Einstein metric which conformally compactifies to a K\"ahler metric. Formulas are derived for the eta invariant of this conformal structure under additional assumptions. One such assumption is that the K\"ahler metric admits a special K\"ahler-Ricci potential in the sense defined by Derdzinski and Maschler. Another is that the K\"ahler metric is part of an ambitoric structure, in the sense defined by Apostolov, Calderbank and Gauduchon, as well as a toric one. The formulas are derived using the Duistermaat-Heckman theorem. This result is closely related to earlier work of Hitchin on the Einstein selfdual case.