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

TL;DR

This paper derives formulas for the eta invariant of conformal structures on 3-manifolds bounding 4-manifolds with special K"ahler and Einstein properties, using the Duistermaat-Heckman theorem and building on Hitchin's work.

Contribution

It provides explicit eta invariant formulas for conformal structures arising from Einstein metrics with K"ahler and ambitoric structures, extending previous results.

Findings

Formulas for eta invariants under special K"ahler-Ricci potential assumptions

Formulas for eta invariants in ambitoric and toric K"ahler cases

Connections to Hitchin's work on Einstein selfdual metrics

Abstract

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.