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 Kähler metric. Formulas are derived for the eta invariant of this conformal structure under additional assumptions. One such assumption is that the Kähler metric admits a special Kähler-Ricci potential in the sense defined by Derdzinski and Maschler. Another is that the Kähler 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Apostolov, V., Calderbank, D.M.J., Gauduchon, P.: Ambikaehler geometry, ambitoric surfaces and Einstein 4-orbifolds. arXiv:1010.0992
Atiyah M.F., Patodi V.K., Singer I.M.: Spectral asymmetry and Riemannian geometry. I. Math. Proc. Cambridge Philos. Soc. 77, 43–69 (1975)
Derdziński A.: Self-dual Kähler manifolds and Einstein manifolds of dimension four. Compos. Math. 49, 405–433 (1983)
Derdzinski A., Maschler G.: Local classification of conformally-Einstein Kähler metrics in higher dimensions. Proc. London Math. Soc. 87, 779–819 (2003)
Derdzinski A., Maschler G.: Special Kähler-Ricci potentials on compact Kähler manifolds. J. reine angew. Math. 593, 73–116 (2006)
Derdzinski A., Maschler G.: A moduli curve for compact conformally-Einstein Kähler manifolds. Compos. Math. 141, 1029–1080 (2005)
Duistermaat J.J., Heckman J.G.: On the variation in the cohomology of the symplectic form of the reduced phase space. Invent. Math. 69, 259–268 (1982)
Eguchi T., Gilkey B.P., Hanson J.A.: Gravitation, gauge theories and differential geometry. Phys. Rep. 66, 213–393 (1980)
Guillemin, V.: Moment maps and combinatorial invariants of hamiltonian T n-spaces. In: Progress in Mathematics, vol. 122 (1994)
Hitchin, N.J.: Einstein metrics and the eta-invariant. Boll. Un. Mat. Ital. B (7) 11(suppl), 95–105 (1997)
LeBrun C.: On complete quaternionic-Kähler manifolds. Duke Math. J. 63, 723–743 (1991)
Maschler G.: Special Kähler-Ricci potentials and Ricci solitons. Ann. Global Anal. Geom. 34, 367–380 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Maschler, G. The eta invariant in the doubly Kählerian conformally compact Einstein case. Math. Z. 271, 1065–1073 (2012). https://doi.org/10.1007/s00209-011-0903-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-011-0903-x