# The Calabi metric and desingularization of Einstein orbifolds

### Peyman Morteza

University of Wisconsin, Madison, USA### Jeff A. Viaclovsky

University of California, Irvine, USA

## Abstract

Consider an Einstein orbifold $(M_{0},g_{0})$ of real dimension $2n$ having a singularity with orbifold group the cyclic group of order n in SU$(n)$ which is generated by an nth root of unity times the identity. Existence of a Ricci-flat Kähler ALE metric with this group at infinity was shown by Calabi. There is a natural “approximate” Einstein metric on the desingularization of M0 obtained by replacing a small neighborhood of the singular point of the orbifold with a scaled and truncated Calabi metric. In this paper, we identify the first obstruction to perturbing this approximate Einstein metric to an actual Einstein metric. If $M_{0}$ is compact, we can use this to produce examples of Einstein orbifolds which do not admit any Einstein metric in a neighborhood of the natural approximate Einstein metric on the desingularization. In the case that $(M_{0},g_{0})$ is asymptotically hyperbolic Einstein and non-degenerate, we show that if the first obstruction vanishes, then there does in fact exist an asymptotically hyperbolic Einstein metric on the desingularization. We also obtain a non-existence result in the asymptotically hyperbolic Einstein case, provided that the obstruction does not vanish. This work extends a construction of Biquard in the case $n=2$, in which case the Calabi metric is also known as the Eguchi–Hanson metric, but there are some key points for which the higher-dimensional case differs.

## Cite this article

Peyman Morteza, Jeff A. Viaclovsky, The Calabi metric and desingularization of Einstein orbifolds. J. Eur. Math. Soc. 22 (2020), no. 4, pp. 1201–1245

DOI 10.4171/JEMS/943