The Calabi metric and desingularization of Einstein orbifolds

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 ${\rm{SU}}(n)$ which is generated by an $n$th root of unity times the identity. Existence of a Ricci-flat K\"ahler ALE metric with this group at infinity was shown by Calabi. There is a natural "approximate" Einstein metric on the desingularization of $M_0$ 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.