Anti-self-dual orbifolds with cyclic quotient singularities

Abstract

Abstract. An index theorem for the anti-self-dual deformation complex on antiself-dual orbifolds with cyclic quotient singularities is proved. We present two applications of this theorem. The first is to compute the dimension of the deformation space of the Calderbank-Singer scalar-flat Kähler toric ALE spaces. A corollary of this is that, except for the Eguchi-Hanson metric, all of these spaces admit nontoric anti-self-dual deformations, thus yielding many new examples of anti-self-dual ALE spaces. For our second application, we compute the dimension of the deformation space of the canonical Bochner-Kähler metric on any weighted projective space CP 2 (r,q,p) for relatively prime integers 1 < r < q < p. A corollary of this is that, while these metrics are rigid as Bochner-Kähler metrics, infinitely many of these admit non-trival self-dual deformations, yielding a large class of new examples of self-dual orbifold metrics on certain weighted projective spaces.

Deformation of LeBrun’s ALE Metrics with Negative Mass

“…After writing this paper, Michael Lock and Jeff Viaclovsky [18] extended the index theorem in [27] to general compact ASD orbifolds with cyclic quotient singularities, and showed for example that the ALE SFK metrics constructed by Calderbank-Singer [2] on the minimal resolution of the quotient C 2 /Γ, Γ being a cyclic group, admit a non-trivial deformation as ALE ASD metrics. But it is not straightforward to see that the method used in this paper can be applied to the twistor spaces of their spaces, since the singularities on the twistor spaces are not Gorenstein any more (i.e.…”

Einstein metrics and Yamabe invariants of weighted projective spaces

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