Journal of the European Mathematical Society


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Volume 17, Issue 11, 2015, pp. 2805–2842
DOI: 10.4171/JEMS/572

Anti-self-dual orbifolds with cyclic quotient singularities

Michael T. Lock[1] and Jeff A. Viaclovsky[2]

(1) The University of Texas at Austin, USA
(2) University of Wisconsin, Madison, United States

An index theorem for the anti-self-dual deformation complex on anti-self-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 non-toric 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 $\mathbb {CP}^2_{(r,q,p)}$ 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-trivial self-dual deformations, yielding a large class of new examples of self-dual orbifold metrics on certain weighted projective spaces.

Keywords: Anti-self-dual metrics, index theory, orbifolds

Lock Michael, Viaclovsky Jeff: Anti-self-dual orbifolds with cyclic quotient singularities. J. Eur. Math. Soc. 17 (2015), 2805-2842. doi: 10.4171/JEMS/572