Anti-self-dual orbifolds with cyclic quotient singularities

Michael T. Lock ; Jeff A. Viaclovsky

Journal of the European Mathematical Society, Tome 17 (2015) no. 11, pp. 2805-2842.

Résumé

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 CP(r,q,p)2​ for relatively prime integers 1corollary 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.

Détail

DOI : 10.4171/jems/572

Classification : 53-XX, 58-XX
Keywords: Anti-self-dual metrics, index theory, orbifolds