No abstract

Recently, Atiyah and LeBrun proved versions of the Gauss-Bonnet and Hirzebruch signature Theorems for metrics with edge-cone singularities in dimension four, which they applied to obtain an inequality of Hitchin-Thorpe type for Einstein edge-cone metrics. Interestingly, many natural examples of edge-cone metrics in dimension four are anti-self-dual (or self-dual depending upon choice of orientation). On such a space there is an important elliptic complex called the anti-self-dual deformation complex, whose index gives crucial information about the local structure of the moduli space of anti-self-dual metrics. In this paper, we compute the index of this complex in the orbifold case, and give several applications.

For a Kähler metric, the Riemannian scalar curvature is equal to twice the Chern scalar curvature. The question we address here is whether this equivalence can hold for a non-Kähler Hermitian metric. For such metrics, if they exist, the Chern scalar curvature would have the same geometric meaning as the Riemannian scalar curvature. Recently, Liu-Yang showed that if this equivalence of scalar curvatures holds even in average over a compact Hermitian manifold, then the metric must in fact be Kähler. However, we prove that a certain class of noncompact complex manifolds do admit Hermitian metrics for which this equivalence holds. Subsequently, the question of to what extent the behavior of said metrics can be dictated is addressed and a classification theorem is proved.

Abstract. There are three main components to this article:• (i) A formula for the eta invariant of the signature complex for any finite subgroup of SO(4) acting freely on S 3 is given. An application of this is a non-existence result for Ricci-flat ALE metrics on certain spaces.• (ii) A formula for the orbifold correction term that arises in the index of the self-dual deformation complex is proved for all finite subgroups of SO(4) which act freely on S 3 . Some applications of this formula to the realm of self-dual and scalar-flat Kähler metrics are also discussed.• (iii) Two infinite families of scalar-flat anti-self-dual ALE spaces with groups at infinity not contained in U(2) are constructed. Using these spaces, new examples of self-dual metrics on n#CP 2 are obtained for n ≥ 3.

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.