Deformation of LeBrun’s ALE Metrics with Negative Mass

Abstract

Abstract. In this article we investigate deformations of a scalar-flat Kähler metric on the total space of complex line bundles over CP 1 constructed by C. LeBrun. In particular, we find that the metric is included in a one-dimensional family of such metrics on the four-manifold, where the complex structure in the deformation is not the standard one.

Local Moduli of Scalar-flat Kähler ALE Surfaces

“…We will next discuss the dimension of the generic orbit in the cases in (1, 1). This case has been studied in [Hon13]. The group of biholomorphic automorphisms is GL(2, C), and the identity component of the holomorphic isometry group is U(2).…”

Quotient singularities, eta invariants, and self-dual metrics

“…For any finite subgroup Γ ⊂ U(2) which acts freely on S 3 , recall the orbifold quotients (CP 2 (1,1,2m) , g BK )/Γ from Theorem 2.2. In [Hon13], Honda discovered the explicit form of the U(2)-action on H 1 SD of the self-dual deformation complex of (CP 2 (1,1,2m) , g BK ). Applying a representation theoretic argument to this, we find the dimension of the space of invariant elements of H 1 SD under the quotient by Γ.…”

“…This is the m = 1 case. For m > 1, Honda showed that the complexification of H 1 SD of the self-dual deformation complex on (CP 2 (1,1,2m) , g BK ) is equivalent to ρ ⊕ρ, where ρ = S 2m−2 (C 2 ) ⊗ det ⊕ S 2m−4 (C 2 ) ⊗ det 2 , (4.4) as a representation space of U(2), see [Hon13]. The dimension of the space of invariant elements of H 1 SD under the action of Γ is equal to that under the action of any subgroup Γ ′ ⊂ Γ that has the same effective action as Γ and is given by…”

A smörgåsbord of scalar-flat Kähler ALE surfaces

“…n−1 j=1 sin 2πk n j cot π n j = −2n k n (6.4) For m > 1, Honda showed that the complexification of the space of infinitesimal scalar-flat Kähler deformations of (O(−2m), g LB ) is equivalent to ρ ⊕ ρ where ρ = S 2m−2 (C 2 ) ⊗ det, (6.5) as a representation space of U(2), see [Hon13]. Using this along with (6.1), we will compute dim(H sf k Γ ′ ) for all appropriate Γ ′ .…”