Blowing up and Desingularizing Constant Scalar Curvature Kähler Manifolds

This paper is concerned with the existence of constant scalar curvature Kähler metrics on blow ups at finitely many points of compact manifolds which already carry constant scalar curvature Kähler metrics. We also consider the desingularization of isolated quotient singularities of compact orbifolds which already carry constant scalar curvature Kähler metrics. Let (M,ω) be either a m-dimensional compact Kähler manifold or a m-dimensional compact Kähler orbifold with isolated singularities. By definition, any point p ∈ M has a neighborhood biholomorphic to a neighborhood of the origin in C/Γ, where Γ is a finite subgroup of U(m) (this last fact is a consequence of the Kähler property) acting freely on C − {0}. Observe that, when p is a smooth point of M , the group Γ reduces to the identity. In the case where M is an orbifold, the Kähler form ω lifts, near any of the singularities of M , to a Kähler form ω̃ on a punctured neighborhood of 0 in C. We will always assume that ω̃ can be smoothly extended through the origin, i.e. that ω is an orbifold metric. If we further assume that the Kähler form ω has constant scalar curvature and if we are given n distinct (smooth) points p1, . . . , pn ∈ M , one of the questions we would like to address in this paper is whether the blow up of M at the points p1, . . . , pn can still be endowed with a constant scalar curvature Kähler form. In this direction, we have obtained the following positive answer : Theorem 1.1. Let (M,ω) be a constant scalar curvature compact Kähler manifold or Kähler orbifold with isolated singularities. Assume that there is no nonzero holomorphic vector field vanishing somewhere on M . Then, given finitely many smooth points p1, . . . , pn ∈ M and positive numbers a1, . . . , an > 0, there exists ε0 > 0 such that the blow up of M at p1, . . . , pn carries constant scalar curvature Kähler forms ωε ∈ π [ω]− ε (a1 PD[E1] + . . .+ an PD[En]), where the PD[Ej ] are the Poincaré dual of the (2m − 2)-homology classes of the exceptional divisors of the blow up at pj and ε ∈ (0, ε0). If the scalar curvature of ω is not zero then the scalar curvatures of ωε and of ω have the same signs.