 
Summary: BLOWING UP AND DESINGULARIZING CONSTANT SCALAR
CURVATURE KšAHLER MANIFOLDS
CLAUDIO AREZZO AND FRANK PACARD
1991 Math. Subject Classification: 58E11, 32C17.
1. Introduction
Abstract. This paper is concerned with the existence of constant scalar curvature Kšahler
metrics on blow ups at finitely many points of compact manifolds which already carry constant
scalar curvature Kšahler metrics. We also consider the desingularization of isolated quotient
singularities of compact orbifolds which already carry constant scalar curvature Kšahler metrics.
Let (M, ) be either a mdimensional compact Kšahler manifold or a mdimensional compact
Kšahler orbifold with isolated singularities. By definition, any point p M has a neighborhood
biholomorphic to a neighborhood of the origin in Cm
/ , where is a finite subgroup of U(m)
(this last fact is a consequence of the Kšahler property) acting freely on Cm
 {0}. Observe that,
when p is a smooth point of M, the group reduces to the identity. In the case where M is
an orbifold, the Kšahler form lifts, near any of the singularities of M, to a Kšahler form ~ on a
punctured neighborhood of 0 in Cm
. We will always assume that ~ can be smoothly extended
through the origin, i.e. that is an orbifold metric.
