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BLOWING UP KAHLER MANIFOLDS WITH CONSTANT SCALAR CURVATURE II
 

Summary: BLOWING UP KšAHLER MANIFOLDS WITH CONSTANT SCALAR
CURVATURE II
CLAUDIO AREZZO AND FRANK PACARD
Abstract. In this paper we prove the existence of Kšahler metrics of constant scalar curvature
on the blow up at finitely many points of a compact manifold that already carries a constant
scalar curvature Kšahler metric. In the case where the manifold has nontrivial holomorphic
vector fields with zeros, we give necessary conditions on the number and locations of the blow
up points for the blow up to carry constant scalar curvature Kšahler metrics.
1991 Math. Subject Classification: 58E11, 32C17.
1. Introduction and statement of the results
1.1. Introduction. Let (M, J, g, ) be a Kšahler manifold of complex dimension m 2, we recall
that the metric g, the complex structure J and the Kšahler form are related by
(X, Y ) = g(J X, Y )
for all X, Y TM. Assume that the scalar curvature of g is constant. Given n distinct points
p1, . . . , pn M, the question we would like to address is whether the blow up of M at the points
p1, . . . , pn can be endowed with a constant scalar curvature Kšahler metric. In the case where the
answer to this question is positive, we would like to characterize the Kšahler classes on the blown
up manifold for which we are able to find such a metric. In [1], we have already given a positive
answer to these questions in the case where the manifold M has no nontrivial holomorphic vector
field with zeros (this condition is for example fulfilled when the group of automorphisms of M is

  

Source: Arezzo, Claudio - Dipartimento di Matematica, Università degli Studi di Parma

 

Collections: Mathematics