 
Summary: arXiv:0706.1838v1[math.DG]13Jun2007
ON THE KšAHLER CLASSES OF CONSTANT SCALAR CURVATURE
METRICS ON BLOW UPS
CLAUDIO AREZZO AND FRANK PACARD
1. Introduction
In this short paper we address the following question :
Problem 1.1. Given a compact constant scalar curvature Kšahler manifold (M, J, g, ),
of complex dimension m := dimC M, and having defined
:= {(p1, . . . , pn) Mn
: a = b pa = pb},
characterize the set PW = {(p1, . . . , pn, 1, . . . , n)} (Mn \ ) Ś (0, +)n for which
~M = Blp1,...,pn M, the blow up of M at p1, . . . , pn has a constant scalar curvature Kšahler
metric (cscK from now on) in the Kšahler class
[]  (1 PD[E1] + · · · + n PD[En]),
where the PD[Ej] are the PoincarŽe duals of the (2m  2)homology classes of the excep
tional divisors of the blow up at pj.
This general problem is too complicated and its solution is likely to pass through
the solution of well known conjectures relating the existence of cscK metrics with the
Kstability of the polarized manifold.
