  We show that the blowup of an extremal Kahler manifold at a relatively stable
point in the sense of GIT admits an extremal metric in Kahler classes that make
the exceptional divisor sufficiently small, extending a result of
Arezzo-Pacard-Singer. We also study the K-polystability of these blowups,
sharpening a result of Stoppa in this case. As an application we show that the
blowup of a Kahler-Einstein manifold at a point admits a constant scalar
curvature Kahler metric in classes that make the exceptional divisor small, if
it is K-polystable with respect to these classes.
