  We give a criterion under which a solution g(t) of the Kahler-Ricci flow
contracts exceptional divisors on a compact manifold and can be uniquely
continued on a new manifold. As t tends to the singular time T from each
direction, we prove convergence of g(t) in the sense of Gromov-Hausdorff and
smooth convergence away from the exceptional divisors. We call this behavior
for the Kahler-Ricci flow a canonical surgical contraction. In particular, our
results show that the Kahler-Ricci flow on a projective algebraic surface will
perform a sequence of canonical surgical contractions until, in finite time,
either the minimal model is obtained, or the volume of the manifold tends to
zero.
