  We investigate the limiting behavior of the unnormalized Kahler-Ricci flow on
a Kahler manifold with a polarized initial Kahler metric. We prove that the
Kahler-Ricci flow becomes extinct in finite time if and only if the manifold
has positive first Chern class and the initial Kahler class is proportional to
the first Chern class of the manifold. This proves a conjecture of Tian for the
smooth solutions of the Kahler-Ricci flow.
