  In this paper we prove a new matrix Li-Yau-Hamilton estimate for
K\"ahler-Ricci flow. The form of this new Li-Yau-Hamilton estimate is obtained
by the interpolation consideration originated in \cite{Ch1}. This new
inequality is shown to be connected with Perelman's entropy formula through a
family of differential equalities. In the rest of the paper, We show several
applications of this new estimate and its linear version proved earlier in
\cite{CN}. These include a sharp heat kernel comparison theorem, generalizing
the earlier result of Li and Tian, a manifold version of Stoll's theorem on the
characterization of `algebraic divisors', and a localized monotonicity formula
for analytic subvarieties.
  Motivated by the connection between the heat kernel estimate and the reduced
volume monotonicity of Perelman, we prove a sharp lower bound heat kernel
estimate for the time-dependent heat equation, which is, in a certain sense,
dual to Perelman's monotonicity of the `reduced volume'. As an application of
this new monotonicity formula, we show that the blow-down limit of a certain
type III immortal solution is a gradient expanding soliton.
  In the last section we also illustrate the connection between the new
Li-Yau-Hamilton estimate and the earlier Hessian comparison theorem on the
`reduced distance', proved in \cite{FIN}.
