In this paper we prove a new matrix Li-Yau-Hamilton (LYH) estimate for K¨ahler-Ricci flow on manifolds with nonnegative bi-
sectional curvature. The form of this new LYH estimate is obtained by the interpolation consideration originated in [Ch] by
Chow. 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 corresponding estimate for linear heat equation.
These include a sharp heat kernel comparison theorem, generalizing the earlier result of Li and Tian [LT], a manifold version
of Stoll’s theorem [St] on the characterization of ‘algebraic divisors’, and a localized monotonicity formula for analytic subvari-
eties, which sharpens the Bishop volume comparison theorem. Motivated by the connection between the heat kernel estimate
and the reduced volume monotonicity of Perelman [P], we prove a sharp lower bound of the fundamental solution to the forward
conjugate heat equation, which 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 of long-time solution is a gradient expanding
soliton, generalizing an earlier result of Cao. We also illustrate the connection between the new LYH estimate and the Hessian
comparison theorem of [FIN] on the forward reduced distance. Localized monotonicity formulae on entropy and forward reduced
volume are also derived.