  This article is concerned with Chern class and Chern number inequalities on
polarized manifolds and nef vector bundles. For a polarized pair $(M,L)$ with
$L$ very ample, our first main result is a family of sharp Chern class
inequalities. Among them the first one is a variant of a classical result and
the equality case of the second one is a characterization of hypersurfaces. The
second main result is a Chern number inequality on it, which includes a reverse
Miyaoka-Yau type inequality. The third main result is that the Chern numbers of
a nef vector bundle over a compact K\"{a}hler manifold are bounded below by the
Euler number. As an application, we classify compact K\"{a}hler manifolds with
nonnegative bisectional curvature whose Chern numbers are all positive. A
conjecture related to the Euler number of compact K\"{a}hler manifolds with
nonpositive bisectional curvature is proposed, which can be regarded as a
complex analogue to the Hopf conjecture.

Differential Geometry;Algebraic Geometry