Positivity notions naturally occur both in algebraic and differential geometry. In this talk, we will introduce the positivity of differential forms hence the positivity of co-homology class. We will follow the literature started by Griffiths. We will also discuss positive polynomials and positive vector bundles. We will show that the curvature polynomial for a holomorphic vector bundle over a complex manifold is positive. In particular, for surface, we will show that $c_2$ is positive.

[Gri69]     Phillip A. Griffiths. Hermitian differential geometry, Chern classes, and positive vector bundles. In Global Analysis (Papers in Honor of K. Kodaira). pages 185-251. Univ. Tokyo Press, Tokyo, 1969.

[LS05]     Reese Harvey and A. W. Knapp. Positive (p, p) forms, Wirtinger's inequality, and currents. In Value distribution theory (Proc. Tulane Univ. Program, Tulane Univ., New Orleans, La., 1972-1973), Part A, pages 43-62. Dekker, New York, 1974.