First, we will talk about the positivity of the second Chern form for a rank two positive bundle over surface.
Next, we will discuss representability of Chern-Weil form.
Positivity of differential form, Griffiths conjecture on positivity, Schur forms
References:
[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.
[Pin18] Vamsi Pritham Pingali.
Representability of Chern-Weil forms.
Math. Z., 288(1-2):629-641, 2018.