We say that a $(p,p)$-form on a smooth projectively variety $X$ of dimension $n$ is positive if its restriction to every $p$-dimensional subspace of the holomorphic tangent space at every point is a volume form.
Suppose we take a Hartshorne-ample vector bundle $V$ over $X$. By a result of Fulton-Lazarsfeld, we know that the Schur polynomials of the Chern classes are numerically positive. My question is : Is there a positive $(p,p)$ representative of the $p$th Chern class $c_p(V)$ ?
Note that this is true for the top Chern class. This is also true for the first Chern class (Nakai-Moizeshon).