I am learning about Nakano positivity of hermitian vector bundles, which is the strongest notion of positivity we can ask. I don't understand what is the geometric meaning of it. Let me briefly summarize the situation:
$\overline E=(E,h)$ is a (holomorphic) hermitian vector bundle on a complex manifold $X$. Then we have a natural notion of curvature $\Theta(E)$ which is a global (1,1)-form of real type on $X$ with values in $\Gamma(E)\otimes \Gamma(E)^\vee$.
At this point, the curvature $\Theta(E)$ induces a hermitian form $\theta_E$ on the vector bundle $T_X\otimes E$. Such a form is defined locally just by taking the local coefficients appearing in $\Theta(E)$ (we are fixing also an orthonormal frame).
We say that $E$ is Nakano positive if $\theta_E$ is actually a hermitian metric for $T_X\otimes E$, i.e. if $\theta_E$ is defined positive.
What is the meaning of $\theta_E$? What does it "measure" when it is a metric? Do we have an intrinsic construction of $\theta_E$ without appealing to local cohordinates?
