It is a well known fact that proper scheme $X$ over $k$ has a up to isomorphism unique dualizing sheaf (EGA I, Hartshorne).

This dualizing sheaf $\omega_X$ comes with two striking properties:

(i) *There is a homomorphism $t : H^n(X, \omega_X ) \to k$ (also called the trace) such that for every coherent
$\mathcal{O}_X$-Module $\mathcal{F}$ the following holds: There exists a canonical bilinear map*

$$ \operatorname{Hom}_{\mathcal{O}_X}(\mathcal{F},\omega_X) \times H^n(X, \mathcal{F}) \to H^n(X, \omega_X) $$

*which gives an isomorphism:*

$$ \operatorname{Hom}_{\mathcal{O}_X}(\mathcal{F},\omega_X) \cong H^n(X, \mathcal{F})^*$$

*by composing with t. Here,* * *means the dual vector space over $k$.*

(ii) *In addition for every integer $i ≥ 0$ and coherent $\mathcal{F}$ , there exists a canonical isomorphism: $\operatorname{Ext}^i(\mathcal{F}, \omega_X) \cong H^{n-i}(X, \mathcal{F})^*$ if and only if $X$ is Cohen–
Macaulay.*

In other words (i) means $H^n(X, \mathcal{F})^*$ is representable. Now $H^n(X, \mathcal{F})^*$ carries structure of a $k$-vector space. If we assume that $\dim_k H^n(X, \mathcal{F}) < \infty$, then $H^n(X, \mathcal{F}) \cong H^n(X, \mathcal{F}) ^*$ and in following we will not differ between $H^n(X, \mathcal{F})$ and it's dual.

A one dimensional $k$- subspace $V_1 \subset H^n(X, \mathcal{F})$ correspond in high tec language to an orbit of a non zero vector $v \in H^n(X, \mathcal{F})$ by action of $k$ on $H^n(X, \mathcal{F})$ via multiplication, ie $V_1= k \cdot v$.

Now my first question is what are the special morphisms in $\operatorname{Hom}_{\mathcal{O}_X}(\mathcal{F},\omega_X) $ which correspond to $V_1$ aka to the orbit of $k$-action on $v$. How are they related to each other as objects in $\operatorname{Hom}_{\mathcal{O}_X}(\mathcal{F},\omega_X) $?

In other words if $\operatorname{Hom}_{\mathcal{O}_X}(\mathcal{F},\omega_X) $ inherits the $k$-action from $H^n(X, \mathcal{F})$, are the elements from the same orbit related to each other in certain "deep" way? Any intuition how one can think about these orbits (except of the boring answer "lines in $H^n(X, \mathcal{F})$")?

The second question is if we take $\mathcal{F}= \omega_X$, then $id_{\omega_X} \in \operatorname{Hom}_(\omega_X,\omega_X)$. Is it's image in $H^n(X, \mathcal{F})$ "special" in certain way? What can we say about this element considered as vector?