Let's fixed a locally ringed space $(X,\mathcal{O}_X)$ (although, this should apply to any ringed topos, but I haven't thought that through). In fact, if it's helpful, you can assume that $X$ is a complex manifold, or a scheme. The question then is:
Is there a natural $\mathcal{O}_X$-module $\mathcal{F}$ such that $\text{Aut}_{\mathcal{O}_X}(\mathcal{F})=\mathcal{O}_X$?
The reason for my interest is due to the fact that $H^1(X,\mathcal{O}_X)$ would then classify twists of $\mathcal{F}$. For example, because $\text{Aut}_{\mathcal{O}_X}(\mathcal{O}_X)=\mathcal{O}_X^\times$, we know that $H^1(X,\mathcal{O}_X^\times)$ classifies twists of $\mathcal{O}_X$, or more colloquially, line bundles.
If there was such a sheaf $\mathcal{F}$, which had some nice geometric flavor, this might give some interesting insight into how to think about genus, or any other statistics associated to $h^1$.
I apologize if I have missed something obvious!
Thanks!
PS: I am not particularly looking for another way to think about $\mathbb{G}_a$-torsors, than the context above. For example, thinking about them as extensions via the isomorphism $H^1(X,\mathcal{O}_X)\cong \text{Ext}^1(\mathcal{O}_X,\mathcal{O}_X)$ isn't particularly helpful to me, in this context.