We study general elements of moduli spaces ${\mathfrak{M}}_{{\mathbb{P}}^2}\left(2,c_1,c_2\right)$ of rank-2 stable holomorphic vector bundles on ${\mathbb{P}}^2$ and their minimal free resolutions. Incidentally, a quite easy proof of the irreducibility of ${\mathfrak{M}}_{{\mathbb{P}}^2}\left(2,c_1,c_2\right)$ is shown.

We describe explicitly the moduli spaces ${ℳ}_g^{\mathrm{pst}}(S,E)$ of polystable holomorphic structures $ℰ$ with $detℰ\cong 𝒦$ on a rank two vector bundle $E$ with $c_1\left(E\right)=c_1\left(K\right)$ and $c_2\left(E\right)=0$ for all minimal class VII surfaces $S$ with $b_2\left(S\right)=1$ and with respect to all possible Gauduchon metrics $g$. These surfaces $S$ are non-elliptic and non-Kähler complex surfaces and have recently been completely classified. When $S$ is a half or parabolic Inoue surface, ${ℳ}_g^{\mathrm{pst}}(S,E)$ is always a compact one-dimensional complex disc. When $S$ is an Enoki surface, one obtains a complex disc with finitely...