*(English)*Zbl 1159.14022

Let $S$ be any class VII surface, that is a compact complex surface with first betti number ${b}_{1}\left(S\right)=1$ and Kodaira dimension $\text{kod}\left(S\right)=-\infty $. Assume furthermore that $S$ is minimal, i.e with second Betti number ${b}_{2}\left(S\right)=1$.

The author describes explicitly the moduli space $M(S,E,g)$ of polystable holomorphic structures $\mathcal{E}$ with $det\left(\mathcal{E}\right)=K$ on a rank two vector bundle $E$ with ${c}_{1}\left(E\right)={c}_{1}\left(K\right)$ and ${c}_{2}\left(E\right)=0$ with respect to any possible Gauduchon metrics $g$ on $S$.

In the non algebraic case, it is known that the moduli space of polystable bundles contain non-filtrable elements and this is a major issue to describe these elements. The authors classify these filtrable bundles and show using gauge theory, that only a particular type can exist. Using a deformation argument and the fact that any class VII surface containing a global spherical shell is the degeneration of a blown-up primary Hopf surface, the author shows that the filtrable bundles are actually generic in the moduli space.

The author studies the local structure of the moduli spaces and their boundary. When $S$ is a half or parabolic Inoue surface, $M(S,E,g)$ is always a compact one-dimensional complex disc. In the general case, when $S$ is an Enoki surface, one obtains a complex disc with finitely many transverse self-intersections whose number becomes arbitrarily large when $g$ varies in the space of Gauduchon metrics. Thus, in that case, the number of singularities is unbounded. In particular, there are infinitely many homeomorphism types of moduli spaces although there are only finitely many topological splittings of the underlying vector bundle. Note that these moduli are not complex space. In the last section, the author dertermines non filtrable bundles which leads to a complete description of the entire moduli space.

The paper underlines the differences between the moduli spaces for algebraic surfaces and minimal class VII surfaces, i.e non Kähler nor elliptic surfaces for which some new phenomena appear. Moreover, in the direction of the pioneering work of [*A. Teleman*, Invent. Math. 162, No. 3, 493–521 (2005; Zbl 1093.32006)], this article is clearly a step further in the classification of class VII surfaces, since some of the methods can be extended to the case ${b}_{2}=2$. Finally, the paper is well written.

##### MSC:

14J60 | Vector bundles on surfaces and higher-order varieties, and their moduli |

14J25 | Special surfaces |

57R57 | Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants |