Compact moduli spaces of surfaces of general type (Q2895539)

The author gives an introduction to the compactification of the moduli space of surfaces of general type and of surfaces with boundary, by stable surfaces. He explains the infinitesimal description of the moduli stack. Then he considers the codimension one boundary components. An important type of boundary component consists of surfaces with a unique cyclic quotient admitting a \(\mu=0\) smoothing, here called Wahl singularities, as they were first studied by \textit{J. Wahl} [Topology 20, 219--246 (1981; Zbl 0484.14012)].NEWLINENEWLINEThe author explains a correspondence between boundary components of Wahl type and rigid holomorphic vector bundles on the general smooth surface \(Y\), in case \(H^{2,0}=H^1=0\). In some detail the motivating example of the projective plane \(Y=\mathbb{P}^2\) together with a curve \(D\) of degree \(d\geq 4\) is described.NEWLINENEWLINEAlso the connection with Donaldson theory is explained.NEWLINENEWLINEFor the entire collection see [Zbl 1236.14001].