On decorated representation spaces associated to spherical surfaces (Q6631304)

The authors study the local properties and loci of singular points of various deformation spaces of surface group representations into the compact Lie group \(\mathrm{SU}(2)\). The surfaces considered are of arbitrary genus and may be punctured. Their work encompasses \textit{spaces of homomorphisms} as well as their quotients by the conjugation action of \(\mathrm{SU}(2)\), referred to as \textit{representation spaces}. Both deformation spaces are analyzed in their absolute and relative versions, where conjugacy classes for peripheral loops are prescribed. The authors also investigate deformation spaces of \textit{decorated representations} -- surface group representations together with a coherent choice of rotation centers and angles for the images of every peripheral loop.\N\NThe paper presents a wide range of results, covering aspects such as the (semi-)algebraicity and symplectic nature of the deformation spaces mentioned above, along with detailed descriptions and dimension counts for singular and smooth loci. In the relative case, the authors carefully specify their results based on the values of the peripheral angles.\N\NFinally, the authors explain that decorated surface group representations into \(\mathrm{SU}(2)\) naturally arise as decorated monodromies of \textit{spherical surfaces} -- surfaces equipped with a metric of constant curvature \(1\) with conical singularities at finitely many points. They raise the question of which non-elementary decorated representations can be realized as the decorated monodromy of a spherical surface with prescribed cone angles.