Geometry of nilpotent orbits: results and conjectures (Q2900279)

Let \(\mathfrak g\) be a simple complex Lie algebra and \(G\) its adjoint Lie group. For a nilpotent element \(x \in \mathfrak g\) one can consider the nilpotent orbit \(\mathcal O_x := G \cdot x\). The minimal orbit \(\mathcal O_{min}\) of \(\mathfrak g\) is the unique orbit which is contained in the closure of any other non-zero nilpotent orbit. Its closure \(\bar{\mathcal O}_{min}\) is normal with an isolated singularity. Minimal orbit closures and their singularities have been studied intensively by many authors over the last ten years. They are expected to be closely related to symplectic singularities, i.e. those singular symplectic varieties which admit a resolution of singularities by a symplectic manifold.NEWLINENEWLINEMore precisely one expects that the following conjecture holds: an isolated symplectic singularity of dimension at least four which admits a symplectic resolution is analytically locally isomorphic to \(\bar{\mathcal O}_{\min} \subset \mathfrak{sl}(V)\). This conjecture was proven in dimension four by \textit{J. Wierzba} and \textit{J. A. Wiśniewski} [Duke Math. J. 120, No. 1, 65--95 (2003; Zbl 1036.14007)], but it is open in general.NEWLINENEWLINEIn the paper under review the author gives a summary of results on nilpotent orbits from the point of view of the minimal model program, but the main focus is an extensive list of related conjectures and open problems.NEWLINENEWLINEFor the entire collection see [Zbl 1235.00045].