Towards a link theoretic characterization of smoothness (Q1651746)

A point on a normal surface is smooth if its link is diffeomorphic to a sphere. In higher dimensions this is no longer true. \textit{M. McLean} [Invent. Math. 204, No. 2, 505--594 (2016; Zbl 1348.53079)] proved that in dimension 3 the contact structure of the link suffices to characterise smooth points among normal isolated singularities. The ingredients, especially the algebro-geometric ones, of this proof are explained. The main result of McLean is that the minimal log-discrepancy \(\text{mld}_x(X)\) of a normal point \(x \in X\) is under certain conditions a contact invariant of the link. He concludes using a conjecture of Shokurov stating that \(\text{mld}_x(X)\leq n=\dim X\) with equality if and only if \((X,x)\) is smooth, which holds for \(n=3\). Using a Theorem of \textit{S. Ishii} on minimal Mather log discrepancies [Ann. Inst. Fourier 63, No. 1, 89--111 (2013; Zbl 1360.14048); corrigendum ibid. 67, No. 4, 1609--1612 (2017)] the authors show that for \(x\) a closed point on a normal \(\mathbb Q\)-Gorenstein variety of dimension \(n\) such that the exceptional divisor of the normalized blow-up of \(X\) at \(x\) has a generically reduced irreducible component one has indeed \(\text{mld}_x(X)\leq n\) with equality if and only if \(X\) is smooth at \(x\). This allows them to extend McLean's result to this class of singularities. If the exceptional divisor of the normalized blow-up has no reduced irreducible components, the approach with Ishii's result does not work and the authors speculate that a more refined invariant of the link can be used to characterise smoothness. To this end they build an invariant out of the CR-structure of the link. They also discuss a strategy using a contact form on shrinking links.