Holomorphic arcs on singularities (Q2342177)

J. F. Nash initiated the study of formal arcs in his 1967 preprint. It has been shown that in many cases, arc spaces provide a lot of information on the singularities of the given space. Let \(X\) be a complex space. The authors initiate the study of \textit{holomorphic arcs}, i.e., holomorphic maps from the closed unit disc to \(X\). The main difference is that the deformation of a formal arc is much more local than the deformation of a holomorphic arc; in particular, the space of holomorphic arcs usually has infinitely many connected components whereas the space of formal arcs always has only finitely many. For any subset \(Z\subset X\), a holomorphic arc \(\Phi:\overline {\mathbb D}\to X\) is called \textit{short} on \((Z\subset X)\) if \({\mathrm{Supp}}\, \Phi^{-1}(Z)=\{0\}\), and the set of all short arcs on \((Z\subset X)\) is denoted by \({\mathrm{ShArc}}(Z\subset X)\). The set of all holomorphic arcs \(\Phi\) such that \(\phi(\partial\mathbb D)\subset X\setminus Z\) is denoted by \({\mathrm{Arc}}^{\circ}(Z\subset X)\). For any arc \(\phi\in{\mathrm{Arc}}^{\circ}(Z\subset X)\) its restriction \(\phi|\partial\mathbb D\) defines an element of \(\pi_1(X\setminus Z)/({\mathrm{conjugation}})\). This map descends to a map \(\pi_0({\mathrm{Arc}}^{\circ}(Z\subset X))\to \pi_1(X\setminus Z)/({\mathrm{conjugation}})\). The authors define and study the \textit{winding number map} \[ w: \pi_0({\mathrm{ShArc}}(Z\subset X))\to \pi_1({\mathrm{link}}(Z\subset X))/({\mathrm{conjugation}}). \] The main result of the paper says that if \((0\in X)\) is a normal surface singularity, then the winding number map is injective. In the case of an isolated quotient singularity \((0\in X)\cong (0\in \mathbb C^n)/G\) there are natural identifications \[ \pi_0({\mathrm{ShArc}}(0\in X))= \pi_1({\mathrm{link}}(0\in X))/({\mathrm{conjugation}})=G/({\mathrm {conjugation}}). \] In particular, for \(G\subset {\mathrm{SL}}(2,\mathbb C)\), this gives a realization of the McKay correspondence between nontrivial conjugacy classes of \(G\) and exceptional curves of the minimal resolution of \(\mathbb C^2/G\). It is also shown that \(\pi_0({\mathrm{ShArc}}(0\in X))\) is infinite for every other surface singularity. The authors give an even more precise description of the winding number map in the case of a surface cusp singularity and that of a normal surface singularity with a good \(\mathbb C^*\)-action. In the proofs the authors use \textit{minimal dlt modifications}, which are tightly connected to the plumbing construction of the links.