Holomorphic arcs on singularities

Abstract

The study of formal arcs was initiated by Nash in a 1967 preprint. Arc spaces of smooth varieties have a rather transparent structure but difficult problems arise for arcs passing through singularities. The Nash conjecture on the irreducible components of such arc spaces was proved for surfaces by Fernandez de Bobadilla and Pe Pereira and for toric singularities by Ishii and Kollar, but counter examples were found in higher dimensions by Ishii and Kollar, de Fernex and Johnson and Kollar. Here we start the study of holomorphic arcs; these are holomorphic maps from the closed unit disk to a complex analytic space. As one expects, there is not much conceptual difference between the set of formal arcs and the set of holomorphic arcs since every formal arc can be approximated by holomorphic arcs. However, a formal deformation of an arc is a much more local object than a holomorphic deformation. Thus, in many cases, the space of holomorphic arcs has infinitely many connected components while the space of formal arcs always has only finitely many. For a complex analytic space we define two variants—the space of arcs and the space of short arcs—and we study their connected components. For short arcs we obtain complete answers for surface singularities and for isolated quotient singularities in all dimensions.