On computing local monodromy and the numerical local irreducible decomposition

Abstract Similarly to the global case, the local structure of a holomorphic subvariety at a given point is described by its local irreducible decomposition. Geometrically, the key requirement for obtaining a local irreducible decomposition is to compute the local monodromy action of a generic linear projection at the given point, which is always well‐defined on any small enough neighborhood. We characterize some of the behavior of local monodromy actions of linear projection maps under analytic continuation thereby allowing computations to be performed beyond a local neighborhood. Using this characterization, we then present an algorithm to compute the local monodromy action following the paradigm of numerical algebraic geometry. A germ of an algebraic subvariety at a point is represented by a numerical local irreducible decomposition comprised of a local witness set for each local irreducible component. The results are illustrated using several examples facilitated by an implementation in an open source software package.