Nonsingular analytic subvariety
= Nonsingular analytic subvariety
A closed analytic subset of a <complex manifold> is nonsingular when every point has a neighbourhood on which the subset is the zero set of a <holomorphic map> of full rank along that zero set. The <holomorphic implicit function theorem> then makes it a local coordinate subspace. This includes a closed complex submanifold, whereas an arbitrary open piece of a complex submanifold need not be globally closed in its ambient manifold.