Local Properties of Analytic Varieties

Algebraic and analytic varieties have become increasingly important in recent years, both in the complex and the real case. Their local structure has been intensively investigated, by algebraic and by analytic means. Local geometric properties are less well understood. Our principal purpose here is to study properties of tangent vectors and tangent planes in the neighborhood of singular points. We study stratifications of an analytic variety into analytic manifolds; in particular, we may require that a transversal to a stratum is also tranversal to the higher dimensional strata near a given point. A conjecture on possible fiberings of the variety (and of surrounding space) is stated; it is proved at points of strata of codimension 2 in the surrounding space. In the last sections, we see that a variety may have numbers or functions intrinsically attached at points or along strata; also an analytic variety may be locally unlike an algebraic variety.