On Rigid Singularities

Abstract

ON RTGTD SINGULARITIES'!'A local complex space X is rigid if every flat family (X,), t E T which contains X is locally trivial, in the sense that the total space V = u X, is cally a product X x T. Equivalently, every sufficiently near deformation of X is isomorphic to X (at least when T is normal).Singularities f this type must therefore be "generic" members of any families which ain them, for no other singularity can specialize to X. is paper exhibits several classes of rigid singularities, many of which already appeared in the literature in one form or another, and, prinly, an extensive class of singularities which cannot be deformed to a singularity.The examples of these last given here are conical singulari-, cones over projective manifolds, every deformation of which is still ical.However, the singularities in these examples tend to be generic, the sense that they cannot be realized as the specialization of other a related paper, pp.11 3-1 17 in this volume, David Mumford extends sharpens these results in the low-dimensional case not treated here.outline, $1 is a brief survey of the cotangent spaces used to construct versa1 deformation of a singularity (Grauert, [dl).$2 contains the basic parison theorem between the deformations of a local complex space the complement of its origin, upon which most of the examples of rigid ularities in $3 are based.$4 compares the deformations of the cone over ojective manifold Y to the projective deformations of Y, and 45 exhibits "generic," but non-rigid singularities.e deal throughout with local complex spaces X , by which is meant the of a complex analytic space, not necessarily reduced, at a point, always n to be origin in some C".Thus X is represented by a complex analytic ace of some convenient neighborhood of the origin, and 8 , denotes structure sheaf of X on this neighborhood.