On Infinitesimal Deformations of the Regular Part of a Complex Cone Singularity

Abstract

This article follows recent work of Miyajima on the complex-analytic approach to deformations of the regular part (i.e. the punctured smooth neighbourhood) of isolated singularities. Attention has previously focused on stably-embeddable infinitesimal deformations as those which correspond to standard algebraic deformations of the germ of a variety, and which also lead to convergent series solutions of the Kodaira-Spencer integrability equation. The emphasis of the present paper, however, is on the subspaces Zₒ of first cohomology classes containing infinitesimal deformations with vanishing Kodaira-Spencer bracket, and Wₒ, consisting more broadly of deformations for which the bracket represents the trivial second cohomology class. Deformations representing classes in Zₒ are automatically integrable, regardless of their analytic behaviour near the singular point. Classes in Wₒ are those for which only the first formal obstruction to integrability is overcome. After some preliminary results on cohomology, the main theorem of this paper gives a partial description of the analytic geometry of Zₒ and Wₒ for affine cones of arbitrary dimension.