Group-circulant singularities and partial desingularization preserving normal crossings

Abstract

The subject is partial desingularization preserving the normal crossings singularities of an algebraic or analytic variety X (over the complex field or over an uncountable algebraically closed field of characteristic zero, in the algebraic case). Our approach has three parts involving distinct techniques: (1) a formal splitting theorem for regular or analytic functions which satisfy a generic splitting hypothesis; (2) a study of singularities in the closure of the normal crossings locus, based on the combinatorics of G-circulant matrices, where G is a finite abelian group, leading to a theorem on reduction to group-circulant normal form; (3) a partial desingularization theorem, proved using (1) and (2) together with weighted blowings-up of group-circulant singularities. Previous results were for partial desingularization preserving simple normal crossings, or preserving general normal crossings when dim X < 5.