Local Normal Forms of Noncommutative Functions

Abstract

This article describes local normal forms of functions in noncommuting variables, up to equivalence generated by isomorphism of noncommutative Jacobi algebras, extending singularity theory in the style of Arnold's commutative local normal forms into the noncommutative realm. This generalisation unveils many new phenomena, including an ADE classification when the Jacobi ring has dimension zero and, by taking suitable limits, a further ADE classification in dimension one. These are natural generalisations of the simple singularities and those with infinite multiplicity in Arnold's classification. We obtain normal forms away from some exceptional Type E cases. Remarkably these normal forms have no continuous parameters, and the key new feature is that the noncommutative world affords larger families. This theory has a range of immediate consequences to the birational geometry of 3-folds. The normal forms of dimension zero are the analytic classification of smooth 3-fold flops, and one outcome of NC singularity theory is the first list of all Type D flopping germs, generalising Reid's famous pagoda classification of Type A, with variants covering Type E. The normal forms of dimension one have further applications to divisorial contractions to a curve. In addition, the general techniques also give strong evidence towards new contractibility criteria for rational curves.