Local Normal Forms of Noncommutative Functions

TL;DR

This paper extends singularity theory to noncommutative functions, revealing new phenomena, classifications, and applications in algebraic geometry, including classifications of 3-fold flops and divisorial contractions.

Contribution

It introduces local normal forms for noncommutative functions, generalizing Arnold's singularity classifications and providing new insights into birational geometry of 3-folds.

Findings

Classification of noncommutative singularities into ADE types.

Normal forms with no continuous parameters, larger families in noncommutative setting.

Application to classification of 3-fold flops and divisorial contractions.

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.