Initial logarithmic Lie algebras of hypersurface singularities

TL;DR

This paper introduces a new Lie algebra structure for hypersurface singularities, extends existing theorems, and establishes convergence and linearization results, providing insights into the automorphism groups and classification of such singularities.

Contribution

It defines the initial Lie algebra of logarithmic vector fields, extends the structure theorem, and relates to automorphism groups, offering new tools for classifying hypersurface singularities.

Findings

The initial Lie algebra can be lifted to a linear Lie algebra of vector fields.

Convergence of linearization is proven for quasihomogeneous singularities.

A lower bound for the dimension of singularities is established based on the initial Lie algebra.

Abstract

We introduce a Lie algebra of initial terms of logarithmic vector fields along a hypersurface singularity. Extending the formal structure theorem in [GS06, Thm. 5.4], we show that the completely reducible part of its linear projection lifts formally to a linear Lie algebra of logarithmic vector fields. For quasihomogeneous singularities, we prove convergence of this linearization. We relate our construction to the work of Hauser and M"uller [M"ul86, HM89] on Levi subgroups of automorphism groups of singularities, which proves convergence even for algebraic singularities. Based on the initial Lie algebra, we introduce a notion of reductive hypersurface singularity and show that any reductive free divisor is linear. As an application, we describe a lower bound for the dimension of hypersurface singularities in terms of the semisimple part of their initial Lie algebra.