The Bruce-Roberts Tjurina number of holomorphic 1-forms along complex analytic varieties

TL;DR

This paper introduces a new invariant called the Bruce-Roberts Tjurina number for holomorphic 1-forms on complex analytic varieties, relating it to existing invariants and applying it to classify certain holomorphic foliations.

Contribution

It defines the Bruce-Roberts Tjurina number for holomorphic 1-forms relative to pairs of varieties and establishes its relation to known invariants, with applications to foliation classification.

Findings

The Bruce-Roberts Tjurina number is related to the Bruce-Roberts number and classical Tjurina numbers.

Established a quasihomogeneity criterion for germs of holomorphic foliations in complex dimension two.

Provided formulas linking the new invariant to existing invariants in the hypersurface case.

Abstract

We introduce the notion of the Bruce-Roberts Tjurina number for holomorphic 1-forms relative to a pair $(X,V)$ of complex analytic subvarieties. When the pair $(X,V)$ consists of isolated complex analytic hypersurfaces, we prove that the Bruce-Roberts Tjurina number is related to the Bruce-Roberts number, the Tjurina number of the 1-form with respect to $V$ and the Tjurina number of $X$, among other invariants. As an application, we present a quasihomogeneity result for germs of holomorphic foliations in complex dimension two.