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

Abstract

We introduce the notion of the \textit{Bruce-Roberts Tjurina number} for holomorphic 1-forms with respect to a pair $(X,V)$ of complex analytic subvarieties. We characterize its finiteness and, when $X$ and $V$ are hypersurfaces with at most isolated singularities, relate it to a determinantal colength, the Tjurina number of the 1-form with respect to $V$, and the Tjurina number of $X$. We also establish a decomposition of the difference between the Bruce-Roberts Milnor and Tjurina numbers and a bound for their quotient. As an application, we prove a relative quasihomogeneity result for germs of non-dicritical generalized curve foliations in complex dimension two.

arXiv comment17 pages