Mixed Bruce–Roberts numbers

Bivià-Ausina, Carles; Ruas, María Aparecida Soares

doi:10.1017/s0013091519000543

Cited by 6 publications

(2 citation statements)

References 18 publications

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“…The main goal of this section is to prove the equality (3). The next lemma is inspired in [3,Proposition 2.8] and the proof follows by using the same ideas.…”

Section: The Relative Bruce-roberts Numbermentioning

confidence: 99%

The relative Bruce-Roberts number of a function on a hypersurface

K.¹,

Nuño–Ballesteros²,

Oréfice-Okamoto³

et al. 2021

Preprint

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We consider the relative Bruce-Roberts number µ − BR (f, X) of a function on an isolated hypersurface singularity (X, 0). We show that µ − BR (f, X) is equal to the sum of the Milnor number of the fibre µ(f −1 (0) ∩ X, 0) plus the difference µ(X, 0) − τ (X, 0) between the Milnor and the Tjurina numbers of (X, 0). As an application, we show that the usual Bruce-Roberts number µ BR (f, X) is equal to µ(f ) + µ − BR (f, X). We also deduce that the relative logarithmic characteristic variety LC(X) − , obtained from the logarithmic characteristic variety LC(X) by eliminating the component corresponding to the complement of X in the ambient space, is Cohen-Macaulay.

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“…The main goal of this section is to prove the equality (3). The next lemma is inspired in [3,Proposition 2.8] and the proof follows by using the same ideas.…”

Section: The Relative Bruce-roberts Numbermentioning

confidence: 99%

The relative Bruce-Roberts number of a function on a hypersurface

K.¹,

Nuño–Ballesteros²,

Oréfice-Okamoto³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…In fact, the Cohen-Macaulayness of LC(X) and LC(X) − implies the conservation of both numbers in any deformation of f . Many authors have recent papers about these issues [1,2,12,13,16,17,19,20].…”

Section: Introductionmentioning

confidence: 99%

The Bruce-Roberts Numbers of a Function on an ICIS

K.¹,

Nuño–Ballesteros²,

Oréfice-Okamoto³

et al. 2022

Preprint

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We give formulas for the Bruce-Roberts number µ BR (f, X) and its relative version µ − BR (f, X) of a function f with respect to an ICIS (X, 0). We show that, where µ and τ are the Milnor and Tjurina numbers, respectively, of the ICIS. The formula for µ BR (f, X) is more complicated and also involves µ(f ) and some lengths in terms of the ideals I X and Jf . We also consider the logarithmic characteristic variety, LC(X), and its relative version, LC(X) − . We show that LC(X) − is Cohen-Macaulay and that LC(X) is Cohen-Macaulay at any point not in X × {0}. We generalize previous results presented by the authors when (X, 0) has codimension one and by Bruce and Roberts when it is weighted homogeneous of any codimension.

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Bruce–Roberts numbers and quasihomogeneous functions on analytic varieties

Bivià-Ausina,

Kourliouros,

Ruas

2024

Res Math Sci

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Given a germ of an analytic variety X and a germ of a holomorphic function f with a stratified isolated singularity with respect to the logarithmic stratification of X, we show that under certain conditions on the singularity type of the pair (f, X), the following relative analog of the well-known K. Saito’s theorem holds true: equality of the relative Milnor and Tjurina numbers of f with respect to X (also known as Bruce–Roberts numbers) is equivalent to the relative quasihomogeneity of the pair (f, X), i.e. to the existence of a coordinate system such that both f and X are quasihomogeneous with respect to the same positive rational weights.

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Mixed Bruce–Roberts numbers

Cited by 6 publications

References 18 publications

The relative Bruce-Roberts number of a function on a hypersurface

The relative Bruce-Roberts number of a function on a hypersurface

The Bruce-Roberts Numbers of a Function on an ICIS

Bruce–Roberts numbers and quasihomogeneous functions on analytic varieties

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