The Bruce-Roberts number of a function on a hypersurface with isolated singularity

TL;DR

This paper establishes a formula relating the Bruce-Roberts number of a function on a hypersurface with isolated singularity to Milnor and Tjurina numbers, and proves the Cohen-Macaulay property of the logarithmic characteristic variety, generalizing previous weighted homogeneous cases.

Contribution

The paper provides a new explicit formula for the Bruce-Roberts number and proves the Cohen-Macaulay property of the logarithmic characteristic variety for general isolated hypersurface singularities.

Findings

ormula relating ruce-Roberts number to Milnor and Tjurina numbers.

ohen-Macaulayness of the logarithmic characteristic variety.

xtension of previous weighted homogeneous results.

Abstract

Let $(X,0)$ be an isolated hypersurface singularity defined by $\phi\colon(\mathbb C^n,0)\to(\mathbb C,0)$ and $f\colon(\mathbb C^n,0)\to\mathbb C$ such that the Bruce-Roberts number $\mu_{BR}(f,X)$ is finite. We first prove that $\mu_{BR}(f,X)=\mu(f)+\mu(\phi,f)+\mu(X,0)-\tau(X,0)$, where $\mu$ and $\tau$ are the Milnor and Tjurina numbers respectively of a function or an isolated complete intersection singularity. Second, we show that the logarithmic characteristic variety $LC(X,0)$ is Cohen-Macaulay. Both theorems generalize the results of a previous paper by some of the authors, in which the hypersurface $(X,0)$ was assumed to be weighted homogeneous.

Equations

$${\mu }_{BR}(f,X)=\mu (f)+\mu (\varphi ,f),$$

$${\mu }_{BR}(f,X)=\mu (f)+\mu (\varphi ,f)+\mu (X,0)-\tau (X,0),$$

$${\Theta }_X:=\{\xi \in {\Theta }_n;dg(\xi )=\xi (g)\in I(X),\forall g\in I(X)\}$$

$$\varphi \frac{\partial }{\partial x_i},\frac{\partial \varphi }{\partial x_j}\frac{\partial }{\partial x_k}-\frac{\partial \varphi }{\partial x_k}\frac{\partial }{\partial x_i},\text{ with }i,j,k=1,...,n;k\ne j.$$

$${\mu }_{BR}(f,X)={\dim }_{\mathbb{C}}\frac{{\mathcal{O}}_n}{df({\Theta }_X)}$$

$${\mu }_{BR}(f,X)=\mu (f)+\mu (\varphi ,f),$$

$$u=(u_1,\dots ,u_q{)}^T=A.v^T,\hat{u}=(u_1,\dots ,u_{q-1}{)}^T=\hat{A}.v^T.$$

$$I^{\prime }$$

$$I^{\prime \prime }$$

$$I$$

$$l\left(R/I\right)=l\left(R/I^{\prime }\right)+l\left(R/I^{\prime \prime }\right).$$

$${\dim }_{\mathbb{C}}\frac{{\mathcal{O}}_n}{df({\Theta }_X^T)}<\infty .$$

$$\varphi (x)\frac{\partial f}{\partial x_i}(x)=0,\forall i=1,\dots ,n.$$

$${\mu }_{BR}(f,X)=\mu (f)+\mu (\varphi ,f)+\mu (X,0)-{\dim }_{\mathbb{C}}\frac{df({\Theta }_X)}{df({\Theta }_X^T)}.$$

$$H:({\mathbb{C}}^n\times \mathbb{C},0)$$

$$(x,t)$$

$$A=\left(\begin{array}{cccc}\frac{\partial g}{\partial x_1}& ...& \frac{\partial g}{\partial x_n}& \frac{\partial g}{\partial t}\text{vspace}0.1cm\\ \frac{\partial H}{\partial x_1}& ...& \frac{\partial H}{\partial x_n}& \frac{\partial H}{\partial t}\end{array}\right)=\left(\begin{array}{cccc}\frac{\partial f}{\partial x_1}& ...& \frac{\partial f}{\partial x_n}& 2t\text{vspace}0.1cm\\ e^t\frac{\partial \varphi }{\partial x_1}& ...& e^t\frac{\partial \varphi }{\partial x_n}& e^t\varphi \end{array}\right)$$

$$v=(0,...,0,1)\in {\mathbb{C}}^{n+1},$$

$$I^{\prime }$$

$$I^{\prime \prime }$$

$$I$$

$${\dim }_{\mathbb{C}}\frac{{\mathcal{O}}_{n+1}}{I}={\dim }_{\mathbb{C}}\frac{{\mathcal{O}}_n}{⟨\varphi \frac{\partial f}{\partial x_i}⟩+J(f,\varphi )}={\dim }_{\mathbb{C}}\frac{{\mathcal{O}}_n}{df({\Theta }_X^T)}<\infty ,$$

$${\dim }_{\mathbb{C}}\left(\frac{{\mathcal{O}}_{n+1}}{I}\right)={\dim }_{\mathbb{C}}\left(\frac{{\mathcal{O}}_{n+1}}{I^{\prime }}\right)+{\dim }_{\mathbb{C}}\left(\frac{{\mathcal{O}}_{n+1}}{I^{\prime \prime }}\right).$$

$${\dim }_{\mathbb{C}}\frac{{\mathcal{O}}_{n+1}}{I^{\prime }}$$

$${\dim }_{\mathbb{C}}\frac{{\mathcal{O}}_{n+1}}{I^{\prime \prime }}$$

$${\mu }_{BR}(f,X)={\dim }_{\mathbb{C}}\frac{{\mathcal{O}}_n}{df({\Theta }_X^T)}-{\dim }_{\mathbb{C}}\frac{df({\Theta }_X)}{df({\Theta }_X^T)},$$

$${\mu }_{BR}(f,X)=\mu (f)+\mu (\varphi ,f)+\mu (X,0)-{\dim }_{\mathbb{C}}\frac{df({\Theta }_X)}{df({\Theta }_X^T)}.$$

$$\overline{E}:\frac{{\Theta }_X}{{\Theta }_X^T}\to \frac{df({\Theta }_X)}{df({\Theta }_X^T)}.$$

$$\tau (X,0)={\dim }_{\mathbb{C}}\frac{dp({\Theta }_X)}{dp({\Theta }_X^T)}.$$

$$0⟶\ker (\alpha )\stackrel{i}{⟶}\frac{{\mathcal{O}}_n}{dp({\Theta }_X^T)}\stackrel{\alpha }{⟶}\frac{{\mathcal{O}}_n}{dp({\Theta }_X^T)}\stackrel{\pi }{⟶}\frac{{\mathcal{O}}_n}{⟨\varphi ⟩+J\varphi }⟶0$$

$$\ker (\pi )=\frac{⟨\varphi ⟩+J\varphi }{⟨\varphi ⟩+J(p,\varphi )},$$

$$\mathrm{Im}(\alpha )=\frac{⟨\frac{\partial \varphi }{\partial x_1}⟩+⟨\varphi ⟩+J(p,\varphi )}{⟨\varphi ⟩+J(p,\varphi )},$$

$${\dim }_{\mathbb{C}}\ker (\alpha )={\dim }_{\mathbb{C}}\frac{{\mathcal{O}}_n}{⟨\varphi ⟩+J\varphi }=\tau (X,0).$$

Full text

The first author has been partially supported by the MICINN Grant PGC2018-094889-B-I00. The second author has been partially supported by FAPESP 2016/25730-0. The third author has been partially supported by CAPES. The fourth author is partially supported by CNPq Grant 309086/2017-5 and FAPESP Grant 2018/22090-5.

1. Introduction

The Milnor number $\displaystyle\mu(f)$ is probably the most important invariant of a holomorphic function germ $\displaystyle f\colon(\mathbb{C}^{n},0)\to\mathbb{C}$. It is defined algebraically as the colength of the Jacobian ideal $\displaystyle Jf$, generated by the partial derivatives of $\displaystyle f$ in $\displaystyle\mathcal{O}_{n}$, the local ring of holomorphic function germs $\displaystyle(\mathbb{C}^{n},0)\to\mathbb{C}$. The finiteness of $\displaystyle\mu(f)$ is equivalent to that $\displaystyle f$ has isolated singularity, which is also equivalent, by Mather’s theorem, to that $\displaystyle f$ is finitely determined with respect to $\displaystyle\mathcal{R}$, the group of coordinate changes in the source. From the topological viewpoint, Milnor showed in [18] that the Milnor fibre of $\displaystyle f$ has the homotopy type of a wedge of $\displaystyle(n-1)$-spheres and that $\displaystyle\mu(f)$ is precisely the number of such spheres. Another important property is the conservation of the Milnor number, which implies that $\displaystyle\mu(f)$ is equal to the number of critical points of a Morsification of $\displaystyle f$.

An interesting and natural extension of this theory appears when we look at function germs defined over singular varieties. In [4], Bruce and Roberts studied function germs $\displaystyle f\colon(\mathbb{C}^{n},0)\to\mathbb{C}$ considered over a germ of analytic variety $\displaystyle(X,0)$ in $\displaystyle(\mathbb{C}^{n},0)$. They defined an invariant, which we call the Bruce-Roberts number of $\displaystyle f$ with respect to $\displaystyle X$, and denote it by $\displaystyle\mu_{BR}(f,X)$. This number is a generalization of the Milnor number in the sense that $\displaystyle\mu_{BR}(f,X)=\mu(f)$ when $\displaystyle X=\mathbb{C}^{n}$. Moreover, it has also the properties that $\displaystyle\mu_{BR}(f,X)$ is finite if and only if $\displaystyle f$ has isolated singularity over $\displaystyle(X,0)$ (in the stratified sense), if and only if $\displaystyle f$ is finitely $\displaystyle\mathcal{R}_{X}$-determined ($\displaystyle\mathcal{R}_{X}$ is the subgroup of $\displaystyle\mathcal{R}$ of coordinate changes which preserve $\displaystyle X$).

Since then, the Bruce-Roberts number has been studied by many authors, see for instance [1, 2, 10, 15, 20, 25]. In particular, in [20] the first, second and fourth authors considered the case that $\displaystyle(X,0)$ is a weighted homogeneous isolated hypersurface singularity. The first main result of [20] is the equality

[TABLE]

where $\displaystyle\phi\colon(\mathbb{C}^{n},0)\to(\mathbb{C},0)$ is the defining equation of $\displaystyle(X,0)$ and $\displaystyle\mu(\phi,f)$ is the Milnor number of the isolated complete intersection singularity (ICIS) defined by $\displaystyle(\phi,f)\colon(\mathbb{C}^{n},0)\to\mathbb{C}^{2}$ (in the sense of Hamm [11, 12]). We remark that this result was produced during the PhD of the second author (see [21]) and, at that time, it was conjectured that if $\displaystyle(X,0)$ is an isolated hypersurface singularity (not necessarily weighted homogeneous), then

[TABLE]

where $\displaystyle\tau(X,0)$ is the Tjurina number. This conjecture is the natural one since $\displaystyle\mu(X,0)=\tau(X,0)$ in the weighted homogeneous case.

In this paper we give a proof of (1), which gives a simple way to compute the Bruce-Roberts number of any function with respect to any isolated hypersurface singularity. In fact, with our method, we only need to compute the Tjurina number and the colength of the image of the differential of $\displaystyle f$ over the trivial tangent vector fields to the hypersurface. As a byproduct of our technique, we also obtain a new formula for the Tjurina number of an isolated hypersurface singularity in terms of its tangent vector fields (see also [25]). We remark that (1) has been also proved recently by Kourliouros [15] by using a totally different approach.

The second main result of [20] was that the logarithmic characteristic variety $\displaystyle LC(X,0)$ is Cohen-Macaulay, again when $\displaystyle(X,0)$ is a weighted homogeneous isolated hypersurface singularity. This is important and has many interesting applications. In fact, this implies the conservation of the Bruce-Roberts number and that $\displaystyle\mu_{BR}(f,X)$ is equal to the number of critical points of a Morsification of $\displaystyle f$ on each logarithmic strata, counted multiplicity (see [4]). Here we also extend this result to the general case that $\displaystyle(X,0)$ is not necessarily weighted homogeneous. We remark that $\displaystyle LC(X,0)$ is never Cohen-Macaulay when $\displaystyle(X,0)$ has codimension $\displaystyle>1$, so the only interesting case is when $\displaystyle(X,0)$ is a hypersurface. We give some applications of this result for $\displaystyle\mu_{BR}$-constant families of functions.

2. The Bruce-Roberts Number

Let $\displaystyle\mathcal{O}_{n}$ be the ring of germs of analytic functions $\displaystyle f\colon(\mathbb{C}^{n},0)\to\mathbb{C}$, and let $\displaystyle(X,0)$ be a germ of hypersurface with isolated singularity in $\displaystyle(\mathbb{C}^{n},0)$. Two germs $\displaystyle f,g\colon(\mathbb{C}^{n},0)\to\mathbb{C}$ are $\displaystyle\mathcal{R}_{X}$-equivalent, (respectively $\displaystyle C^{0}$-$\displaystyle\mathcal{R}_{X}$-equivalent) if there exists a germ of diffeomorphism (respectively homeomorphism) $\displaystyle\psi\colon(\mathbb{C}^{n},0)\to(\mathbb{C}^{n},0)$, such that $\displaystyle\psi(X)=X$ and $\displaystyle f=g\circ\psi$.

We denote by $\displaystyle\Theta_{n}$ the $\displaystyle\mathcal{O}_{n}$-module of germs of vector fields on $\displaystyle(\mathbb{C}^{n},0)$, and by $\displaystyle\Theta_{X}$ the subset of $\displaystyle\Theta_{n}$ of vector fields which are tangent to $\displaystyle X$, that is,

[TABLE]

where $\displaystyle I(X)$ is the ideal in $\displaystyle\mathcal{O}_{n}$ consisting of germs of functions vanishing on $\displaystyle X$. We denote by $\displaystyle df(\Theta_{X})$ the ideal in $\displaystyle{\mathcal{O}}_{n}$ generated by $\displaystyle\xi(f):=df(\xi)$ such that $\displaystyle\xi\in\Theta_{X}$ and $\displaystyle df$ is the differential of $\displaystyle f$. We consider $\displaystyle\Theta_{X}^{T}$ the $\displaystyle\mathcal{O}_{n}$-submodule of $\displaystyle\Theta_{X}$ generated by the trivial vector fields, that is, $\displaystyle\Theta_{X}^{T}$ is generated by

[TABLE]

Therefore, for any function germ $\displaystyle f\colon(\mathbb{C}^{n},0)\to\mathbb{C}$, $\displaystyle df(\Theta_{X}^{T})=\langle\phi\tfrac{\partial f}{\partial x_{i}}\rangle+J(f,\phi)$, where $\displaystyle J(f,\phi)$ is the ideal in $\displaystyle\mathcal{O}_{n}$ generated by the maximal minors of the Jacobian matrix of $\displaystyle(f,\phi)$.

Definition 2.1**.**

Let $\displaystyle f\in\mathcal{O}_{n}$. The number

[TABLE]

is the Bruce-Roberts number of $\displaystyle f$ with respect to $\displaystyle X$.

It follows from Nakayama’s Lemma and the Hilbert Nullstellenstaz that the Bruce-Roberts number of $\displaystyle f$ with respect to $\displaystyle(X,0)$ is finite if and only if the variety of zeros of the ideal $\displaystyle df(\Theta_{X})$ is equal to the origin or empty. Moreover, we have other important equivalences, for instance, that $\displaystyle\mu_{BR}(f,X)$ is finite if and only if $\displaystyle f$ is finitely $\displaystyle\mathcal{R}_{X}$-determined (see [4]).

Since $\displaystyle df(\Theta_{X})\subset Jf$, the Bruce-Roberts number is greater or equal to the Milnor number. In [20] we show that if $\displaystyle(X,0)$ is a weighted homogeneous hypersurface with isolated singularity and $\displaystyle f$ is a finitely $\displaystyle\mathcal{R}_{X}$-determined function germ, then

[TABLE]

where $\displaystyle\mu(\phi,f)$ is the Milnor number of the ICIS $\displaystyle(X\cap f^{-1}(0),0)$, as defined by Hamm in [11, 12]. The main goal of this work is to extend this formula to the general case that $\displaystyle(X,0)$ is hypersurface with isolated singularity (not necessarily weighted homogeneous). We use the following two lemmas.

Lemma 2.2**.**

[4]** Let $\displaystyle R$ be a local ring, $\displaystyle A$ a $\displaystyle q\times p$ matrix, with $\displaystyle p\geq q$ and $\displaystyle v$ a sequence of $\displaystyle p$ elements in $\displaystyle R$. Let $\displaystyle\hat{A}$ denote the $\displaystyle(q-1)\times p$ matrix obtained by deleting the last row of $\displaystyle A$ and put

[TABLE]

Denote the ideal in $\displaystyle R$ generated by the $\displaystyle q\times q$ minors of $\displaystyle A$ by $\displaystyle I_{q}(A)$ and define $\displaystyle I_{q-1}(\hat{A})$ in a similar way. Finally let

[TABLE]

Then, if $\displaystyle R$ is Cohen-Macaulay of dimension $\displaystyle p$ and $\displaystyle l\left(R/I\right)<\infty$,

[TABLE]

Lemma 2.3**.**

Let $\displaystyle(X,0)$ be an isolated hypersurface singularity in $\displaystyle(\mathbb{C}^{n},0)$ and $\displaystyle f\in{\mathcal{O}_{n}}$. Then, $\displaystyle\mu_{BR}(f,X)<\infty$ if and only if

[TABLE]

Proof.

Assume that $\displaystyle\mu_{BR}(f,X)<\infty$ but $\displaystyle\operatorname{dim}_{\mathbb{C}}\mathcal{O}_{n}/df(\Theta_{X}^{T})=\infty$. By Nakayama’s Lemma and the Hilbert Nullstelllensatz, the variety $\displaystyle V(\langle\phi\tfrac{\partial f}{\partial x_{i}}\rangle+J(f,\phi))$ has dimension $\displaystyle\geq 1$. Let $\displaystyle x\neq 0$ such that $\displaystyle x\in V(\langle\phi\tfrac{\partial f}{\partial x_{i}}\rangle+J(f,\phi))$. In particular,

[TABLE]

Since $\displaystyle\mu(f)\leq\mu_{BR}(f,X)<\infty$, $\displaystyle V(Jf)=\{0\}$. Hence $\displaystyle\phi(x)=0$, and therefore, $\displaystyle x\in V(\langle\phi\rangle+J(f,\phi))$. But this is in contradiction with [2, Proposition 2.8]. The converse follows because $\displaystyle df(\Theta_{X}^{T})\subset df(\Theta_{X}).$ ∎

The following theorem is our first new characterization of the Bruce-Roberts number.

Theorem 2.4**.**

Let $\displaystyle(X,0)$ be an isolated hypersurface singularity in $\displaystyle(\mathbb{C}^{n},0)$ and $\displaystyle f\in{\mathcal{O}_{n}}$ such that $\displaystyle\mu_{BR}(f,X)<\infty$. Then,

[TABLE]

Proof.

We consider the following function germs

[TABLE]

In order to apply Lemma 2.2, let

[TABLE]

and

[TABLE]

then

[TABLE]

By Lemma 2.3,

[TABLE]

so we can use Lemma 2.2:

[TABLE]

On the other hand,

[TABLE]

by the Lê-Greuel formula [3]. Moreover,

[TABLE]

which gives

[TABLE]

∎

3. The Tjurina number

In this section we present a characterization of the Tjurina number of an isolated hypersurface singularity $\displaystyle(X,0)$ in terms of the tangent vectors fields to $\displaystyle(X,0)$.

Lemma 3.1**.**

Let $\displaystyle\phi\colon(\mathbb{C}^{n},0)\to(\mathbb{C},0)$ be an analytic germ with isolated singularity and let $\displaystyle X=\phi^{-1}(0)$. Assume that $\displaystyle\eta\in\Theta_{X}$ and $\displaystyle d\phi(\eta)=\lambda\phi$, for some $\displaystyle\lambda\in\mathcal{O}_{n}$. Then $\displaystyle\eta\in\Theta_{X}^{T}$ if and only if $\displaystyle\lambda\in J\phi$.

Proof.

If $\displaystyle\eta\in\Theta_{X}^{T}$, obviously $\displaystyle\lambda\in J\phi$. Conversely, if $\displaystyle\lambda\in J\phi$ there exists $\displaystyle\xi\in\Theta_{n}$ such that $\displaystyle d\phi(\xi)=\lambda$. Thus, $\displaystyle d\phi(\eta-\phi\xi)=0$. Since $\displaystyle\phi$ has isolated singularity, $\displaystyle\partial\phi/\partial x_{1},\dots,\partial\phi/\partial x_{n}$ form a regular sequence in $\displaystyle\mathcal{O}_{n}$. Therefore, the module of syzygies of $\displaystyle\partial\phi/\partial x_{1},\dots,\partial\phi/\partial x_{n}$ is generated by the trivial relations. We have that $\displaystyle\eta-\phi\xi\in\Theta_{X}^{T}$. But $\displaystyle\phi\xi$ is also in $\displaystyle\Theta_{X}^{T}$, so $\displaystyle\eta\in\Theta_{X}^{T}$.

∎

The following theorem shows that, surprisingly, $\displaystyle\operatorname{dim}_{\mathbb{C}}df(\Theta_{X})/df(\Theta_{X}^{T})$ does not depend on the function germ $\displaystyle f\in\mathcal{O}_{n}$, provided it is finitely $\displaystyle\mathcal{R}_{X}$-determined. As a consequence, we show that this dimension is equal to the Tjurina number of $\displaystyle(X,0)$.

Theorem 3.2**.**

Let $\displaystyle(X,0)\subset(\mathbb{C}^{n},0)$ be an isolated hypersurface singularity and let $\displaystyle f\in\mathcal{O}_{n}$ be finitely $\displaystyle\mathcal{R}_{X}$-determined. Let $\displaystyle E:\Theta_{X}\to df(\Theta_{X})$ be the evaluation map given by $\displaystyle E(\xi)=df(\xi).$ Then $\displaystyle E$ induces an isomorphism

[TABLE]

Proof.

Obviously, $\displaystyle E$ is an epimorphism and $\displaystyle E(\Theta_{X}^{T})=df(\Theta_{X}^{T})$. We only need to show that $\displaystyle\ker E\subset\Theta_{X}^{T}$. That is, we have to show that if $\displaystyle\eta\in\Theta_{X}$ such that $\displaystyle df(\eta)\equiv 0$, then $\displaystyle\eta\in\Theta_{X}^{T}$.

We have $\displaystyle\mu_{BR}(f,X)<\infty$. Let $\displaystyle\phi\colon(\mathbb{C}^{n},0)\to(\mathbb{C},0)$ be an analytic germ with isolated singularity such that $\displaystyle X=\phi^{-1}(0)$. By [2, Proposition 2.8], $\displaystyle(\phi,f)\colon(\mathbb{C}^{n},0)\to(\mathbb{C}^{2},0)$ defines an ICIS. Therefore, the variety determined by $\displaystyle J(\phi,f)$, $\displaystyle V(J(\phi,f))$, has dimension 1 (see [16]). Moreover, since $\displaystyle f$ and $\displaystyle\phi$ have isolated singularity, the rank of $\displaystyle(J(\phi,f)(x))$ is equal to $\displaystyle 1$ for $\displaystyle x\neq 0$ in a neighborhood of the origin. Hence, $\displaystyle V(J(\phi,f))$ is an isolated determinantal singularity (IDS) of dimension 1 and, therefore, it is reduced, by [19, lemma 2.5].

Suppose that $\displaystyle\eta\notin\Theta_{X}^{T}$. By Lemma 3.1, $\displaystyle d\phi(\eta)=\lambda\phi$, for some $\displaystyle\lambda\notin J\phi$. On the other hand, $\displaystyle f$ has also isolated singularity, so $\displaystyle\partial f/\partial x_{1},\dots,\partial f/\partial x_{n}$ form a regular sequence in $\displaystyle\mathcal{O}_{n}$. Therefore, the module of syzygies of $\displaystyle\partial f/\partial x_{1},\dots,\partial f/\partial x_{n}$ is generated by the trivial relations. Since $\displaystyle df(\eta)\equiv 0$, we have that $\displaystyle\eta\in\displaystyle\langle\tfrac{\partial f}{\partial x_{i}}\tfrac{\partial}{\partial x_{j}}-\tfrac{\partial f}{\partial x_{j}}\tfrac{\partial}{\partial x_{i}}\displaystyle\rangle$.

This implies that $\displaystyle\lambda\phi=d\phi(\eta)\in J(\phi,f)$ and hence $\displaystyle V(J(\phi,f))\subset V(\lambda\phi)=V(\lambda)\cup V(\phi)$. Let $\displaystyle C_{1},\dots,C_{k}$ be the irreducible components of $\displaystyle V(J(\phi,f))$. Since $\displaystyle V(J(\phi,f))$ is reduced of dimension 1, each $\displaystyle C_{i}$ is an irreducible curve in $\displaystyle(\mathbb{C}^{n},0)$. In particular, either $\displaystyle C_{i}\subset V(\lambda)$ or $\displaystyle C_{i}\subset V(\phi)$, for each $\displaystyle i=1,\dots,k$.

If $\displaystyle C_{i}\not\subset V(\phi)$ for all $\displaystyle i=1,...,k$, then $\displaystyle C_{i}\subset V(\lambda)$ for all $\displaystyle i=1,...,k$, hence $\displaystyle V(J(\phi,f))\subset V(\lambda)$, therefore $\displaystyle\lambda\in\sqrt{\lambda}\subset\sqrt{J(\phi,f)}=J(\phi,f)\subset J\phi$, which is a contradiction with the way $\displaystyle\lambda$ was chosen. Otherwise, if $\displaystyle C_{i}\subset V(\phi)$ for some $\displaystyle i$, then $\displaystyle C_{i}\subset V(\langle\phi\rangle+J(f,\phi))$. This gives $\displaystyle\operatorname{dim}_{\mathbb{C}}\mathcal{O}_{n}/(\langle\phi\rangle+J(f,\phi))=\infty$, in contradiction with the Lê-Greuel formula for the ICIS $\displaystyle V(\phi,f)$.

∎

We are ready now to characterize the Tjurina number in terms of the tangent vector fields to the hypersurface $\displaystyle(X,0)$.

Theorem 3.3**.**

Let $\displaystyle(X,0)\subset(\mathbb{C}^{n},0)$ be an isolated hypersurface singularity and let $\displaystyle p:\mathbb{C}^{n}\to\mathbb{C}$ be a non-zero linear function. Then

[TABLE]

Proof.

Since $\displaystyle p$ is a non-zero linear function, we may suppose that $\displaystyle p(x_{1},...,x_{n})=a_{1}x_{1}+\dots+a_{n}x_{n}$, with $\displaystyle a_{1}\neq 0$. We consider the following sequence

[TABLE]

where $\displaystyle\alpha$ is given by multiplication by $\displaystyle\overline{\partial\phi/\partial x_{1}}$, $\displaystyle i$ is the inclusion and $\displaystyle\pi$ is the projection.

The sequence above is exact. Indeed, $\displaystyle dp(\Theta_{X}^{T})=\langle\phi\rangle+J(p,\phi)\subset\langle\phi\rangle+J\phi$. Moreover,

[TABLE]

and

[TABLE]

so $\displaystyle\ker(\pi)\subset\operatorname{Im}(\alpha)$. The opposite inclusion follows because $\displaystyle\pi\circ\alpha=0$. Hence,

[TABLE]

We claim that

[TABLE]

In fact, given $\displaystyle g\in dp(\Theta_{X})$ there exists $\displaystyle\xi\in\Theta_{X}$ such that $\displaystyle g=dp(\xi)$. We have $\displaystyle g=d\phi(\xi)=\displaystyle\sum_{i=1}^{n}\xi_{i}\tfrac{\partial\phi}{\partial x_{i}}=\lambda\phi$, for some $\displaystyle\lambda\in{\mathcal{O}}_{n}$, and $\displaystyle dp(\xi)=\displaystyle\sum_{i=1}^{n}a_{i}\xi_{i}$. Therefore,

[TABLE]

This shows the inclusion

[TABLE]

Conversely, let $\displaystyle g\in\mathcal{O}_{n}$ be such that $\displaystyle g\frac{\partial\phi}{\partial x_{1}}\in\langle\phi\rangle+J(p,\phi)$. There exist $\displaystyle b,b_{ij}\in{\mathcal{O}_{n}}$ such that

[TABLE]

where $\displaystyle b^{\prime}_{jk}=-b_{jk}$, if $\displaystyle j<k$ and $\displaystyle b^{\prime}_{jk}=b_{kj}$, if $\displaystyle k<j$. Thus,

[TABLE]

It follows that

[TABLE]

and a simple calculation shows that $\displaystyle dp\left(\tfrac{1}{a_{1}}\xi\right)=g$. This gives the opposite inclusion

[TABLE]

∎

The following corollary is an immediate consequence of Theorems 3.2 and 3.3.

Corollary 3.4**.**

Let $\displaystyle(X,0)\subset(\mathbb{C}^{n},0)$ be an isolated hypersurface singularity and let $\displaystyle f\in\mathcal{O}_{n}$ be finitely $\displaystyle\mathcal{R}_{X}$-determined. Then

[TABLE]

We remark that Tajima in [25] has proved the same result with a different approach.

4. Main result and some applications

Finally, we prove the main result which gives the relationship between the Milnor number and the Bruce-Roberts number of $\displaystyle f$ with respect to an isolated hypersurface singularity $\displaystyle(X,0)$.

Corollary 4.1**.**

Let $\displaystyle(X,0)\subset(\mathbb{C}^{n},0)$ be the an isolated hypersurface singularity defined by $\displaystyle\phi\colon(\mathbb{C}^{n},0)\to(\mathbb{C},0)$ and let $\displaystyle f\in\mathcal{O}_{n}$ be finitely $\displaystyle\mathcal{R}_{X}$-determined. Then,

[TABLE]

Proof.

It follows from Theorem 2.4 and Corollary 3.4. ∎

The following results are direct applications of Corollary 4.1.

Corollary 4.2**.**

Let $\displaystyle f,\phi\colon(\mathbb{C}^{n},0)\to(\mathbb{C},0)$ be function germs with isolated singularity, and let $\displaystyle(X,0)$ and $\displaystyle(Y,0)$ be the hypersurfaces determined by $\displaystyle\phi$ and $\displaystyle f$, respectively. If $\displaystyle\mu_{BR}(f,X)<\infty$ and $\displaystyle\mu_{BR}(\phi,Y)<\infty$, then

[TABLE]

Proof.

The hypothesis imply that $\displaystyle(f,\phi)$ defines an ICIS, and $\displaystyle\mu(\phi,f)=\mu(f,\phi)$. By Corollary 4.1, we have then $\displaystyle\mu_{BR}(f,X)-\mu_{BR}(\phi,Y)=\tau(Y,0)-\tau(X,0).$

∎

The next corollary shows that the Bruce-Roberts number is a topological invariant when $\displaystyle(X,0)$ is an isolated hypersurface singularity.

Corollary 4.3**.**

Let $\displaystyle(X,0)\subset(\mathbb{C}^{n},0)$ be an isolated hypersurface singularity. Let $\displaystyle f,g\in\mathcal{O}_{n}$ be finitely $\displaystyle\mathcal{R}_{X}$-determined function germs such that $\displaystyle f$ is $\displaystyle C^{0}$-$\displaystyle{\mathcal{R}}_{X}$-equivalent to $\displaystyle g$. Then $\displaystyle\mu_{BR}(f,X)=\mu_{BR}(g,X).$

Proof.

Let $\displaystyle h:(\mathbb{C}^{n},0)\to(\mathbb{C}^{n},0)$ be a homeomorphism such that $\displaystyle h(X)=X$ and $\displaystyle g=f\circ h$. We have $\displaystyle h(X\cap f^{-1}(0))=X\cap g^{-1}(0)$. If $\displaystyle(X,0)$ is defined by $\displaystyle\phi\colon(\mathbb{C}^{n},0)\to(\mathbb{C},0)$, then $\displaystyle\mu(f,\phi)=\mu(g,\phi)$, because the Milnor number of an ICIS is a topological invariant. Moreover, $\displaystyle f$ and $\displaystyle g$ are also $\displaystyle C^{0}$-$\displaystyle\mathcal{R}$-equivalent, so $\displaystyle\mu(f)=\mu(g)$ by [18]. By Corollary 4.1, we get $\displaystyle\mu_{BR}(f,X)=\mu_{BR}(g,X)$. ∎

From Corollary 4.1 and the Lê-Greuel formula [3], we also have the following:

Corollary 4.4**.**

Let $\displaystyle(X,0)$ be the isolated hypersurface singularity defined by $\displaystyle\phi\colon(\mathbb{C}^{n},0)\to(\mathbb{C},0)$, and let $\displaystyle f\in\mathcal{O}_{n}$ be finitely $\displaystyle\mathcal{R}_{X}$-determined. Then

[TABLE]

We can use our results to relate the Bruce-Roberts number of a generic linear projection to the top polar multiplicity of an hypersurface as defined by Gaffney [7]. Let $\displaystyle(X,0)\subset(\mathbb{C}^{n},0)$ be an isolated hypersurface singularity defined as the zero set of a function germ $\displaystyle\phi\colon(\mathbb{C}^{n},0)\to\mathbb{C}$, the ($\displaystyle n-1$)-th polar multiplicity is defined as

[TABLE]

where $\displaystyle p\colon(\mathbb{C}^{n},0)\to(\mathbb{C},0)$ is a generic linear projection. From the proofs of Theorems 2.4 and 3.2, we observe that

[TABLE]

therefore,

[TABLE]

On the other hand, in [14], Perez and Saia show that

[TABLE]

where $\displaystyle m_{i}(X,0)$ is the $\displaystyle i$-th polar multiplicity. From Lê-Tessier’s formula, we have

[TABLE]

where $\displaystyle\operatorname{Eu}(X,0)$ denotes the Euler obstruction of $\displaystyle(X,0)$ (see [17]). Hence,

[TABLE]

We can conclude, hence, the following corollary.

Corollary 4.5**.**

Let $\displaystyle p\colon(\mathbb{C}^{n},0)\to(\mathbb{C},0)$ a generic linear projection and $\displaystyle(X,0)\subset(\mathbb{C}^{n},0)$ be a hypersurface with isolated singularity. Then:

1. (1)

$\displaystyle m_{n-1}(X,0)=\mu_{BR}(p,X)+\tau(X,0)$; 2. (2)

$\displaystyle\operatorname{Eu}(X,0)=\mu_{BR}(p,X)+\tau(X,0)-\mu(X,0)+(-1)^{n}.$**

We remark, by Corollary 4.5, that the Bruce-Roberts number of a generic linear projection on an isolated hypersurface singularity does not depend on the projection.

5. The logarithmic characteristic variety

We recall the definitions of logarithmic stratification and of logarithmic characteristic variety due to Saito [23]. Let $\displaystyle(X,0)\subset(\mathbb{C}^{n},0)$ be a germ of a reduced analytic subvariety. Take a representative $\displaystyle X$ on some small open neighbourhood $\displaystyle U$ of 0 in $\displaystyle\mathbb{C}^{n}$. For each $\displaystyle x\in U$, we denote by $\displaystyle\Theta_{X}(x)$ the linear subspace of $\displaystyle T_{x}U$ generated by the vectors $\displaystyle\delta(x)$, with $\displaystyle\delta\in\Theta_{X,x}$.

Lemma 5.1**.**

[4, 23]** There is a unique stratification $\displaystyle\{X_{\alpha};\;\alpha\in I\}$ of $\displaystyle U$ that satisfies the following properties:

1. (1)

Each stratum $\displaystyle X_{\alpha}$ is a smooth connected immersed submanifold of $\displaystyle U$ and $\displaystyle U$ is the disjoint union $\displaystyle\cup_{\alpha\in I}X_{\alpha}.$ 2. (2)

If $\displaystyle x\in U$ lies in a stratum $\displaystyle X_{\alpha}$, then the tangent space $\displaystyle T_{x}X_{\alpha}$ coincides with $\displaystyle\Theta_{X}(x).$ 3. (3)

If $\displaystyle X_{\alpha}$ and $\displaystyle X_{\beta}$ are two distinct strata with $\displaystyle X_{\alpha}$ meeting the closure of $\displaystyle X_{\beta}$ then $\displaystyle X_{\alpha}$ is contained in the frontier of $\displaystyle X_{\beta}$.

Definition 5.2**.**

The stratification $\displaystyle\{X_{\alpha}:\alpha\in I\}$ of the previous lemma is called the logarithmic stratification of $\displaystyle X$ and the strata $\displaystyle X_{\alpha}$, the logarithmic strata. The germ $\displaystyle(X,0)$ is holonomic if, for some neighborhood $\displaystyle U$ of $\displaystyle 0$ in $\displaystyle\mathbb{C}^{n}$, the logarithmic stratification has only finitely many strata.

Definition 5.3**.**

The logarithmic characteristic variety, $\displaystyle LC(X,0)$, is defined as follows. Suppose the vector fields $\displaystyle\delta_{1},\dots,\delta_{m}$ generate $\displaystyle\Theta_{X}$ on some neighborhood $\displaystyle U$ of $\displaystyle 0$ in $\displaystyle\mathbb{C}^{n}$. Let $\displaystyle T^{*}_{U}\mathbb{C}^{n}$ be the restriction of the cotangent bundle of $\displaystyle\mathbb{C}^{n}$ to $\displaystyle U$. We define $\displaystyle LC_{U}(X)$ to be

[TABLE]

Then $\displaystyle LC(X,0)$ is the germ of $\displaystyle LC_{U}(X)$ in $\displaystyle T^{*}\mathbb{C}^{n}$ along $\displaystyle T^{*}_{0}\mathbb{C}^{n}$, the cotangent space to $\displaystyle\mathbb{C}^{n}$ at $\displaystyle 0$.

We observe that $\displaystyle LC(X,0)$ is a well defined germ of analytic subvariety in $\displaystyle T^{*}\mathbb{C}^{n}$ which is independent of the choice of the vector fields $\displaystyle\delta_{i}$ (see [4, 23] for details). If $\displaystyle(X,0)$ is holonomic with logarithmic strata $\displaystyle X_{0},\dots,X_{k}$ then $\displaystyle LC(X,0)$ has dimension $\displaystyle n$, with irreducible components $\displaystyle Y_{0},\dots,Y_{k}$, where $\displaystyle Y_{i}=\overline{N^{*}X_{i}}$, the closure of the conormal bundle $\displaystyle N^{*}X_{i}$ of $\displaystyle X_{i}$ in $\displaystyle\mathbb{C}^{n}$ (see [4, Proposition 1.14]).

There exists a deep connection between the logarithmic characteristic variety and the Bruce-Roberts number. If $\displaystyle LC(X,0)$ is Cohen-Macaulay, then $\displaystyle\mu_{BR}(f,X)$ is equal to the number of critical points of a Morsification of $\displaystyle f$ on each logarithmic stratum $\displaystyle X_{\alpha}$, counted with multiplicity (see [4, Corollary 5.8]). However, it also is well known (see [4, Proposition 5.8]) that $\displaystyle LC(X,0)$ is never Cohen-Macaulay when $\displaystyle(X,0)$ has codimension $\displaystyle>1$. Thus, the only interesting case to look at is when $\displaystyle(X,0)$ is hypersurface.

In a previous paper [20], we showed that $\displaystyle LC(X,0)$ is Cohen-Macaulay when $\displaystyle(X,0)$ is a weighted homogeneous isolated hypersurface singularity. This fact had several interesting corollaries (see [1, 9, 10, 20]). In the next theorem, we extend this result to the general case that $\displaystyle(X,0)$ is an isolated hypersurface singularity (not necessarily weighted homogeneous).

Theorem 5.4**.**

Let $\displaystyle(X,0)$ be any isolated hypersurface singularity. Then $\displaystyle LC(X,0)$ is Cohen-Macaulay.

Proof.

We assume that $\displaystyle X=\phi^{-1}(0)$, with $\displaystyle\phi:(\mathbb{C}^{n},0)\to(\mathbb{C},0)$. We have to prove that $\displaystyle LC(X,0)$ is Cohen-Macaulay at every point $\displaystyle(0,p)\in LC(X,0)$. Let $\displaystyle f\in\mathcal{O}_{n}$ be finitely $\displaystyle\mathcal{R}_{X}$-determined such that $\displaystyle df(0)=p$. Let $\displaystyle C_{1},\dots,C_{k}$ be the irreducible components of $\displaystyle X$. The logarithmic strata of $\displaystyle X$ are $\displaystyle X_{0}=\mathbb{C}^{n}\smallsetminus X$, $\displaystyle X_{i}=C_{i}\smallsetminus\{0\}$ for $\displaystyle i=1,\dots,k$ and $\displaystyle X_{k+1}=\{0\}$. Let $\displaystyle F\colon(\mathbb{C}^{n}\times\mathbb{C},0)\to\mathbb{C}$ be given by $\displaystyle F(x,t)=f_{t}(x)$ a Morsification of $\displaystyle f$. It follows from [4, Corollary 5.8] that $\displaystyle LC(X,0)$ is Cohen-Macaulay at $\displaystyle(0,p)$ if and only if

[TABLE]

where $\displaystyle n_{i}$ is the number of critical points of $\displaystyle f_{t}$ on the stratum $\displaystyle X_{i}$ and $\displaystyle m_{i}$ is the multiplicity of the corresponding irreducible component $\displaystyle Y_{i}$ of $\displaystyle LC(X,0)$.

By Corollary 4.4, we know that

[TABLE]

We write $\displaystyle\mathcal{O}_{n+1}/(\langle F\rangle+J(f_{t},\phi))=R/I$, where $\displaystyle R=\mathcal{O}_{n+1}/\langle F\rangle$ and $\displaystyle I=(\langle F\rangle+J(\phi,f_{t}))/\langle F\rangle$. The ring $\displaystyle R$ is Cohen-Macaulay of dimension $\displaystyle n$ and $\displaystyle I$ is generated by the 2-minors of a matrix of size $\displaystyle n\times 2$. Since $\displaystyle\operatorname{dim}R/I=1$ and $\displaystyle 1=n-(n-2+1)(2-2+1)$, it follows that $\displaystyle R/I$ is determinantal and, therefore, Cohen-Macaulay, by Eagon-Hochster result [6]. In particular, we have conservation of multiplicity, that is, for all $\displaystyle t\neq 0$, we have

[TABLE]

We remark the sum in the right hand side has only a finite number of terms, corresponding to the critical points of $\displaystyle f_{t}$ on $\displaystyle X_{i}$.

When $\displaystyle i=1,\dots,k$ and $\displaystyle x\in X_{i}$, $\displaystyle X$ is smooth at $\displaystyle x$ and $\displaystyle m_{i}=1$, hence

[TABLE]

since $\displaystyle f_{t}$ is a Morse function (see [4, Proposition 5.12]). For $\displaystyle i=0$, if $\displaystyle x\in X_{0}$ then $\displaystyle x\notin X$. We have

[TABLE]

since $\displaystyle f_{t}$ is a Morse function and in this case $\displaystyle m_{0}=1$. Finally, for $\displaystyle i=k+1$ we have just one critical point $\displaystyle x=0$ in $\displaystyle X_{k+1}=\{0\}$. Thus,

[TABLE]

since $\displaystyle f_{t}$ is a Morse function and in this case $\displaystyle n_{k+1}=1$. Summing up for all $\displaystyle i=0,\dots,k+1$ we get

[TABLE]

and hence $\displaystyle\mu_{BR}(f,X)=\sum_{i=0}^{k+1}n_{i}m_{i}$.

∎

Corollary 5.5**.**

Let $\displaystyle(X,0)$ be an isolated hypersurface singularity and let $\displaystyle f\in\mathcal{O}_{n}$ be finitely $\displaystyle\mathcal{R}_{X}$-determined. Then,

[TABLE]

where $\displaystyle N$ is the number of critical points of a Morsification of $\displaystyle f$ on $\displaystyle X\smallsetminus\{0\}$.

Proof.

By Theorem 5.4 and [4, Corollary 5.8, Propositions 5.12 and 5.14], we have

[TABLE]

where $\displaystyle p:(\mathbb{C}^{n},0)\to(\mathbb{C},0)$ is a generic linear projection, $\displaystyle(*)$ follows from Corollary 4.4, $\displaystyle(**)$ follows from the Lê-Greuel formula and the last equality follows again from Lê-Greuel formula and from the definition of the polar multiplicity. ∎

The constancy of the Milnor number in a family of holomorphic functions is controlled by means of the integral closure of the Jacobian ideal (see [8]). Similar results have been obtained for the Bruce-Roberts number in [1], but there the authors need the additional hypothesis that $\displaystyle LC(X,0)$ is Cohen-Macaulay. It follows from our Theorem 5.4 that the results in [1] are true for any isolated hypersurface singularity.

More specifically, let $\displaystyle(X,0)\subset(\mathbb{C}^{n},0)$ be an isolated hypersurface singularity and let $\displaystyle f\colon(\mathbb{C}^{n},0)\to\mathbb{C}$ be finitely $\displaystyle\mathcal{R}_{X}$-determined. Given $\displaystyle F\colon(\mathbb{C}^{n}\times\mathbb{C},0)\to(\mathbb{C},0)$ a deformation of $\displaystyle f$, we put $\displaystyle f_{t}(x)=F(x,t)$. We say that $\displaystyle F$ is a $\displaystyle\mu_{BR}$-constant deformation of $\displaystyle f$ if $\displaystyle\mu_{BR}(f_{t},X)=\mu_{BR}(f,X)$ for $\displaystyle t$ small enough.

We also recall that the polar curve of $\displaystyle F$ is defined as

[TABLE]

where $\displaystyle\xi_{1},\dots,\xi_{p}$ are generators of $\displaystyle\Theta_{X}$. The following corollary is an immediate consequence of the results of [1] and Theorem 5.4.

Corollary 5.6**.**

With the above notation, the following statements are equivalent:

• (i)

$\displaystyle F$* is a $\displaystyle\mu_{BR}$-constant deformation of $\displaystyle f$;*

• (ii)

The polar curve of $\displaystyle F$ with respect to $\displaystyle\{t=0\}$ does not split, that is, $\displaystyle C=\{0\}\times\mathbb{C}.$

Moreover, if $\displaystyle\frac{\partial F}{\partial t}\in\overline{df(\Theta_{X})}$ then $\displaystyle F$ is a $\displaystyle\mu_{BR}$-constant deformation of $\displaystyle f$ (where $\displaystyle\overline{I}$ is the integral closure of an ideal $\displaystyle I$).

Other interesting results for $\displaystyle\mu_{BR}$-constant families of functions are obtained by Grulha in [10]. Again, these results need the additional hypothesis that $\displaystyle LC(X,0)$ is Cohen-Macaulay, which is not necessary any more by our Theorem 5.4. In the following corollary we denote by $\displaystyle\operatorname{Eu}(f,X)$ the relative local Euler obstruction of a function $\displaystyle f$ on $\displaystyle(X,0)$ (see [10] for details).

Corollary 5.7**.**

With the same notation as in Corollary 5.6:

• (i)

If $\displaystyle\mu_{BR}(f_{t},X)$ is constant, then $\displaystyle\mu(f_{t})$, $\displaystyle\mu(X\cap f_{t}^{-1}(0),0)$ and $\displaystyle\operatorname{Eu}(f_{t},X)$ are constant in the family.

• (ii)

If $\displaystyle\mu(f_{t})$ is constant, then the constancy of $\displaystyle\operatorname{Eu}(f_{t},X)$ or $\displaystyle\mu(X\cap f_{t}^{-1}(0),0)$ implies that $\displaystyle\mu_{BR}(f_{t},X)$ is constant in the family,

Index