On deformations of isolated singularity functions

Abstract

We study multi-parameter deformations of isolated singularity function-germs on either a subanalytic set or a complex analytic space. We prove that if such a deformation has no coalescing of singular points, then it has weak constant topological type. This extends some classical results due to Lê and Ramanujam (Am J Math 98:67–78, 1976) and Parusiński (Bull Lond Math Soc 31(6):686–692, 1999), as well as a recent result due to Jesus-Almeida and the first author (Int Math Res Notices 2023(6):4869–4886, 2023). It also provides a sufficient condition for a one-parameter family of complex isolated singularity surfaces in ${\mathbb {C}}^3$ to have weak constant topological type. On the other hand, for complex isolated singularity families defined on an isolated determinantal singularity, we prove that $\mu$-constancy implies weak constant topological type.

1 Introduction

The study of deformations of maps and spaces plays a central role in Singularity Theory and in Complex Geometry. For holomorphic isolated singularity function-germs on a complex affine space ${\mathbb {C}}^n$, the topological behavior of small perturbations is related to the Milnor number, which is an analytic invariant with a very clear topological and algebraic meaning. In fact, when $n\ne 3$ there is a classical result due to Lê and Ramanujam [1] in 1976, which states that a family $f_s: ({\mathbb {C}}^n,0) \rightarrow ({\mathbb {C}},0)$ is topologically trivial if and only if the Milnor number $\mu (f_s)$ is constant, for $s \in {\mathbb {C}}$ sufficiently small.

The case $n=3$ is one of the main open problems in Singularity Theory, known as the $\mu$-constant conjecture. Here, topological triviality means that for each s small enough there exists a homeomorphism of germs $h_s: ({\mathbb {C}}^n,0) \rightarrow ({\mathbb {C}}^n,0)$ that takes the germ of $V(f_s):= f_s^{-1}(0)$ at the origin onto the germ of $V(f_0):= f_0^{-1}(0)$ at the origin.

On the other hand, King [2] and [3] studied the more general setting of families of real analytic affine map-germs, in several parameters, having an isolated singularity at the origin. He gave sufficient conditions for such a family to be topologically trivial. In particular, he introduced the concept of curve of coalescing of critical points, which is central in our paper.

In 1999, Parusiński [4] extended Lê & Ramanujam’s result for ${\mathbb {K}}$-analytic isolated singularity families $f_s: ({\mathbb {K}}^n,0) \rightarrow ({\mathbb {K}},0)$, with ${\mathbb {K}}={\mathbb {C}}$ or ${\mathbb {R}}$, even when $n=3$, provided that the family depends linearly on the parameter $s \in {\mathbb {K}}$. This means that it has the form $f_s = f_0 + s \varphi$, for some ${\mathbb {K}}$-analytic isolated singularity function-germ $\varphi : ({\mathbb {K}}^n,0) \rightarrow ({\mathbb {K}},0)$.

Recently, Jesus-Almeida and the first author [5] studied ${\mathbb {K}}$-analytic families $f_s: (X,0) \rightarrow ({\mathbb {K}},0)$ depending linearly on one parameter $s \in {\mathbb {K}}$, where $(X,0) \subset ({\mathbb {K}}^N,0)$ is the germ of a subanalytic set, when ${\mathbb {K}}={\mathbb {R}}$, or a complex analytic space, when ${\mathbb {K}}={\mathbb {C}}$. In particular, it was proved that if such a family has no coalescing of singular points, then it has weak constant topological type. Here, having no coalescing of singular points means that there exist real numbers $\delta >0$ and ${\epsilon }>0$ such that $\Sigma (f_s) \cap {\mathbb {B}}_{{\epsilon }} = \{0\}$ for every $s \in \mathbb {D}_\delta$, where $\Sigma (f_s)$ denotes the singular locus of $f_s$ (in the stratified sense, with respect to a given (w)-regular stratification) and ${\mathbb {B}}_{\epsilon }$ denotes the closed ball around 0 in ${\mathbb {K}}^N$. Also, having weak constant topological type means that for each $s \in ({\mathbb {K}},0)$ there exists a homeomorphism of germs $h_s: \big ( V(f_s),0 \big ) \rightarrow \big ( V(f_0),0 \big )$. Clearly, this is weaker than having topological triviality in the sense of Lê & Ramanujam.

The first goal of this paper is to study ${\mathbb {K}}$-analytic isolated singularity families $f_{s_1, \dots , s_k}: (X,0) \rightarrow ({\mathbb {K}},0)$ depending linearly on k-many parameters $s_1, \dots , s_k \in {\mathbb {K}}$, with $k>1$. Then this will allow us to study one-parameter ${\mathbb {K}}$-analytic isolated singularity families $\mathtt f_s: (X,0) \rightarrow ({\mathbb {K}},0)$ that depend polynomially on $s \in {\mathbb {K}}$, which means it has the form

$$\begin{aligned} \mathtt f_s = f_0 + s^{a_1} \varphi _1 + s^{a_2} \varphi _2 + \dots + s^{a_k} \varphi _k \,, \end{aligned}$$

with $0<a_1< \dots < a_k$. This will be done by setting

$$\begin{aligned} f_{s_1, \dots , s_k}:= f_0 + s_1 \varphi _1 + s_2 \varphi _2 + \dots + s_k \varphi _k \,, \end{aligned}$$

so that $\mathtt f_s = f_{s^{a_1}, \dots , s^{a_k}}$.

Our main results are the following:

In Sect. 1, we give an easy criterium to assure that a ${\mathbb {K}}$-analytic family $f_{s_1, \dots , s_k}: (X,0) \rightarrow ({\mathbb {K}},0)$ has isolated singularity or even no coalescing of critical points. This is Proposition 1.2. In Sect. 2, we prove:

Theorem 0.1

Let $(X,0) \subset ({\mathbb {K}}^N,0)$ be the germ of either a subanalytic set, when ${\mathbb {K}}={\mathbb {R}}$, or a complex analytic space, when ${\mathbb {K}}={\mathbb {R}}$. Let ${\mathcal {S}}$ be a (w)-regular stratification of a representative X of (X, 0). Let $(f_{s_1, \dots , s_k})$ be a family of ${\mathbb {K}}$-analytic isolated singularity function-germs on (X, 0) depending linearly on the parameters $s_1, \dots , s_k \in {\mathbb {K}}$. If $(f_{s_1, \dots , s_k})$ has no coalescing of singular points (with respect to ${\mathcal {S}}$) then it has weak constant topological type.

As consequence, we have:

Theorem 0.2

Let $\mathtt f_s: (X,0) \rightarrow ({\mathbb {K}},0)$ be a one-parameter ${\mathbb {K}}$-analytic isolated singularity family depending polynomially on $s \in {\mathbb {K}}$. Let $f_{s_1, \dots , s_k}: (X,0) \rightarrow ({\mathbb {K}},0)$ be the multi-parameter family associated to $(\mathtt f_s)$, defined as above. If $(f_{s_1, \dots , s_k})$ has no coalescing of singular points, then $(\mathtt f_s)$ has weak constant topological type.

Corollary 0.2 relates to the $\mu$-constant conjecture, as it provides a sufficient condition for a one-parameter family of complex isolated singularity surfaces in ${\mathbb {C}}^3$ to have weak constant topological type. Precisely, we have:

Corollary 0.3

Let $\mathtt f_s: ({\mathbb {C}}^3,0) \rightarrow ({\mathbb {C}},0)$ be an analytic isolated singularity family depending polynomially on $s \in {\mathbb {C}}$. Let $f_{s_1, \dots , s_k}: ({\mathbb {C}}^3,0) \rightarrow ({\mathbb {C}},0)$ be the multi-parameter family associated to $(\mathtt f_s)$, as above. If $(f_{s_1, \dots , s_k})$ is $\mu$-constant, then $(\mathtt f_s)$ has weak constant topological type.

The case when the family $\mathtt f_s: ({\mathbb {C}}^3,0) \rightarrow ({\mathbb {C}},0)$ depends analytically on $s \in {\mathbb {C}}$ is analogous, by a well-known result of Samuel [6] (see also [1], Theorem 1.7).

In Sect. 3, we restrict our attention to the particular case of complex analytic isolated singularity families $f_{s_1, \dots , s_k}: (X,0) \rightarrow ({\mathbb {C}},0)$ depending linearly on the parameter $s_1, \dots , s_k \in {\mathbb {C}}$, with $k \ge 1$, where $(X,0) \subset ({\mathbb {C}}^N,0)$ is the germ of a complex isolated determinantal singularity (IDS). In this case, there is a well-defined Milnor number $\mu (f_s)$ in the sense of [7].

We prove:

Theorem 0.4

Let $(X,0) \subset ({\mathbb {C}}^N,0)$ be the germ of a complex isolated determinantal singularity. Let $f_{s_1, \dots , s_k}: (X,0) \rightarrow ({\mathbb {C}},0)$ be a family of isolated singularity function-germs depending linearly on $s_1, \dots , s_k \in {\mathbb {C}}$, with $k \ge 1$. If $(f_{s_1, \dots , s_k})$ is $\mu$-constant, then it has weak constant topological type.

As a consequence, like before, we have:

Corollary 0.5

Let $(X,0) \subset ({\mathbb {C}}^N,0)$ be the germ of a complex isolated determinantal singularity. Let $\mathtt f_s: (X,0) \rightarrow ({\mathbb {C}},0)$ be an analytic isolated singularity family depending polynomially on $s \in {\mathbb {C}}$, and let $f_{s_1, \dots , s_k}: (X,0) \rightarrow ({\mathbb {C}},0)$ be the multi-parameter family associated to $(\mathtt f_s)$, as above. If $(f_{s_1, \dots , s_k})$ is $\mu$-constant, then $(\mathtt f_s)$ has weak constant topological type.

2 Isolated singularity families and coalescing of singular points

Let $(X,0) \subset ({\mathbb {K}}^N,0)$ be as above, and let ${\mathcal {S}} = ({\mathcal {S}}_{\alpha })_{{\alpha }\in \Lambda }$ be a (w)-regular stratification of a representative X of (X, 0), in the sense of [8]. Without losing generality, we can suppose that $\{0\}$ is a stratum of ${\mathcal {S}}$.

Let ${\tilde{f}}_0: ({\mathbb {K}}^N,0) \rightarrow ({\mathbb {K}},0)$ be an analytic function-germ and let $f_0: (X,0) \rightarrow ({\mathbb {K}},0)$ be the restriction of ${\tilde{f}}_0$ to (X, 0). Clearly, the singular locus $\Sigma (f_0)$ of $f_0$ is formed by $X \cap \Sigma ({\tilde{f}}_0)$, the intersection of X with the critical set of ${\tilde{f}}_0$, together with the points $x \in X {\setminus } \Sigma ({\tilde{f}}_0)$ such that the smooth manifold $(\tilde{f}_0)^{-1}({\tilde{f}}_0(x))$ intersects ${\mathcal {S}}_{{\alpha }(x)}$ not transversally at x in ${\mathbb {K}}^N$, where ${\mathcal {S}}_{{\alpha }(x)}$ denotes the stratum of the stratification ${\mathcal {S}}$ that contains the point x.

Notice that since we choose a stratification ${\mathcal {S}}$ such that $\{0\}$ is a stratum, the origin $0 \in X \subset {\mathbb {K}}^N$ is a singular point of $f_0$, even if ${\tilde{f}}_0$ is a submersion at $0 \in {\mathbb {K}}^N$.

We say that $f_0: (X,0) \rightarrow ({\mathbb {K}},0)$ has an isolated singularity if there exists a positive real number ${\epsilon }>0$ such that $\Sigma (f_0) \cap {\mathbb {B}}_{\epsilon }= \{0\}$. This means that the restriction of $f_0$ to each stratum ${\mathcal {S}}_{\alpha }\ne \{0\}$ is a submersion.

Now, if ${\tilde{f}}_s: ({\mathbb {K}}^N,0) \rightarrow ({\mathbb {K}},0)$ is a deformation of ${\tilde{f}}_0$ depending continuously on the parameter $s \in ({\mathbb {K}}^k,0)$, it induces a deformation $f_s: (X,0) \rightarrow ({\mathbb {K}},0)$ of $f_0$ that obviously depends continuously on s. In the case when $f_0$ has an isolated singularity, the next lemma relates the critical set of $f_s$ with the critical set of ${\tilde{f}}_s$ inside some small neighborhood of $0 \in {\mathbb {K}}^N$, whenever $\Vert s\Vert$ is small enough.

We have:

Lemma 1.1

Suppose that $f_0: (X,0) \rightarrow ({\mathbb {K}},0)$ has an isolated singularity, and let ${\epsilon }>0$ be such that $\Sigma (f_0) \cap {\overline{{\mathbb {B}}}}_{\epsilon }= \{0\}$. Then there exists $\delta >0$ sufficiently small such that for every $s \in \mathbb {D}_\delta$ one has that

$$\begin{aligned} \Sigma (f_s) \cap {\overline{{\mathbb {B}}}}_{\epsilon }= \left( \Sigma ({\tilde{f}}_s) \cap X \cap {\overline{{\mathbb {B}}}}_{\epsilon }\right) \cup \{0\} \,. \end{aligned}$$

Proof

Clearly, one has that $\Sigma ({\tilde{f}}_s) \cap X \subset \Sigma (f_s)$, so one inclusion is obvious. On the other hand, for every $x \in X \cap {\overline{{\mathbb {B}}}}_{\epsilon }$ fixed, one has that either x is the origin or $({\tilde{f}}_0)^{-1}({\tilde{f}}_0(x))$ is smooth at x and it intersects ${\mathcal {S}}_{{\alpha }(x)}$ transversally at x in ${\mathbb {K}}^N$. Since transversality is a stable property, there exists $\delta (x)>0$ sufficiently small such that for every $s \in \mathbb {D}_{\delta (x)}$ one has that either $x \in \Sigma ({\tilde{f}}_s) \cup \{0\}$ or $({\tilde{f}}_s)^{-1}({\tilde{f}}_s(x))$ is smooth at x and it intersects ${\mathcal {S}}_{{\alpha }(x)}$ transversally at x in ${\mathbb {K}}^N$. Moreover, there exists a small neighborhood $V_x$ of x in $X \cap {\overline{{\mathbb {B}}}}_{\epsilon }$ such that for every $x' \in V_x$ and for every $s \in \mathbb {D}_{\delta (x)}$ one has that either $x' \in \Sigma ({\tilde{f}}_s) \cup \{0\}$ or $({\tilde{f}}_s)^{-1}({\tilde{f}}_s(x'))$ is smooth at $x'$ and it intersects ${\mathcal {S}}_{{\alpha }(x)}$ transversally at $x'$ in ${\mathbb {K}}^N$. Therefore, since $X \cap {\overline{{\mathbb {B}}}}_{\epsilon }$ is compact, there exists $\delta >0$ such that for every $x \in X \cap {\overline{{\mathbb {B}}}}_{\epsilon }$ and for every $s \in \mathbb {D}_\delta$ one has that either $x \in \Sigma ({\tilde{f}}_s) \cup \{0\}$ or $x \notin \Sigma (f_s)$. Hence $\Sigma (f_s) \cap {\overline{{\mathbb {B}}}}_{\epsilon }\subset \left( \Sigma ({\tilde{f}}_s) \cap X \cap {\overline{{\mathbb {B}}}}_{\epsilon }\right) \cup \{0\}$. $\square$

We say that a family $(f_s)$ as above is an isolated singularity family if for each $s \in {\mathbb {K}}$ with $\Vert s\Vert$ sufficiently small, the function-germ $f_s: (X,0) \rightarrow ({\mathbb {K}},0)$ has isolated singularity. That is, if there exists $\delta >0$ such that for every $s \in \mathbb {D}_\delta$ there is ${\epsilon }(s)>0$ such that $\Sigma (f_s) \cap {\mathbb {B}}_{{\epsilon }(s)} = \{0\}$.

An isolated singularity family $(f_s)$ as above is said to have no coalescing of singular points if there exist real numbers $\delta >0$ and ${\epsilon }>0$ such that $\Sigma (f_s) \cap {\mathbb {B}}_{{\epsilon }} = \{0\}$ for every $s \in \mathbb {D}_\delta$.

As an immediate consequence of Lemma 1.1, we have:

Proposition 1.2

Suppose that $f_0: (X,0) \rightarrow ({\mathbb {K}},0)$ has isolated singularity. Then:

(i):

If $({\tilde{f}}_s)$ is an isolated singularity family, then $(f_s)$ is also an isolated singularity family.

(ii):

If $({\tilde{f}}_s)$ has no coalescing of critical points, then $(f_s)$ has no coalescing of singular points.

Example 1.3

Consider $X=\{x^2-y^2=0\} \subset {\mathbb {K}}^3$ with the stratification given by $\mathcal {S}_0= X \setminus \{x=y=0\}$ and $\mathcal {S}_1= \{x=y=0\}$. Consider the family of function-germs $f_s: (X,0) \rightarrow ({\mathbb {K}},0)$ given by

$$\begin{aligned} f_s(x,y,z)=x^3+y^4+z^2+sx^4 + s^2y^5 \,. \end{aligned}$$

Since $f_0$ has an isolated singularity and since the family $({\tilde{f}}_s)$ on ${\mathbb {K}}^3$ is an isolated singularity family with no coalescing of critical points, it follows from Theorem 1.2 that $(f_s)$ is an isolated singularity family with no coalescing of singular points.

3 One-parameter families of isolated singularities

Let $(X,0) \subset ({\mathbb {K}}^N,0)$ and ${\mathcal {S}} = ({\mathcal {S}}_{\alpha })_{{\alpha }\in \Lambda }$ be as above. Let $g_s: (X,0) \rightarrow ({\mathbb {K}},0)$ be an isolated singularity family of ${\mathbb {K}}$-analytic function-germs depending linearly on the parameter $s \in {\mathbb {K}}$. This means that each $g_s$ has the form

$$\begin{aligned} g_s(z) = g_0(z) + s \varphi (z) \,, \end{aligned}$$

where $g_0: (X,0) \rightarrow ({\mathbb {K}},0)$ and $\varphi : (X,0) \rightarrow ({\mathbb {K}},0)$ are ${\mathbb {K}}$-analytic isolated singularity function-germs.

Define the map-germ

$$\begin{aligned} \begin{array}{cccc} \mathcal {F}: &{} \quad (X,0) &{} \quad \rightarrow &{} \quad ({\mathbb {K}}^2,0) \\ &{} \quad z &{} \quad \mapsto &{} \quad \big ( g_0(z), \varphi (z) \big ) \\ \end{array} \,, \end{aligned}$$

and for each $s \in {\mathbb {K}}$ define the line $\mathcal {H}_{s}$ in ${\mathbb {K}}^2$ given by

$$\begin{aligned} \mathcal {H}_{s}:= \{(y_0,y_1) \in {\mathbb {K}}^2; \ y_0 + s y_1 =0\}\,. \end{aligned}$$

Notice that

$$\begin{aligned} (\mathcal {F})^{-1}(\mathcal {H}_{s}) = V(g_{s}) \,. \end{aligned}$$

Following [5] (Definition 2.1), we say that the family $(g_{s})$ is $\Delta$-regular (with respect to the stratification ${\mathcal {S}}$) if the line $\mathcal {H}_0$ is not a limit of secant lines of the discriminant set $\Delta (\mathcal {F})$ of $\mathcal {F}$ at $0 \in {\mathbb {K}}^2$. This means that there exist a neighborhood U of 0 in ${\mathbb {K}}^2$ and a real number $\delta >0$ such that

$$\begin{aligned} \mathcal {H}_s \cap \Delta (\mathcal {F}) \cap U \subset \{0\} \end{aligned}$$

whenever $\Vert s\Vert <\delta$.

Now we give:

Definition 2.1

Let $(g_s)$ be an isolated singularity family as above, and let ${\epsilon }>0$ be a small real number. We say that a real number $\delta >0$ is a good parameter-radius for $(g_s)$ in ${\mathbb {B}}_{\epsilon }$ if for every $s \in \mathbb {D}_\delta$ one has that:

• (i) $\Sigma (g_s) \cap {\mathbb {B}}_{\epsilon }= \{0\}$;

• (ii) $V(g_s) \setminus V(\varphi )$ intersects the sphere $\mathbb {S}_{\epsilon }$ transversally in ${\mathbb {K}}^n$, in the stratified sense;

• (iii) $\mathcal {H}_s \cap \Delta (\mathcal {F}) \cap U \subset \{0\}$.

It follows from [5] (Lemma 3.2 together with Proposition 4.4) that if $(g_s)$ has no coalescing of singular points, then for every ${\epsilon }>0$ small enough there exists a good parameter-radius $\delta >0$ for $(g_s)$ in ${\mathbb {B}}_{\epsilon }$.

Then the next proposition easily follows from the proof of ( [5], Theorem 1.1).

Proposition 2.2

Let $(g_s)$ be an isolated singularity family as above. If ${\epsilon }>0$ is small enough and if $\delta >0$ is a good parameter-radius for $(g_s)$ in ${\mathbb {B}}_{\epsilon }$, then for each $s \in \mathbb {D}_\delta$ there exists a homeomorphism

$$\begin{aligned} h_s: V(g_s) \cap {\mathbb {B}}_{\epsilon }\rightarrow V(g_0) \cap {\mathbb {B}}_{\epsilon }\end{aligned}$$

that fixes the origin.

4 Multi-parameter families of isolated singularities

Let $(X,0) \subset ({\mathbb {K}}^N,0)$ and ${\mathcal {S}}$ be as above. Let $f_{s_1, \dots , s_k}: (X,0) \rightarrow ({\mathbb {K}},0)$ be a family of ${\mathbb {K}}$-analytic function-germs depending linearly on the parameters $s_1, \dots , s_k$, with $k>1$. This means that each $f_{s_1, \dots , s_k}$ has the form

$$\begin{aligned} f_{s_1, \dots , s_k}(z) = f_0(z) + s_1 \varphi _1(z) + \dots + s_k \varphi _k(z) \,, \end{aligned}$$

where $f_0: (X,0) \rightarrow ({\mathbb {K}},0)$ and $\varphi _i: (X,0) \rightarrow ({\mathbb {K}},0)$ for $i=1, \dots , k$ are ${\mathbb {K}}$-analytic function-germs. Suppose that $f_{s_1, \dots , s_k}$ has an isolated singularity, for every $(s_1, \dots , s_k) \in ({\mathbb {K}}^k,0)$.

If ${\mathbb {K}}={\mathbb {R}}$ set $\xi :=1$, and if ${\mathbb {K}}={\mathbb {C}}$ set $\xi :=2$. Then let $\mathbb {S}^{\xi k-1}$ denote the unit sphere around 0 in ${\mathbb {K}}^k$.

Now, for each point $r = (r_1, \dots , r_{k}) \in \mathbb {S}^{\xi k-1}$ fixed, consider the one-parameter isolated singularity family $(g^r_{s})$ given by

$$\begin{aligned} g^r_{s}(z):= f_{s r_1, \dots , s r_k}(z) \,. \end{aligned}$$

As before, we define the map-germ

$$\begin{aligned} \begin{array}{cccc} \mathcal {F}^r: &{} \quad (X,0) &{} \quad \rightarrow &{} \quad ({\mathbb {K}}^2,0) \\ &{} \quad z &{} \quad \mapsto &{} \quad \big ( f_0(z), r_1 \varphi _1(z) + \dots + r_k \varphi _k(z) \big ) \\ \end{array} \,, \end{aligned}$$

so that

$$\begin{aligned} (\mathcal {F}^r)^{-1}(\mathcal {H}_{s}) = V(g^r_{s}) \,. \end{aligned}$$

We say that the family $(f_{s_1, \dots , s_k})$ is $\Delta$-regular if for each $r \in \mathbb {S}^{\xi k-1}$ the corresponding one-parameter family $(g^r_{s})$ is $\Delta$-regular. This means that for each $r \in \mathbb {S}^{\xi k-1}$ there exist a neighborhood $U_r$ of 0 in ${\mathbb {K}}^2$ and a real number $\delta _{r}>0$ such that

$$\begin{aligned} \mathcal {H}_s \cap \Delta (\mathcal {F}^r) \cap U_r \subset \{0\} \end{aligned}$$

whenever $\Vert s\Vert <\delta _{r}$.

We have:

Lemma 3.1

If the family $(f_{s_1, \dots , s_k})$ is $\Delta$-regular, then there exist a neighborhood U of 0 in ${\mathbb {K}}^2$ and a real number $\delta >0$ such that for every $r \in \mathbb {S}^{\xi k-1}$ and for every $s \in {\mathbb {K}}$ with $\Vert s\Vert <\delta$ one has

$$\begin{aligned} \mathcal {H}_s \cap \Delta (\mathcal {F}^r) \cap U \subset \{0\} \,. \end{aligned}$$

Proof

By continuity, for each $r \in \mathbb {S}^{\xi k-1}$ fixed, there exists some small open neighborhood $N_r$ of r in $\mathbb {S}^{\xi k-1}$ such that, for every $w \in N_r$, one has that $\mathcal {H}_s \cap \Delta (\mathcal {F}^{w}) \cap U_r \subset \{0\}$ whenever $\Vert s\Vert <\delta _{r}$. Then the result follows from the fact that $\mathbb {S}^{\xi k-1}$ is compact. $\square$

Now suppose that the family $(f_{s_1, \dots , s_k})$ has no coalescing of singular points. Then it is clear that there exists $\lambda >0$ sufficiently small such that for every $r \in \mathbb {S}^{\xi k-1}$ and for every $\Vert s\Vert \le \lambda$ one has that $\Sigma (g_{s}^r) \cap {\mathbb {B}}_{{\epsilon }} = \{0\}$.

Moreover, it is not difficult to see that $\lambda >0$ can be chosen small enough so that it is also true that $V(g^r_s) {\setminus } V(r_1 \varphi _1 + \dots + r_k \varphi _k)$ intersects the sphere $\mathbb {S}_{\epsilon }$ transversally in ${\mathbb {K}}^n$, in the stratified sense, for every $s \in \mathbb {D}_\lambda$ and for every $r = (r_1, \dots , r_{k}) \in \mathbb {S}^{\xi k-1}$.

So if we set $\delta ':= \min \{\delta , \lambda \}$, it follows that for every ${\epsilon }>0$ small enough, $\delta '$ is a good parameter-radius for $(g_s^r)$ in ${\mathbb {B}}_{\epsilon }$, for every $r \in \mathbb {S}^{\xi k-1}$. Then it follows from Proposition 2.2 that for each $s = (s_1, \dots , s_k) \in {\mathbb {K}}^k$ with $\Vert s\Vert < \delta '$ there exists a homeomorphism

$$\begin{aligned} h_s: V(f_{s_1, \dots , s_k}) \cap {\mathbb {B}}_{\epsilon }\rightarrow V(f_0) \cap {\mathbb {B}}_{\epsilon }\end{aligned}$$

that fixes the origin.

This proves Theorems 0.1, and 0.2 follows easily.

Example 3.2

Recall the one-parameter isolated singularity family

$$\begin{aligned} \mathtt f_s(x,y,z)=x^3+y^4+z^2+sx^4 + s^2y^5 \end{aligned}$$

on $X=\{x^2-y^2=0\} \subset {\mathbb {K}}^3$, as in Example 1.3. Consider the two-parameters family

$$\begin{aligned} f_{s_1,s_2}(x,y,z)=x^3+y^4+z^2+s_1x^4 + s_2y^5 \end{aligned}$$

on X. Using Theorem 1.2, one can easily see that the family $(f_{s_1,s_2})$ has no coalescing of singular points. So it follows from Theorem 0.2 that the family $(\mathtt f_s)$ has weak constant topological type.

Finally, Corollary 0.3 follows from Theorem 0.2 together with the following:

Lemma 3.3

In the case when $(X,0) = ({\mathbb {C}}^n,0)$, an isolated singularity family $(f_{s_1, \dots , s_k})$ has no coalescing of singular (critical) points if and only if it is $\mu$-constant.

Proof

First notice that in the case when $k=1$ the result is well-known (see [9] for instance). So, in our case, where $k>1$, it follows that $(f_{s_1, \dots , s_k})$ has no coalescing of critical points if and only for each $r \in \mathbb {S}^{2k-1}$ there exists a real number $\lambda _r>0$ such that $\mu (g_s^r)$ is constant for every $s \in {\mathbb {C}}$ with $\Vert s\Vert <\lambda$. But since $\mathbb {S}^{2k-1}$ is compact, that happens if and only if there is a real number $\lambda >0$ such that $\mu (f_{s_1, \dots , s_k})$ is constant for every $s \in {\mathbb {C}}^k$ with $\Vert s\Vert <\lambda$. $\square$

Example 3.4

Consider the two-parameters isolated singularity family $(f_{s_1,s_2})$ on ${\mathbb {C}}^2$ given by

$$\begin{aligned} f_{s_1,s_2}(x,y):= x^3 + y^2 + x^2y + s_1(x^2-x^2y) + s_2 x^2 \,. \end{aligned}$$

Note that it is not $\mu$-constant. In fact, $\mu (f_0)=2$, while $\mu (f_{0,s_2})=1$ for $s_2 \ne 0$. However, the one-parameter family $(g_s)$ given by

$$\begin{aligned} g_s(x,y):= f_{s_1,-s_1}(x,y) = x^3 + y^2 + x^2y + s_1(-x^2y) \end{aligned}$$

is clearly $\mu$-constant.

Clearly, if the multi-parameter family $(f_{s_1, \dots , s_k})$ associated to a one-parameter family $(\mathtt f_s)$ on ${\mathbb {C}}^n$ is $\mu$-constant, then $(\mathtt f_s)$ is also $\mu$-constant. But our criterium for constancy of the topological type of $(\mathtt f_s)$ is weaker then the criterium suggested by Le-Ramanujam’s Conjecture also in the sense that $(\mathtt f_s)$ may be $\mu$-constant while the associated family $(f_{s_1, \dots , s_k})$ is not.

5 Deformations of isolated singularity function-germs on IDS

Now let $(X,0) \subset ({\mathbb {C}}^N,0)$ be the germ of a complex analytic isolated determinantal singularity (IDS), and let ${\tilde{f}}_s: ({\mathbb {C}}^N,0) \rightarrow ({\mathbb {C}},0)$ be a family of complex function-germs depending holomorphically on the parameter $s = (s_1, \dots , s_k) \in {\mathbb {C}}^k$. This means that the map ${\tilde{F}}: ({\mathbb {C}}^N \times {\mathbb {C}}^k, (0,0)) \rightarrow ({\mathbb {C}},0)$ defined by $F(x,s):= {\tilde{f}}_s(x)$ is holomorphic. The restriction of ${\tilde{F}}$ to (X, 0) defines a family of complex function-germs

$$\begin{aligned} f_s: (X,0) \rightarrow ({\mathbb {C}},0) \,. \end{aligned}$$

If $(f_s)$ is an isolated singularity family, then for each $s \in ({\mathbb {C}}^k,0)$ there is a well-defined Milnor number $\mu (f_s)$ in the sense of [7]. It is the number of points in the critical locus of a morsification $f_s': X' \rightarrow {\mathbb {C}}$ of $f_s$, where $X'$ is a smoothing of X near the origin.

In the particular case when X is an ICIS, it is well-know that $\mu (f_s) \le \mu (f_0)$ whenever $\Vert s\Vert$ is small enough. Moreover, the family $(f_s)$ has constant Milnor number if and only if it has no coalescing of singular points [10]. This means that there exist positive real numbers ${\epsilon }>0$ and $\delta >0$ such that 0 is the only critical point of $f_s$ in ${\mathbb {B}}_{\epsilon }$, for every $s \in \mathbb {D}_\delta$. These two facts come from the fact that when X is an ICIS, the Milnor number $\mu (f_0,0)$ of $f_0$ at 0 has an algebraic expression as the dimension of a quotient ring that is Cohen–Macaulay. Then the principle of conservation of number gives that

$$\begin{aligned} \mu (f_0,0) = \sum _{x \in \Sigma (f_s)} \mu (f_s,x) \,, \end{aligned}$$

(1)

where $\mu (f_s,x)$ denotes the Milnor number of $f_s$ at the singular point $x \in \Sigma (f_s)$.

When X is an arbitrary IDS there is no such algebraic expression for the Milnor number of $f_s: (X,0) \rightarrow ({\mathbb {C}},0)$, so one cannot apply the same argument. However, a simple topological argument gives the same equality. Precisely, we have:

Lemma 4.1

Let $f_s: (X,0) \rightarrow ({\mathbb {C}},0)$ be an isolated singularity family defined on an IDS. Then there exists $\delta >0$ such that for every $s \in \mathbb {D}_\delta$ equality (1) above holds.

Proof

Let $f_s': X' \rightarrow {\mathbb {C}}$ be a morsification (in family) of $f_s$, which means that $f_s'$ is a Morse function for every $s \in ({\mathbb {C}}^k,0)$. Since X is an IDS and since $f_0$ has an isolated singularity, we can choose ${\epsilon }>0$ small enough such that $X \cap {\mathbb {B}}_{\epsilon }{\setminus } \{0\}$ is smooth and the restriction of $f_0$ to $X \cap {\mathbb {B}}_{\epsilon }{\setminus } \{0\}$ is a submersion. We can also assume that $f_0'$ has exactly $\mu (f_0)$-many (Morse) critical points in $X' \cap {\mathbb {B}}_{\epsilon }$. Then there exists $\delta >0$ sufficiently small such that for every $s \in \mathbb {D}_\delta$ one has that $f_s'$ has exactly $\mu (f_0)$-many critical points in $X' \cap {\mathbb {B}}_{\epsilon }$. This follows from the fact that Morse points are stable under small perturbations. Thus equality (1) above follows. $\square$

As an immediate consequence, we have:

Proposition 4.2

Let $f_s: (X,0) \rightarrow ({\mathbb {C}},0)$ be an isolated singularity family defined on an IDS. Then there exists $\delta >0$ such that:

(i):

for every $s \in \mathbb {D}_\delta$ one has $\mu (f_s) \le \mu (f_0)$.

(ii):

$\mu (f_s) = \mu (f_0)$ for every $s \in \mathbb {D}_\delta$ if and only if $(f_s)$ has no coalescing of singular points.

On the one hand, Proposition 4.2 gives a characterization for the $\mu$-constancy of deformations of functions on an IDS, in the spirit of [10].

On the other hand, if $(X,0) \subset ({\mathbb {C}}^N,0)$ is an IDS and if $f_s: (X,0) \rightarrow ({\mathbb {C}},0)$ is an isolated singularity family depending linearly on the parameters $s_1, \dots , s_k \in {\mathbb {C}}$, Proposition 4.2 together with Theorem 0.2 give Theorem 0.4.

Notice that Propositions 4.2 and 1.1 combined give the following Corollary, which is a good criterium to decide if the family $(f_s)$ is $\mu$-constant:

Corollary 4.3

Let ${\tilde{f}}_s: ({\mathbb {C}}^N,0) \rightarrow ({\mathbb {C}},0)$ be an isolated singularity family depending holomorphically on $s \in {\mathbb {C}}^k$. Let $(X,0) \subset ({\mathbb {C}}^N,0)$ be an IDS and suppose that the family $f_s: (X,0) \rightarrow ({\mathbb {C}},0)$ given by restriction to (X, 0) is also an isolated singularity family. If $({\tilde{f}}_s)$ is $\mu$-constant then $(f_s)$ is $\mu$-constant.

Example 4.4

Let $X \subset {\mathbb {C}}^4$ be the 2-dimensional IDS defined by the matrix

$$\begin{aligned} \begin{pmatrix} w &{} \quad y &{} \quad x \\ z &{} \quad w &{} \quad y \\ \end{pmatrix}. \end{aligned}$$

Consider the $\mu$-constant family ${\tilde{f}}_s: ({\mathbb {C}}^4,0) \rightarrow ({\mathbb {C}},0)$ given by

$$\begin{aligned} {\tilde{f}}_s(x,y,z,w):= x^2+y^2+z^2+w^2+sx^3 \,, \end{aligned}$$

and let $f_s: (X,0) \rightarrow ({\mathbb {C}},0)$ be its restriction to X. For each $s \in {\mathbb {C}}$ fixed, the singular points of $f_s$ are those points in X at which all the $3 \times 3$-minors of the following matrix vanish:

$$\begin{aligned} \begin{pmatrix} {\alpha }(x) &{} \quad 2y &{} \quad 2z &{} \quad 2w \\ 0 &{} \quad z &{} \quad y &{} - \quad 2w \\ z &{} \quad -w &{} \quad x &{} \quad -y \\ w &{} \quad -2y &{} \quad 0 &{} \quad x \\ \end{pmatrix},\end{aligned}$$

where ${\alpha }(x):= 2x+3sx^2$. Then some calculation shows that the singular locus of $f_s$ is the finite set

$$\begin{aligned} \Sigma (f_s) = \{{\alpha }(x) = y=z=w=0\} = \left\{ \left( 0,0,0,0 \right) , \left( -\frac{2}{3s},0,0,0 \right) \right\} \,. \end{aligned}$$

Hence the family $(f_s)$ has no coalescing of singular points. Thus if follows from Proposition 4.2 that $f_s$ is a $\mu$-constant deformation of the function-germ $f_0: (X,0) \rightarrow ({\mathbb {C}},0)$.

On the other hand, consider the family $g_s: (X,0) \rightarrow ({\mathbb {C}},0)$ given by restriction to X of the family ${\tilde{g}}_s: ({\mathbb {C}}^4,0) \rightarrow ({\mathbb {C}},0)$ defined by

$$\begin{aligned} {\tilde{g}}_s(x,y,z,w):= x^3+y^2+z^2+w^2+sx^2 \,, \end{aligned}$$

which has non-constant Milnor number. Setting ${\beta }(x):= 3x^2+2sx$ one has that

$$\begin{aligned} \Sigma (g_s) = \{{\beta }(x) = y=z=w=0\} = \left\{ \left( 0,0,0,0 \right) , \left( -\frac{2s}{3},0,0,0 \right) \right\} \,. \end{aligned}$$

Then the family $(g_s)$ has coalescing of singular points and thus it is not a $\mu$-constant deformation.