# Finiteness of the space of n-cycles for a reduced (n – 2)-concave complex space

Daniel Barlet

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Daniel Barlet. Finiteness of the space of n-cycles for a reduced (n – 2)-concave complex space. Épijour-
nal de Géométrie Algébrique, 2017, Volume 1, 10.46298/epiga.2017.volume1.1521 . hal-01184765v2

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# Finiteness of the space of $n$-cycles for a reduced $(n-2)$-concave complex space

Daniel Barlet

pour Yum-Tong Siu, avec mon amitié

**Abstract.** We show that for $n \ge 2$ the space of closed $n$-cycles in a strongly $(n-2)$-concave complex space has a natural structure of reduced complex space locally of finite dimension and represents the functor “analytic family of $n$-cycles” parametrized by Banach analytic sets.

**Keywords.** Closed $n$-cycles; strongly $q$-concave space; Hartogs figure; $f$-analytic family of cycles

**2010 Mathematics Subject Classification.** 32G13; 32G10; 32F10; 32D15

[Français]

**Titre.** Finitude de l'espace des $n$-cycles pour un espace complexe $(n-2)$-concave réduit

**Résumé.** Nous montrons que, pour $n \ge 2$, l'espace des $n$-cycles fermés dans un espace complexe fortement $(n-2)$-concave a une structure naturelle d'espace complexe réduit localement de dimension finie et que cet espace représente le foncteur “famille analytique de $n$-cycles” paramétrée par des ensembles analytiques banachiques.

<footnote>Received by the Editors on June 27, 2016, and in final form on April 3, 2017.
Accepted on June 19, 2017.</footnote>

<footnote>Daniel Barlet
Institut Elie Cartan, Université de Lorraine, CNRS UMR 7502, and Institut Universitaire de France
BP 239, F-54506 Vandoeuvre-lès-Nancy cedex, France
e-mail: daniel.barlet@univ-lorraine.fr</footnote>
# Contents

<table><tbody><tr><td>1. Introduction</td><td>2</td></tr><tr><td>2. Hartogs figures</td><td>4</td></tr><tr><td>2.A. Banachization</td><td>4</td></tr><tr><td>2.B. n-Hartogs figure on a complex space</td><td>5</td></tr><tr><td>2.C. q-concave open sets.</td><td>11</td></tr><tr><td>2.D. Convexity-concavity</td><td>12</td></tr><tr><td>2.E. Boxed Hartogs figures</td><td>13</td></tr><tr><td>3. The extension and finiteness theorem</td><td>14</td></tr><tr><td>3.A. Some useful lemmas</td><td>14</td></tr><tr><td>3.B. Adjusted scales.</td><td>15</td></tr><tr><td>3.C. Shrinkage.</td><td>19</td></tr><tr><td>3.D. Excellent family.</td><td>20</td></tr><tr><td>3.E. The extension and finiteness theorem.</td><td>21</td></tr><tr><td>4. Finiteness of the space of n-cycles of a reduced strongly (n − 2)-concave space (n ≥ 2).</td><td>23</td></tr><tr><td>4.A. The global extension theorem.</td><td>23</td></tr><tr><td>4.B. Some consequences.</td><td>24</td></tr><tr><td>4.C. An analytic extension criterion.</td><td>25</td></tr><tr><td>4.D. Proof of Theorem 1.2 and its corollary</td><td>27</td></tr><tr><td>4.E. A compactness criterion for the connected components of the reduced complex space C<sup>f</sup><sub>n</sub>(Z) when Z is strongly (n − 2)-concave</td><td>28</td></tr></tbody></table>

# 1. Introduction

The aim of this article is to show that in a reduced strongly (n − 2)-concave<fnref n="1" /> complex space Z with n ≥ 2, the space of closed n-cycles is in a natural way endowed with a structure of a reduced complex space locally of finite dimension. With its tautological family of n-cycles it represents the functor “analytic family of n-cycles in Z” and also the functor “f-analytic family of n-cycles in Z” introduced in [Bar08] (see also [Bar13] and [Bar15]) parametrized by a Banach analytic set.

This answers a question asked to me by Y-T. Siu forty years ago.

I was able to solve this question thanks to the notion of f-analytic family introduced in loc. cit. and using the space $C_n^f(Z)$ of finite type cycles with its natural topology.

We obtain the following results.

**Theorem 1.1.** Let $n \ge 2$ be an integer. Let $Z$ be a strongly $(n-2)$-concave reduced complex space of pure dimension $n+p$ that is to say admitting a $\mathscr{C}^2$ exhaustion function $\varphi: Z \to [0, 2]$ which is strongly $(n-2)$-convex outside the compact set $K := \varphi^{-1}([1, 2])$. For any $\alpha \in ]0, 1[$ and any n-cycle $X_0$ in an open neighbourhood of the compact set $\varphi^{-1}([\alpha, 2])$ there exists $\beta \in ]0, \alpha[$ such that, if $Z_\beta := \{z \in Z \mid \varphi(z) > \beta\}$, the cycle $X_0$ extends in a unique way to the open set $Z_\beta$ and admits an open neighbourhood $\mathcal{U}$ in the space $C_n^f(Z_\beta)$ such that the ringed space defined by $\mathcal{U}$ and the sheaf of holomorphic functions on $\mathcal{U}$ is a (reduced) complex space locally of finite dimension.

Recall that a holomorphic function $h: \mathcal{U} \to \mathbb{C}$ on an open set in $C_n^f(Z_\beta)$ is a continuous function on $\mathcal{U}$ such that for any holomorphic map $f: S \to \mathcal{U}$ (corresponding to an f-analytic family of n-cycles in Z, see loc. cit.) of a Banach analytic set S to U the composed function $h \circ f$ is holomorphic.

<footnote n="1">¹ $\uparrow$ Our conventions will be precised below.</footnote>
**Theorem 1.2.** Consider the same situation as in the previous theorem, and let now $X_0 \in C_n^f(Z)$ be a finite type n-cycle in $Z$. Then there exists $\beta \in ]0, 1[$ and open neighbourhoods $V$ and $U$ respectively of $X_0$ in $C_n^f(Z)$ and of $X_0 \cap Z_\beta$ in $C_n^f(Z_\beta)$ such that the restriction map

$r : \mathcal{V} \to \mathcal{U}$

is well defined and bi-holomorphic.

We are going to recall briefly the notion of f-analytic family of finite type n-cycles in a complex space Z.

Firstly the notion of an analytic family of n-cycles in a reduced complex space Z parametrized by a reduced complex space S is defined as follows (see<fnref n="2" /> [Bar75, Chapter I, p. 33] or [BM14, Chapter IV, Section 3]), using the following notion of an adapted scale (see [BM14, Chapter IV, Section 2.1]).

**Definition 1.3.** We call $E := (U, B, j)$ a $n$-scale on a complex space $Z$ when $U$ and $B$ are open relatively compact polydiscs respectively in $\mathbb{C}^n$ and $\mathbb{C}^p$ and where $j : Z_E \to W$ is a closed embedding of an open set $Z_E$ in $Z$ into an open neighbourhood $W$ of $\bar{U} \times \bar{B}$ in $\mathbb{C}^{n+p}$. The open set $Z_E$ is called the **domain** of $E$ and the open set $j^{-1}(U \times B)$ the **center** of $E$. When $X$ is a $n$-cycle in $Z$, the $n$-scale $E$ is **adapted** to $X$ when $|X| \cap j^{-1}(\bar{U} \times \partial B) = \emptyset$.

Note that when $E$ is a $n$-scale adapted to a $n$-cycle $X$ in $Z$, the projection of $U \times B$ on $U$ restricted to $j_*(X \cap j^{-1}(U \times B))$ gives a finite proper map of degree $k \ge 0$ and the fibers of this map are classified by a holomorphic map $f : U \to \text{Sym}^k(B)$. In this case we shall say that $f$ is the **classifying map** of the cycle $X$ in the adapted scale $E$.

**Definition 1.4.** Let $Z$ be a complex space and let $(X_s)_{s \in S}$ be a family of $n$-cycles in $Z$ parametrized by a reduced complex space $S$. We shall say that this family is **analytic at a point** $s_0 \in S$ if for any $n$-scale $E := (U, B, j)$ on $Z$ which is adapted to the cycle $X_{s_0}$ there exists an open neighbourhood $S_0$ of $s_0$ in $S$ satisfying the following properties:

i) For each $s \in S_0$ the scale $E$ is adapted to $X_s$.

ii) Assume that $k := \deg_E(X_{s_0})$. Then for each $s \in S_0$ we have $\deg_E(X_s) = k$.

iii) There exists a holomorphic map $f : S_0 \times U \to \text{Sym}^k(B)$ such that for each $s \in S_0$ the restriction of $f$ to $\{s_0\} \times U$ classifies the cycle $X_s$ in the scale $E$.

It is easy to see that an analytic family of cycles has a “set theoretic” graph

$$|G| := \{(s, x) \in S \times Z / x \in |X_s|\}$$

which is a closed analytic subset in $S \times Z$ and that its projection on $S$ has pure $n$-dimensional fibers (which are the supports of the cycles). When we have an analytic family $(X_s)_{s \in S}$ and when the projection of its graph $pr : |G| \to S$ is quasi-proper<fnref n="3" /> we shall say that $(X_s)_{s \in S}$ is a **f-analytic family** of (finite type) $n$-cycles in $Z$. Of course this condition implies that each cycle $X_s$ is a finite type $n$-cycle (it means that each cycle admits only finitely many irreducible components) but this condition contains this fact in a local uniform manner on $S$.

In an analogous way, when we have an analytic family $(X_s)_{s \in S}$ and when the projection of its graph $pr : |G| \to S$ is proper, we shall say that it is a **proper analytic family of compact cycles** in $Z$.

The following corollary is of course the main result.

<footnote n="2">² ↑ See Chapter 3 Section 4 in [Bar75] for the case when $S$ is a Banach analytic set.</footnote>

<footnote n="3">³ ↑ This means, by definition, that for any $s_0 \in S$ there exists an open neighbourhood $S_1$ of $s_0$ in $S$ and a compact set $K$ in $|G|$ such that any irreducible component of any fiber $pr^{-1}(s)$ for any $s \in S_1$ meets $K$.</footnote>
**Corollary 1.5.** Consider the same situation as in the previous theorems. Then the ringed space given by the sheaf of holomorphic functions on $C_n^f(Z)$ is a reduced complex space locally of finite dimension. Moreover, endowed with its tautological family of (finite type) $n$-cycles it represents the functor

$$ (\text{reduced complex spaces}) \to (\text{sets}) \\ S \mapsto \{\text{f-analytic family of } n\text{-cycles in } Z \text{ parametrized by } S\}. $$

**Remark.** Let $X$ be a (non empty) irreducible analytic subset of dimension $n$ in a strongly $(n-2)$-concave complex space $Z$ as in Theorem 1.1. Let $x_0 \in X$ be a point in $X$ where the supremum of the restriction of the exhaustion function $\varphi$ to $X$ is obtained. Then the point $x_0$ is in the compact set $K = \varphi^{-1}([1, 2])$ because the Levi form of $\varphi$ at the point $x_0$ has at least $n$ non positive eigenvalues. So any (non empty) irreducible analytic subset of dimension $n$ in $Z$ has to meet the compact set $K$. Then any $n$-cycle in $Z$ is of finite type and any analytic family of $n$-cycles in $Z$ has a quasi-proper graph so is a f-analytic family. This implies that, in the previous corollary, the obvious map$^4 C_n^f(Z) \to C_n^{\text{loc}}(Z)$ is an isomorphism of ringed spaces and $C_n^f(Z)$ represents also the functor “analytic family of $n$-cycles in $Z$”.

**Question.** As it appears in the previous remark, we may expect the same result for $(n-1)$-cycles under our assumption of strong $(n-2)$-concavity. But our way to use Hartogs figures in the present article needs one more positive eigenvalue than one can expect. Is the result also true for $(n-1)$-cycles under our hypothesis? It would probably be interesting, for instance, to have this kind of result for 1-cycles in a strongly 0-concave complex space.

**Acknowledgments.** I want to thank the referees for many interesting comments and suggestions to improve this article.

## 2. Hartogs figures

### 2.A. Banachization

For the analytic extension via Hartogs figures, the use of the Banach spaces $H(\bar{U}, \mathbb{C})$ of continuous functions, holomorphic inside on a compact polydisc $\bar{U}$ is not well adapted. We shall use the Banach space $B(U, \mathbb{C})$ of bounded holomorphic functions on $U$ with the “sup” norm on $U$. Of course, $H(\bar{U}, \mathbb{C})$ is a closed Banach subspace in $B(U, \mathbb{C})$.

**Proposition 2.1.** Let $U$ be a relatively compact polydisc in $\mathbb{C}^n$ and let $S$ be a Banach analytic set. Let $F: S \to B(U, \mathbb{C})$ be a holomorphic map. Then the corresponding function $f: S \times U \to \mathbb{C}$ defined by $f(s,t) := F(s)[t]$ for $(s,t)$ in $S \times U$ is holomorphic (and locally on $S$ uniformly bounded on $U$).

Conversely, if we have a holomorphic function $f: S \times U \to \mathbb{C}$ and an open polydisc $U'' \subset U$, the associated map $F: S \to B(U'', \mathbb{C})$, defined for $(s,t)$ in $S \times U''$ by $F(s)[t] := f(s,t)$, is holomorphic.

*Proof.* The evaluation function $ev: B(U, \mathbb{C}) \times U \to \mathbb{C}$ is holomorphic as one may easily see by differentiation. Then the function $f$ associated to $F$ is the composition of the holomorphic maps $F \times id_U$ and $ev$. So it is holomorphic.

The converse is consequence of the linear isometric inclusion of $H(\bar{U}'', \mathbb{C})$ in $B(U'', \mathbb{C})$ as $F'': S \to H(\bar{U}'', \mathbb{C})$ is holomorphic as soon as $f$ is holomorphic (see [Bar75] or [BM]). □

<footnote>$^4$ ↑ The set $C_n^{\text{loc}}(Z)$ is the set of all closed cycles of dimension $n$ in the complex space $Z$; its natural topology which is associated to the adapted scales is described in [BM14, Chapter IV]. In general, the inclusion map $C_n^f(Z) \to C_n^{\text{loc}}(Z)$ is continuous but is not a homeomorphism onto its image. See [Bar15] for the comparison between compact sets in these two spaces.</footnote>
## 2.B. $n$-Hartogs figure on a complex space

For $\alpha \in (\mathbb{R}_+^*)^2$ let $M(\alpha)$ be the open set in $\mathbb{C}^2$ defined by

$$ 
\begin{align*}
M(\alpha) &:= M^P(\alpha) \cup M^C(\alpha) \quad \text{with} \\
M^P(\alpha) &:= \{ |t_1 - \alpha_1/2| < \alpha_1/4, \quad |t_2| < \alpha_2 \} \\
M^C(\alpha) &:= \{ |t_1| < \alpha_1, \quad \alpha_2/2 < |t_2| < \alpha_2 \}
\end{align*}
$$

and let also $\mathcal{M}(\alpha) := \{(t_1, t_2) \in \mathbb{C}^2 / |t_i| < \alpha_i, i = 1, 2\}$.

For $\varepsilon > 0$ small enough, we define

$$ 
\begin{align*}
M(\alpha)^{\varepsilon} &:= M^P(\alpha)^{\varepsilon} \cup M^C(\alpha)^{\varepsilon} \quad \text{with} \\
M^P(\alpha)^{\varepsilon} &:= \{ |t_1 - \alpha_1/2| < \alpha_1/4 - \varepsilon/4, \quad |t_2| < \alpha_2 - \varepsilon \} \\
M^C(\alpha)^{\varepsilon} &:= \{ |t_1| < \alpha_1 - \varepsilon, \quad \alpha_2/2 + \varepsilon/2 < |t_2| < \alpha_2 - \varepsilon \}
\end{align*}
$$

and also $\mathcal{M}(\alpha)^{\varepsilon} := \{(t_1, t_2) \in \mathbb{C}^2 / |t_i| < \alpha_i - \varepsilon, i = 1, 2\}$.

For a polydisc of radius $R$ in $\mathbb{C}^m$ we shall denote by $P^\varepsilon$ the polydisc with same center and radius $R - \varepsilon$ for $0 < \varepsilon < R$.

**Definition 2.2.** Let $n \ge 2$ and $p \ge 1$ two integers and let $\Delta \subset \subset \Delta'$ be two open sets in a reduced complex space $Z$. We shall call $\mathcal{H} := (M, M, B, j)$ a $n$-Hartogs figure in $Z$ relative to the boundary of $\Delta$, the following data

* an embedding $j$ of an open set $Z'$ in $\Delta'$ into an open set in $\mathbb{C}^{n+p}$,

* open sets $M \subset \subset \Delta'$ and $M \subset \subset \Delta$ relatively compact in $Z'$,

* a relatively compact polydisc $B$ in $\mathbb{C}^p$

such that there exists $\alpha \in (\mathbb{R}_+^*)^2$ and a relatively compact polydisc $V$ in $\mathbb{C}^{n-2}$ with the following conditions:

i) The map $j$ induces a closed embedding of $M$ in $M(\alpha) \times V \times B$.

ii) The map $j$ induces a closed embedding of $M$ in $M(\alpha) \times V \times B$.

iii) We have $j^{-1}(\bar{M}(\alpha) \times \bar{V} \times \partial B) \subset \Delta$.

**Definition 2.3.** Let $n \ge 2$ and $p \ge 1$ two integers and let $\Delta \subset \subset \Delta'$ be two open sets in a reduced complex space $Z$. Let $\mathcal{H} = (M, M, B, j)$ be a $n$-Hartogs figure in $Z$ relative to the boundary of $\Delta$ and let $X_0$ be a $n$-cycle in $\Delta$. We shall say that $\mathcal{H}$ is adapted to $X_0$ when the following condition is satisfied :

$$ j^{-1}(\bar{M}(\alpha) \times \bar{V} \times \partial B) \cap |X_0| = \emptyset. \qquad (@) $$

## Remarks.

1. Let $\mathcal{H}$ be a $n$-Hartogs figure in $Z$ relative to the boundary of $\Delta$. If the open set $\Delta_1 \subset \subset \Delta'$ has a boundary $\partial \Delta_1$ near enough to $\partial \Delta$, then $\mathcal{H}$ is again a $n$-Hartogs figure relative to the boundary of $\Delta_1$. For instance, if $\Delta := \{\varphi > 0\}$ where $\varphi$ is a continuous proper function on $Z$, we may choose $\Delta_1 := \{\varphi > \varepsilon\}$ for $\varepsilon > 0$ small enough.

2. Note that the $n$-scale $E_\mathcal{H} := (M(\alpha) \times V, B, j)$ on $\Delta$ associated to $\mathcal{H}$ is adapted to $X_0$ as soon as the $n$-Hartogs figure $\mathcal{H}$ is adapted to $X_0$.
3. If $\tilde{X}_0$ is a $n$-cycle in $\Delta'$ such that its restriction to $\Delta$ is equal to $X_0$, the $n$-scale $E_{\tilde{H}} := (\mathcal{M}(\alpha) \times V, B, j)$ on $\Delta'$ is adapted to $\tilde{X}_0$ if and only if the $n$-Hartogs figure $\mathcal{H}$ is adapted to $X_0$.

Note that the $n$-scale $E_{\tilde{H}}$ is not a $n$-scale on $\Delta$ although the subset $\bar{\mathcal{M}}(\alpha) \times \bar{V} \times \partial B$ is contained in $\Delta$.

**Definition 2.4.** In the situation above we define, for $\varepsilon > 0$ small enough, the $n$-Hartogs figure $\mathcal{H}^\varepsilon$ on $\Delta$ as follows:

$$ \mathcal{H}^{\varepsilon} := (\mathcal{M}^{\varepsilon}, M^{\varepsilon}, B, j) $$

where we use the notations

$$ \mathcal{M}^{\varepsilon} := j^{-1}(\mathcal{M}(\alpha)^{\varepsilon} \times V^{\varepsilon} \times B) \quad \text{and also} \quad M^{\varepsilon} := j^{-1}(M(\alpha)^{\varepsilon} \times V^{\varepsilon} \times B). $$

It is obvious to see that when $\mathcal{H}$ is a $n$-Hartogs figure relative to the boundary of $\Delta$, then $\mathcal{H}^\varepsilon$ is again a $n$-Hartogs figure relative to the boundary of $\Delta$ for all $\varepsilon > 0$ small enough.

Moreover, if $\mathcal{H}$ is adapted to the $n$-cycle $X_0$ of $\Delta$, the same is true for $\mathcal{H}^\varepsilon$ for all $\varepsilon > 0$ small enough.

**Lemma 2.5.** Let $V$ be a relatively compact open polydisc in $\mathbb{C}^q$. The restriction map

$$ \mathrm{res} : \mathcal{B}(\mathcal{M}(\alpha) \times V, \mathbb{C}) \to \mathcal{B}(M(\alpha) \times V, \mathbb{C}) $$

is a linear isometry of Banach spaces.

Note that the restriction map

$$ \mathrm{res} : \mathcal{B}(\mathcal{M}(\alpha)^{\varepsilon} \times V^{\varepsilon}, \mathbb{C}) \to \mathcal{B}(M(\alpha)^{\varepsilon} \times V^{\varepsilon}, \mathbb{C}) $$

induces also an isometry for all $\varepsilon > 0$ small enough.

*Proof.* Let $f(v, t_1, t_2) := \sum_{m \in \mathbb{Z}} a_m(v, t_1) t_2^m$ the Laurent expansion of the holomorphic function $f : M^C(\alpha) \times V \to \mathbb{C}$. The holomorphic functions $a_m, m \in \mathbb{Z}$ on the product of $V$ by the disc $\{|t_1| < \alpha_1\}$ are given by the formula

$$ a_m(v, t_1) := \frac{1}{2i\pi} \int_{|z|=r} f(v, t_1, z) \cdot \frac{dz}{z^{m+1}} \quad \text{with} \ r \in [\alpha_2/2, \alpha_2[. $$

As the holomorphy of $f$ on $M^P(\alpha) \times V$ implies that $a_m \equiv 0$ for each negative $m$ on the open set $\{|t_1 - \alpha_1/2| < \alpha_1/4\} \times V$, we conclude that the functions $a_m$ are identically zero for $m < 0$ and so $f$ is holomorphic on $\mathcal{M}(\alpha) \times V$. This shows that the restriction map $\mathrm{res}$ is bijective (and also it is linear continuous) between the two Fréchet spaces $\mathcal{O}(\mathcal{M}(\alpha) \times V)$ and $\mathcal{O}(M(\alpha) \times V)$; so it is an isomorphism of Fréchet spaces.

Let us show that if $f$ is in $\mathcal{B}(\mathcal{M}(\alpha) \times V, \mathbb{C})$, then $\mathrm{res}(f)$, which belongs to the space $\mathcal{B}(M(\alpha) \times V, \mathbb{C})$, has the same “sup” norm. For this purpose fix $\varepsilon > 0$ small enough. As $\bar{\mathcal{M}}(\alpha)^{\varepsilon} \times \bar{V}^{\varepsilon}$ is a compact polydisc in $\mathcal{M}(\alpha) \times V$, the maximum of $f$ on this compact is obtained at some point $z$ in the distinguish boundary of it. But as $z$ is also in the boundary of $M^C(\alpha)^{\varepsilon} \times \bar{V}^{\varepsilon}$, the desired equality follows. Conversely, if $g$ is in $\mathcal{B}(M(\alpha) \times V, \mathbb{C})$, its analytic extension $f$ to $\mathcal{M}(\alpha) \times V$ will be bounded on the boundary of $\mathcal{M}(\alpha)^{\varepsilon} \times V^{\varepsilon}$ by the sup of $g$ on $M^C(\alpha)^{\varepsilon} \times V^{\varepsilon}$. So we obtain the equality of the “sup” norms for $g$ and $f$ respectively on $M(\alpha) \times V$ and $\mathcal{M}(\alpha) \times V$. $\square$
The Banach analytic set $\mathcal{B}(U, \text{Sym}^k(\mathbb{C}^p))$. Recall that if $p \ge 1$ and $k \ge 1$ are integers, there exists a closed embedding (in fact given by a polynomial map) of $\text{Sym}^k(\mathbb{C}^p) := (\mathbb{C}^p)^k/\mathfrak{S}_k$ into the vector space $E(k) := \bigoplus_{h=1}^k S^h(\mathbb{C}^p)$ given by the elementary tensorial symetric functions$^5$. If $U$ is an open relatively compact polydisc in $\mathbb{C}^n$, the subset $\mathcal{B}(U, \text{Sym}^k(\mathbb{C}^p))$ is closed and Banach analytic in the Banach space $\mathcal{B}(U, E(k))$. Indeed, if $Q : E(k) \to \mathbb{C}^N$ is a polynomial map such that $Q^{-1}(0) = \text{Sym}^k(\mathbb{C}^p)$, then the holomorphic map

$$
Q : \mathcal{B}(U, E(k)) \to \mathcal{B}(U, \mathbb{C}^N) \quad \text{defined by} \quad f \mapsto Q \circ f
$$

satisfies $\mathcal{Q}^{-1}(0) = \mathcal{B}(U, \text{Sym}^k(\mathbb{C}^p))$.

Nevertheless, be aware that for an open set $\Omega$ in $E(k)$ the subset $\mathcal{B}(U, \Omega)$ of elements in $\mathcal{B}(U, E(k))$ taking their values in $\Omega$ is not, in general, open in $\mathcal{B}(U, E(k))$; so, for an open polydisc $B \subset \subset \mathbb{C}^p$, the subset $\mathcal{B}(U, \text{Sym}^k(B))$ is not open in $\mathcal{B}(U, \text{Sym}^k(\mathbb{C}^p))$ in general.

**Remark.** The obvious map $H(\bar{U}, E(k)) \to \mathcal{B}(U, E(k))$ is a closed (linear) isometry and induces a holomorphic inclusion map

$$
i_U : H(\bar{U}, \mathrm{Sym}^k(\mathbb{C}^p)) \hookrightarrow \mathcal{B}(U, \mathrm{Sym}^k(\mathbb{C}^p))
$$

and for all $\varepsilon > 0$ the restriction

$$
r : \mathcal{B}(U, E(k)) \to H(\bar{U}^{\varepsilon}, E(k))
$$

is a (linear and continuous) compact map which induces a holomorphic restriction map

$$
\mathcal{B}(U, \mathrm{Sym}^k(\mathbb{C}^p)) \to H(\bar{U}^\epsilon, \mathrm{Sym}^k(\mathbb{C}^p)).
$$

This remark will allow us to use Lemma 2.5 with Banach analytic sets like $H(\bar{U}, \text{Sym}^k(B))$.

**Notations.** Let $k \in \mathbb{N}$ and let $U' \subset \subset U \subset \subset \mathbb{C}^n$ and $B \subset \subset \mathbb{C}^p$ be polydisks. We shall note $\Sigma_{U,U'}(k)$
the Banach analytic set classifying the couples of an element in $H(\bar{U}, \text{Sym}^k(B))$ with its isotropy data
on $\bar{U}'$. Recall that for a holomorphic map $f : U \to \text{Sym}^k(B)$ the isotropy data on the polydisc $U'$ is
the map

$$
T(f) : U' \to F \otimes E'
$$

where

$$
F := \bigoplus_i \left( L(\Lambda^i(\mathbb{C}^n), \Lambda^i(\mathbb{C}^p)) \right) \quad \text{and} \quad E' := \bigoplus_{m=0}^{k-1} S^m(\mathbb{C}^p).
$$

It corresponds to the collection of holomorphic maps

$$
T_m^i(f) : U' \to L(\Lambda^i(\mathbb{C}^n), \Lambda^i(\mathbb{C}^p)) \otimes S^m(\mathbb{C}^p)
$$

for all $i \in [1, \min(n, p)]$ and $m \in [0, k-1]$ which are given, near a point in $U'$ where the multiform
graph $X_f$ associated to $f$ has local branches $f_1, \dots, f_k$, by the formula

$$
T_m^i(f)(t) := \sum_{j=1}^{k} \Lambda^i(Df_j(t)) \otimes f_j(t)^m.
$$

These maps are always holomorphic on all $U$ and determine the trace map for the projection $\pi : X_f \to U$ of the holomorphic differential forms on $U \times B$.

The subset

$$
\Sigma_{U,U'}(k) \subset H(\bar{U}, \mathrm{Sym}^k(B)) \times H(\bar{U}', F \otimes E')
$$

<footnote>$^5$ ↑ See for instance [BM14, Chapter I, §4].</footnote>
is defined as the graph of the map $f \mapsto T(f) := \bigoplus_{i,m} T_m^i(f)$ which is not holomorphic in general. Nevertheless it is a Banach analytic subset and its natural projection

$$ \Sigma_{U,U'}(k) \to H(\bar{U}, \mathrm{Sym}^k(B)) $$

is a holomorphic homeomorphism (see [Bar75, Chapter III, Proposition 2, p. 81] or [BM, Chapter V]).

The important point which motivates the introduction of this Banach analytic subset is the fact that, when $S$ is a reduced complex space, an analytic family of multiform graphs given by a holomorphic map $f: S \times U \to \mathrm{Sym}^k(B)$ will give an analytic family of cycles in $U \times B$ parametrized by $S$ if and only if the corresponding isotropy data (given by the maps $T_m^i(f/S)$ on $S \times U$) are holomorphic on $S \times U$. This is the isotropy condition; see [Bar75, Chapter II] or [BM14, Chapter IV, Section 5].

Our next result is the main tool for performing the analytic extension of $n$-cycles near a $(n-2)$-concave boundary.

**Proposition 2.6.** Consider the open sets $\mathcal{M}(\alpha) \times V$ and $M(\alpha) \times V$ in $\mathbb{C}^n$. The inverse of the restriction map is a holomorphic isomorphism of analytic extension

$$ \mathrm{prlgt}: \mathcal{B}(M(\alpha) \times V, \mathrm{Sym}^k(\mathbb{C}^p)) \to \mathcal{B}(\mathcal{M}(\alpha) \times V, \mathrm{Sym}^k(\mathbb{C}^p)). $$

Composed with the restriction to the compact set $\bar{\mathcal{M}}(\alpha)^\varepsilon \times \bar{V}^\varepsilon$ it sends the subset $H(\bar{\mathcal{M}}(\alpha) \times \bar{V}, \mathrm{Sym}^k(B))$ into $H(\bar{\mathcal{M}}(\alpha)^\varepsilon \times \bar{V}^\varepsilon, \mathrm{Sym}^k(B))$ for $\varepsilon > 0$ small enough.

Moreover, this holomorphic map induces a holomorphic map, again for $\varepsilon > 0$ small enough

$$ \Sigma_{\mathcal{M}(\alpha) \times V, \mathcal{M}(\alpha)^{\varepsilon/3} \times V^{\varepsilon/3}}(k) \to \Sigma_{\mathcal{M}(\alpha)^{\varepsilon/3} \times V^{\varepsilon/3}, \mathcal{M}(\alpha)^{2\varepsilon/3} \times V^{2\varepsilon/3}}(k) $$

which factorizes the restriction map

$$ \Sigma_{\mathcal{M}(\alpha) \times V, \mathcal{M}(\alpha)^{\varepsilon/3} \times V^{\varepsilon/3}}(k) \to \Sigma_{\mathcal{M}(\alpha)^{\varepsilon/3} \times V^{\varepsilon/3}, \mathcal{M}(\alpha)^{2\varepsilon/3} \times V^{2\varepsilon/3}}(k) $$

through the restriction

$$ \Sigma_{\mathcal{M}(\alpha) \times V, \mathcal{M}(\alpha)^{\varepsilon/3} \times V^{\varepsilon/3}}(k) \to \Sigma_{\mathcal{M}(\alpha) \times V, \mathcal{M}(\alpha)^{\varepsilon/3} \times V^{\varepsilon/3}}(k). $$

*Proof.* Lemma 2.5 gives that the map

$$ \mathrm{prlgt}: \mathcal{B}(M(\alpha) \times V, E(k)) \to \mathcal{B}(\mathcal{M}(\alpha) \times V, E(k)) $$

is an isometry of Banach spaces. It is clear that its inverse sends $\mathcal{B}(\mathcal{M}(\alpha) \times V, \mathrm{Sym}^k(\mathbb{C}^p))$ into the Banach analytic subset $\mathcal{B}(M(\alpha) \times V, \mathrm{Sym}^k(\mathbb{C}^p))$ and that if the map $f \in \mathcal{B}(\mathcal{M}(\alpha) \times V, \mathrm{Sym}^k(\mathbb{C}^p))$ takes values in $\mathrm{Sym}^k(B)$, the same is true for its restriction. This proves the first part of the proposition.

In order to prove the second part, it is enough to show that a holomorphic map of a Banach analytic set $S$ with values in the subset

$$ H(\bar{\mathcal{M}}(\alpha) \times \bar{V}, \mathrm{Sym}^k(B)) $$

which is isotropic on the product of $S$ with any relatively compact subset in the open set $\mathcal{M}(\alpha)^{\varepsilon/3} \times V^{\varepsilon/3}$ will have an analytic extension which will be isotropic on any relatively compact open set in $\mathcal{M}(\alpha)^{\varepsilon/3} \times V^{\varepsilon/3}$. So it will be isotropic on the closure of the open set $\mathcal{M}(\alpha)^{2\varepsilon/3} \times V^{2\varepsilon/3}$. $\square$

**Proposition 2.7.** Let $n \ge 2$ and $p \ge 1$ be integers and let $U_1 \times B_1$ the product of two polydiscs with centers 0 respectively in $\mathbb{C}^n$ and $\mathbb{C}^p$. Denote by $(t_1, \dots, t_n, x_1, \dots, x_p)$ coordinates on $U_1 \times B_1$. Let $\varphi$ be a real valued function of class $\mathcal{C}^2$ on $U_1 \times B_1$, such that

$$ \varphi(t, x) = \mathrm{Re}(t_1) + \sum_{i=1}^{n} \rho_i |t_i|^2 + \sum_{j=1}^{p} \sigma_j |x_j|^2 + o(||(t, x)||^2) \quad (@@) $$
where the real numbers $\rho_2, \sigma_1, \dots, \sigma_p$ are positive (so $\varphi$ is $(n-2)$-convex near $(0,0)$ and $d\varphi_{0,0} \neq 0$).
Let $\Delta$ be the open set $\{\varphi > 0\}$ in $U_1 \times B_1$ and let $\Delta'$ be an open neighbourhood of the compact set $\bar{\Delta}$ in $\mathbb{C}^{n+p}$. Let $X_0$ be a closed analytic subset of pure dimension $n$ in $\Delta'$ such that each irreducible component of $X_0$ meets $\Delta$ and such that

$$
|X_0| \cap \{t_1 = \dots = t_n = 0\} \subset \{0\}. \qquad (*)
$$

Then there exists $\alpha \in (\mathbb{R}_+^*)^2$ and polydiscs $V$ and $B \subset \subset B_1$ with centers $0$ respectively in $\mathbb{C}^{n-2}$ and $\mathbb{C}^p$ such that the following conditions are satisfied :

1. $\mathcal{M}(\alpha) \times V \times B \subset \subset \Delta'$;

2. $M(\alpha) \times V \times B \subset \subset \Delta$;

3. $\overline{M(\alpha)} \times \bar{V} \times \partial B \subset \Delta$;

4. $|X_0| \cap (\overline{M(\alpha)} \times \bar{V} \times \partial B) = \emptyset$ (this implies $|X_0| \cap (\overline{M(\alpha)} \times \bar{V} \times \partial B) = \emptyset$).

*Proof.* Choose the polydisc $B \subset \subset B_1$ small enough in order that we have

$$
X_0 \cap (\{0\} \times \bar{B}) \subset \{0\} \quad \text{and} \quad \{0\} \times \partial B \subset \Delta.
$$

This is possible as we have $|X_0| \cap \{t_1 = \cdots = t_n = 0\} \subset \{0\}$ and as $\varphi$ is positive on a small enough punctured neighbourhood of the origin in the $p$-plane $\{t_1 = \cdots = t_n = 0\} \times \mathbb{C}^p$. So we shall have

$$
|X_0| \cap (\bar{W} \times \partial B) = \emptyset \quad \text{and} \quad \bar{W} \times \partial B \subset \Delta
$$

for any small enough open neighbourhood $W$ of the origin in $\mathbb{C}^n$. A immediate consequence is that
Conditions 1, 3 and 4 will be satisfied as soon as $\alpha$ and $V$ are small enough.

In order to check Condition 2, let us remark first that, up to choosing the real numbers $\rho'_1$ and $\rho'_2$
such that $\rho'_1 > |\rho_1|$, $\rho'_2 \in ]0, \rho_2[$ and $r > \sup_{i \ge 3} |\rho_i|$, we obtain on $W \times B$ chosen small enough

$$
\varphi(t, x) \geq \mathrm{Re}(t_1) - \rho'_1 |t_1|^2 + \rho'_2 |t_2|^2 - r \cdot \left( \sum_{i=3}^{n} |t_i|^2 \right) \quad (**) \\
\text{with strict inequality as soon as } x \neq 0. \text{ Then for}
$$

with strict inequality as soon as $x \neq 0$. Then for

$$
V_{\varepsilon} = \{(t_3, \dots, t_n) / \sum_{3}^{n} |t_i|^2 < \varepsilon^2\}
$$

the following inequalities hold

$$
\varphi(t, x) \geq \frac{1}{4} \alpha_1 - \rho'_1 \alpha_1^2 - r \cdot \varepsilon^2 \quad \text{on} \quad \overline{M^P(\alpha) \times V_\varepsilon \times B} \qquad (1)
$$

$$
\varphi(t, x) \geq -\alpha_1 - \rho'_1 \alpha_1^2 + \frac{1}{4} \rho'_2 \alpha_2^2 - r \cdot \varepsilon^2 \quad \text{on} \quad \overline{M^C(\alpha) \times V_\varepsilon \times B} \quad (2)
$$

for $\alpha$ and $\varepsilon$ small enough in order that $M(\alpha) \times V_\varepsilon$ is contained in $W$.

This allows to fix $\alpha_1, \alpha_2$ and $\varepsilon$.

Now we shall choose $\alpha_1$ and $\varepsilon$ smaller in order to satisfy the following conditions :

$$
8\alpha_1 < \rho'_2 \alpha_2^2, \quad \alpha_1 < \frac{1}{8\rho'_1} \quad \text{and} \quad \varepsilon^2 < \frac{1}{8r}\alpha_1. \qquad (3)
$$
To obtain $M^P(\alpha) \times V_\varepsilon \times B \subset \Delta$ it is enough to show that on $M^P(\alpha) \times V_\varepsilon \times B$ we have, if we let $\alpha_1 = u.\alpha_2$ and $\varepsilon^2 = v.\alpha_1 = uv.\alpha_2$

$$ \frac{1}{4} > \rho'_1 \cdot \alpha_1 + r.v. $$

Indeed, as we assumed $\rho'_1 \cdot \alpha_1 < \frac{1}{8}$ and $r.\varepsilon^2 < \frac{1}{8} \cdot \alpha_1$ (so $r.v < 1/8$) the first condition holds.

In order to satisfy $M^C(\alpha) \times V_\varepsilon \times B \subset \Delta$ it is enough to show that on $M^C(\alpha) \times V_\varepsilon \times B$ we have

$$ \frac{1}{4} \rho'_2 \cdot \alpha_2 > u + \rho'_1 u^2 \alpha_2 + r.uv. $$

But our condition implies

$$ \rho'_1 u^2 \alpha_2 < \frac{1}{8} u \qquad r.uv < \frac{1}{8} u $$

which gives $u + \rho'_1 u^2 \alpha_2 + r.uv < 2.u$. The condition $8\alpha_1 < \rho'_2 \cdot \alpha_2^2$ which implies $2u < \frac{1}{4} \rho'_2 \cdot \alpha_2$, allows to conclude. $\square$

## Remarks.

1. We only used Condition (*) for $X_0$ and Inequality (**) for $\varphi$ in a neighbourhood of the origin in the proof above.

2. Sufficient conditions on $\varphi \in \mathscr{C}^2$ to satisfy (@@) are:

i) The origin is not a critical point of $\varphi$.

ii) The Levi form of $\varphi$ at 0 has, at most, $(n-2)$ non positive eigenvalues in the complex tangent hyperplane to the real hypersurface $\{\varphi(z) = 0\}$; the existence of real function $\varphi \in \mathscr{C}^2$ such that $\Delta = \{\varphi > 0\}$ and satisfying these two conditions is equivalent to the fact that the open set $\Delta$ has a strongly $(n-2)$-concave smooth boundary near the origin (see Definition 2.10 given below). Indeed, if $\varphi$ is not critical at 0 and has a Levi form at 0 with, at most, $(n-2)$ non positive eigenvalues in the complex hyperplane tangent to the real hypersurface $\{\varphi(z) = 0\}$, its order 2 Taylor expansion at the origin is written, in suitable local holomorphic coordinates $(\tau, x)$

$$ \varphi(\tau, x) = \mathrm{Re}(\tau_1) + \mathrm{Re}(Q(\tau, x)) + \sum_{i=1}^{n} \rho_i \cdot |\tau_i|^2 + \sum_{j=1}^{p} \sigma_j \cdot |x_j|^2 + o(||(\tau, x)||^2) $$

where $Q$ is a holomorphic homogeneous degree 2 polynomial and where the real numbers $\rho_2$ and $\sigma_j, j \in [1, p]$ are positive. Define new local holomorphic coordinates

$$ t_1 := \tau_1 + Q(\tau, x), \quad t_i := \tau_i \quad \text{for } i \in [2, n] \quad \text{and} \quad x_j := x_j \quad \text{for } j \in [1, p]. $$

Then we obtain (@@).

3. Condition (*) implies that $X_0$ has no local irreducible component at 0 contained in the hyper-plane $\{t_1 = 0\}$. In fact, as the coordinate $t_1$ is choosen in order to suppress the real part of the holomorphic homogeneous degree 2 term in the order 2 Taylor expansion of $\varphi$ at 0 (see the previous remark), we want that no local irreducible component of $X_0$ at the origin is contained in the complex hypersurface $\tau_1 + Q(\tau, x) = 0$ locally defined near 0 for $\varphi$ given. Then, as soon as the restriction $\varphi|_{X_0}$ has not 0 as a critical point, Condition (*) could be realized when the Levi form of $\varphi$ has at most $(n-2)$ non positive eigenvalues on the complex tangent hyperplane at the origin of the hypersurface $\{\varphi = 0\}$.
4. One may easily see that under our hypothesis, the cycle $X_0$ meets the open set $\Delta$ when it contains 0. Indeed, the analytic subset

$$ \{t_1 = t_3 = \cdots = t_n = 0\} \cap |X_0| $$

is nonempty, has dimension at least 1 and meets the complement of $\Delta$ only at the origin.

Of course, assuming that $0 \in X_0$, the proposition shows that, in fact, $X_0$ contains a branched covering of degree $k \ge 1$ of $M^P(\alpha) \times V_\varepsilon$ inside

$$ M^P(\alpha) \times V_\varepsilon \times B \subset \subset \Delta. $$

5. In the situation of Proposition 2.7, for any continuous family $(X_s)_{s \in S}$ of $n$-cycles in $\Delta$ parametrized by a Banach analytic set $S$ such that $X_{s_0} = X_0 \cap \Delta$, there exists an open neighbourhood $S'$ of $s_0$ in $S$, such that for each $s \in S'$ Condition 4 remains true after analytic extension of the cycles (see Proposition 2.6), because, thanks to Condition 3, $\overline{M(\alpha)} \times \overline{V} \times \partial B$ is a compact subset in $\Delta$.

## 2.C. $q$-concave open sets.

**Definition 2.8.** Let $\varphi : U \to \mathbb{R}$ be real valued $\mathcal{C}^2$ function on an open set $U$ in $\mathbb{C}^N$. We shall say that $\varphi$ is **strongly $q$-convex** when its Levi form at each point of $U$ has at most $q$ non positive eigenvalues.

So, with this definition a strongly 0-convex function is a strongly plurisubharmonic function.

**Definition 2.9.** Let $\varphi : Z \to \mathbb{R}$ a real valued $\mathcal{C}^2$ function on a reduced complex space $Z$. We shall say that $\varphi$ is **strongly $q$-convex** if locally near each point of $Z$ it can be induced by a $\mathcal{C}^2$ strongly $q$-convex function in a local embedding in an open set of an affine space.

Remark that a strongly $q$-convex function on an irreducible complex space of dimension at least equal to $q+1$ has no local maximum because there exists at any point a germ of curve on which the restriction of $\varphi$ is strongly p.s.h.

**Definition 2.10.** Let $Z$ be a reduced complex space and let $\Delta$ be a relatively compact open set in $Z$. We shall say that $\Delta$ has a **smooth $\mathcal{C}^2$ boundary** when for each point $z$ in $\partial\Delta$ there exists a local holomorphic embedding $j : W \to U$ of an open neighbourhood $W$ of $z$ in an open set $U$ of the Zariski tangent space of $Z$ at $z$ and an open set $D$ with smooth $\mathcal{C}^2$ boundary in $U$ such that $W \cap j^{-1}(\partial D) = W \cap \partial\Delta$.

We shall say that the open set $\Delta \subset Z$ with smooth $\mathcal{C}^2$ boundary is **strongly $q$-concave at a point** $z \in \partial\Delta$ if, in some local holomorphic embedding $j : W \to U$ of $Z$ around $z$ as above, one can define $\Delta$ in $W$ as the subset $\{j \circ \varphi > 0\} \cap W$ where $\varphi$ is a real valued $\mathcal{C}^2$ function on $U$ such that

1. $d\varphi_{j(z)} \neq 0$ on the tangent space $T_{U,j(z)}$ of $U$ at $j(z)$.

2. The restriction of the Levi form at $j(z)$ of $\varphi$ to the complex hyperplane tangent at $j(z)$ to the real hypersurface $\{\varphi(x) = \varphi(j(z))\}$ in $U$ has at most $q$ non positive eigenvalues.

We shall say that $\Delta$ is **strongly $q$-concave** if $\Delta$ is strongly $q$-concave near each point in $\partial\Delta$.
**Remark.** Assume that $Z$ is of pure dimension $q + p$. If the defining function $\varphi$ of $\Delta$ satisfies Conditions 1 and 2 above, we can compose $\varphi$ with a real strictly increasing (non critical) convex $\mathcal{C}^2$ function (this does not change the level sets $\{\varphi = \text{constant}\}$), in order that $c \circ \varphi$ is $\mathcal{C}^2$ strongly $q$-convex (and non critical) near $z$.

Conversely, if $\varphi$ is a real valued $\mathcal{C}^2$ function which is strongly $q$-convex and not critical near at a point $z \in Z$, the open set $\{\varphi(x) > \varphi(z)\}$ has a strongly $q$-concave boundary in a neighbourhood of $z$.

With this terminology, using the remarks above, we may give the following reformulation of Proposition 2.7:

**Corollary 2.11.** Let $n \ge 2$ and $p \ge 1$ be integers, let $Z$ be a reduced complex space of pure dimension $n+p$ and let $\Delta := \{\varphi > 0\}$ be an open set with $\mathcal{C}^2$ smooth boundary in $Z$. Let $X_0$ be a $n$-cycle in an open neighbourhood of a point $z \in \partial\Delta$ such that the function $\varphi|_{X_0}$ is not critical at $z$.

Assume that $\Delta$ is strongly $(n-2)$-concave near $z$; then there exists a $n$-Hartogs figure $\mathcal{H} := (\mathcal{M}, M, B, j)$ relative to the boundary of $\Delta$, adapted to $X_0$, and such that the point $z$ lies in $\mathcal{M}$.

*Proof.* Using a local embedding of an open neighbourhood of $z$ in an open set of the Zariski tangent space $T_{Z,z}$, it is enough to prove the corollary in the case where $Z$ is an open set in $\mathbb{C}^{n+p'}$, with $p' \ge p$ an integer. As we may choose the function $\varphi$ strongly $(n-2)$-convex such that $d\varphi_z \ne 0$ thanks to the previous remarks, we can choose local coordinates near $z$ in order to be in the situation of Proposition 2.7 in the case $z \in X_0$, as we assumed that $\varphi|_{X_0}$ is not critical at $z$. In this case the proposition gives the result.

If $z$ is not in $X_0$, the same construction in an open neighbourhood of $z$ with no limit point in $X_0$ allows to conclude, and in this case the degree of $X_0$ in the (adapted) scale $E_{\mathcal{H}}$ will be zero. $\square$

## 2.D. Convexity–concavity

In this paragraph we want to have a brief discussion about $q$-convexity and $q$-concavity.

Let us consider in an open set $U$ of $\mathbb{C}^{n+p}$ a $\mathcal{C}^2$ function $\varphi : U \to \mathbb{R}$ and a non critical zero $z_0$ on $\varphi$. So $\varphi(z_0) = 0$ and $d\varphi_{z_0} \ne 0$. Let $D := \{z \in U / \varphi(z) < 0\}$ and let $H$ be the complex hyperplane tangent at $z_0$ to the real hypersurface $\{\varphi = 0\}$ which is smooth near $z_0$.

Our terminology (Norguet–Siu convention, see [NS77]) is to say that the open set $D$ is strongly $q$-convex near $z_0 \in \partial D$ if the restriction to $H$ of the Levi form of $\varphi$ at $z_0$ has at most $q$ non positive eigenvalues.

Looking now at the same open set $D$ but asking for some strong concavity condition, we write $D := \{z \in U / -\varphi(z) > 0\}$. Then we shall say that $D$ is strongly $q$-concave at the point $z_0$ if the restriction to $H$ of Levi form of $-\varphi$ at $z_0$ has at most $q$ non positive eigenvalues.

If the signature of the restriction to $H$ of the Levi form of $\varphi$ at $z_0$ is given by $(p-1)$ “plus” and $n$ “minus” we see that that near $z_0$ our open set $D$ will be strongly $n$-convex near $z_0$ and strongly $(p-1)$-concave near $z_0$. So $D$ will be strongly $(p-1)$-concave near $z_0$ if the function $-\varphi$ is strongly $(p-1)$-convex at the point $z_0$.

In order that a $\mathcal{C}^2$ exhaustion function $\varphi : Z \to ]0, 2]$ on a reduced complex space $Z$ gives relatively compact $q$-concave subsets $Z_\alpha := \{\varphi(z) > \alpha\}$ for each $\alpha \in ]0, 1[$ which is not critical for $\varphi$, we see that it is enough that the Levi form of $\varphi$ at each point in $\varphi^{-1}([0, 1])$ has at most $q$ non positive eigenvalues. That is to say that $\varphi$ is strongly $q$-convex on this open set.

In order to reach the key situation given in Proposition 2.7 with a $n$-cycle, we need to dispose of a $\mathcal{C}^2$-exhaustion $\varphi : Z \to ]0, 2]$ which is $(n-2)$-strongly convex on the open set $\varphi^{-1}([0, 1])$. So we need to assume that $n \ge 2$.
## 2.E. Boxed Hartogs figures

**Definition 2.12.** Let $n \ge 2$ and $p \ge 1$ be integers and let $\Delta \subset \subset \Delta'$ be two open sets in a reduced complex space $Z$. Let $\mathcal{H} = (\mathcal{M}, M, B, j)$ and $\mathcal{H}' := (\mathcal{H}', M', B, j)$ be two $n$-Hartogs figures in $Z$ relative to the boundary of $\Delta$ given by the same (local) embedding $j$ and having the same polydisc $B \subset \subset \mathbb{C}^p$. We shall say that these two $n$-Hartogs figures are **boxed** when we can choose $\alpha, \alpha', V, V'$ in Definition 2.2 in order to have

* $\mathcal{M}'(\alpha') \subset \subset \mathcal{M}(\alpha)$,

* $M'(\alpha') \subset \subset \mathcal{M}(\alpha)$,

* $V' \subset \subset V$.

For instance, if $\varepsilon > 0$ is small enough, the $n$-Hartogs figures $(\mathcal{H}, \mathcal{H}^\varepsilon)$ are boxed (see Definition 2.4).

**Proposition 2.13.** Let $n \ge 2$ and $p \ge 1$ be integers, let $Z$ be a reduced complex space of pure dimension $n+p$ and let $\Delta \subset \subset Z$ be an open set with smooth $\mathscr{C}^2$ boundary in $Z$ which is strongly $(n-2)$-concave. Assume that $\Delta := \{\varphi > 0\}$ and let $X_0$ be a $n$-cycle in an open neighbourhood $\Delta'$ of the compact set $\bar{\Delta}$, such that any irreducible component of $X_0$ meets $\Delta$. Then there exists a finite family of boxed $n$-Hartogs figures $(\mathcal{H}'_a, \mathcal{H}_a)_{a \in A}$ relative to the boundary of $\Delta$, such that the following conditions hold:

1. The open sets $\mathcal{M}'_a$ for $a \in A$ cover the boundary $\partial \Delta$.

2. For each $a \in A$ the Hartogs figures $\mathcal{H}_a$ and $\mathcal{H}'_a$ are adapted to $X_0$.

3. For each $a \in A$ any irreducible component of $X_0$ meeting $\mathcal{M}_a$ meets the open set $\mathcal{M}'_a$.

4. No compact irreducible component of $X_0 \cap \Delta$ meets the union of the compact sets $\bar{\mathcal{M}}_a, a \in A$.

**Remark.** Let $X_0$ be any $n$-cycle in $\Delta'$. Choosing the open set $\Delta'$ small enough around the compact set $\bar{\Delta}$, we can assume that the cycle $X_0$ has only finitely many irreducible components in $\Delta'$ and that each of them which is not compact meets $\partial \Delta$ (see Remark 3 following Proposition 2.7).

**Corollary 2.14.** In the situation of Proposition 2.13, if we assume that the open set $\Delta'$ containing $\bar{\Delta}$ is small enough, any irreducible component $\Gamma$ of the cycle $X_0$ in $\Delta'$ satisfies for all $a \in A$ and all $\eta > 0$ small enough:

$$ \Gamma \cap (\mathcal{M}_a^\eta \times V_a^\eta \times B_a) = \text{prlgt}_a [\Gamma \cap (M_a \times V_a \times B_a)] $$

where $\text{prlgt}_a : H(\bar{\mathcal{M}}_a \times \bar{V}_a, \text{Sym}^k(B_a)) \to H(\bar{\mathcal{M}}_a^\eta \times \bar{V}_a^\eta, \text{Sym}^k(B_a))$ is the holomorphic map of analytic extension built in Proposition 2.6.

**Remark.** In the situation of the previous corollary, choosing $\varepsilon > 0$ small enough, there exists, for each $a \in A$, a holomorphic extension map which lifts the map $\text{prlgt}_a$:

$$ \text{iprlgt}_a : \Sigma_{M_a, \mathcal{M}_a^\varepsilon}(k) \to \Sigma_{\mathcal{M}_a^\eta, \mathcal{M}_a^{\eta+\varepsilon}}(k). $$

It allows to extend in this setting an analytic family of branched coverings in $M_a$ parametrized by a Banach analytic set $S$ and which is isotropic on $S \times \mathcal{M}_a^\varepsilon$ to an analytic family of branched coverings in $\mathcal{M}_a^\eta$ which is isotropic on $S \times \mathcal{M}_a^{\eta+\varepsilon}$.

**Proof of Proposition 2.13.** Corollary 2.11 and the remark following it implies the existence, for each $z \in \partial \Delta$ of a $n$-Hartogs figure $\mathcal{H}_z$ relative to the boundary of $\Delta$, contained in $\Delta'$ and satisfying the following properties:
i) $z \in \mathcal{M}_z$;

ii) $\mathcal{H}_z$ is adapted to $X_0$;

iii) each irreducible component of $X_0$ meeting $\mathcal{M}_z$ meets the open set $\mathcal{M}_z$;

iv) No compact irreducible component of $X_0 \cap \Delta$ meets $\bar{\mathcal{M}}_z$.

As the open sets $\mathcal{M}_z$ cover the compact set $\partial\Delta$ we can find a finite sub-covering. Then the properties 1, 2, 3 and 4 are consequences of i), ii), iii) and iv) by letting $\mathcal{H}'_a := \mathcal{H}^\varepsilon_a$ and choosing $\varepsilon > 0$ small enough. $\square$

*Proof of Corollary 2.14.* Let $\Gamma$ be an irreducible component of $X_0$ meeting $\mathcal{M}_a$ for some $a \in A$. Then $\Gamma$ meets $\mathcal{M}_a$. As $\Gamma$ does not meet $\bar{M}(\alpha)_a \times \bar{V}_a \times \partial B_a$ because $\mathcal{H}_a$ is adapted to $X_0$, the intersection $\Gamma \cap \bar{\mathcal{M}}_a$ is the graph of an element $\gamma \in H(\bar{M}(\alpha)_a \times \bar{V}_a, \text{Sym}^{k_a}(B_a))$ with $k_a \in \mathbb{N}^* {}^6$.

The closed analytic subset $Y$ of the open set $\mathcal{M}_a^\eta$ defined by $Y := \text{prlgt}_a[\Gamma \cap \mathcal{M}_a]$ is not empty, of pure dimension $n$ and is contained in $\Gamma$. So it is a union of irreducible components of $\Gamma \cap \mathcal{M}_a^\eta$. But it contains a non empty open set in each irreducible component of this branched covering. So these two analytic subsets coincide.

If an irreducible component of $X_0$ does not meet any $\bar{\mathcal{M}}_a$ it has to be compact and contained in $\Delta$. In this case the desired equality is obvious. $\square$

# 3. The extension and finiteness theorem

## 3.A. Some useful lemmas

The version below of Sard's lemma is more or less classical.

**Lemma 3.1.** Let $Z$ be a reduced complex space and let $\varphi : Z \to \mathbb{R}$ be a real valued $\mathscr{C}^1$ function. Then the set of critical values of $\varphi$ has Lebesgue measure 0.

*Proof.* Firstly note that a point $z \in Z$ is critical for $\varphi$ if, by definition, the differential of $\varphi$ vanishes on $T_{Z,z}$, the Zariski tangent space of $Z$ at $z$. Remember also that a complex space is, by definition, countable at infinity ; so $Z$ and its singular locus have only countably many irreducible components. As a countable union of sets of measure 0 is again of measure 0, it is enough to prove the lemma when $Z$ is irreducible. We shall prove the lemma by induction on the integer $\dim Z$. The case $\dim Z = 0$ is obvious. Assume the lemma true for $\dim Z \le n-1$ for some integer $n \ge 1$ and take an irreducible complex space $Z$ of dimension $n$. The singular set $S$ of $Z$ has dimension at most $(n-1)$, and for each irreducible component $S_i$ of $S$ the image of the critical set of $\varphi|_{S_i}$ has measure 0. So the critical set of $\varphi|_S$ is again of measure 0. But a critical point of $\varphi$ which belongs to $S$ is a critical point of $\varphi|_S$. So it is enough to show that the set of critical values of $\varphi$ restricted to the complex connected manifold $Z \setminus S$ has measure 0. This is the classical Sard's lemma. $\square$

**Lemma 3.2.** Let $V$ be an open set and $K$ be a compact set in $\bar{U} \times \bar{B}$. The subset $V$ in $H(\bar{U}, \text{Sym}^k(B))$ consisting of the $X$ such that any irreducible component meeting $K$ meets $V$ is an open set in $H(\bar{U}, \text{Sym}^k(B))$.

*Proof.* Let us clarify the meaning of an irreducible component of an element $X$ in $H(\bar{U}, \text{Sym}^k(B))$: we call irreducible component of such a $X$ the closure in $\bar{U} \times B$ of an irreducible component of the branched covering of $U$ defined by the projection of $X \cap (U \times B)$ on $U$.

<footnote n="6">⁶ $\uparrow$ as it exists some $(m, v) \in \bar{M}(\alpha)_a \times \bar{V}_a$ such that $\Gamma \cap (\{m, v\} \times B_a) \neq \emptyset$.</footnote>
Let X$_{0}$ be such that any irreducible component of X$_{0}$ which meets K meets V , and assume that ( X$_{ν}$ )$_{ν}$$_{≥}$$_{1}$ is a sequence converging to X$_{0}$ such that for each ν ≥ 1 there exists an irreducible component Γ$_{ν}$ of ¯ X$_{ν}$ meeting K but not V . Passing to a subsequence, we may assume that the sequence (Γ$_{ν}$)$_{ν}$$_{≥}$$_{1}$ converges uniformly on any compact of U × B to a non empty n -cycle Γ with closure contained in X$_{0}$ and which is a branched covering of U . Then ¯ Γ meets K and not V . Indeed, if ( t$_{0}$, x$_{0}$ ) would be in ¯ Γ ∩ V , there exists open neighbourhoods U$_{1}$ and B$_{1}$ of t$_{0}$ and x$_{0}$ respectively in ¯ U and ¯ B such that U$_{1}$ × B$_{1}$ is contained in V . But then, as U$_{2}$ := U$_{1}$ ∩ U and B$_{2}$ := B$_{1}$ ∩ B are non empty open sets, for t$_{2}$ in U$_{2}$ the fibers of the Γ$_{ν}$ at t$_{2}$ for ν big enough will meet { t$_{2}$ } × B$_{2}$ and so V . As at least one irreducible component of Γ meets K without meeting V and as its closure is an irreducible component of X$_{0}$ , this gives a contradiction. □

Of course, in the case V =  , we get back the fact that the subset in H ( ¯ U , Sym $^{k}$( B )) of elements which do not meet K is open.

Lemma 3.3. Let Z be a complex space and let ( U$_{i}$ )$_{i}$$_{∈}$$_{I}$ be an open covering of Z . Assume that for each i ∈ I a closed n -cycle X$_{i}$ is given in U$_{i}$ . Assume that the following patching condition holds:

$$
\forall (i,j) \in I^2 \quad X_i \cap \mathcal{U}_j = X_j \cap \mathcal{U}_i
$$

as an equality of cycles in U$_{i}$ ∩ U$_{j}$ . Then there exists a unique closed n -cycle X in Z such that for each i ∈ I we have X ∩ U$_{i}$ = X$_{i}$ .

For the easy proof see [BM14, Chapter IV, Proposition 1.3.1].

The following variant will be used.

Lemma 3.4. (Variant) In the situation of the previous lemma replace the patching condition by the following two conditions:

1. For each couple ( i, j ) ∈ I 2 an open subset W$_{i,j}$ ⊂ ⊂ U$_{i}$ ∩ U$_{j}$ is given and we ask that X$_{i}$ ∩ W$_{i,j}$ = X$_{j}$ ∩ W$_{i,j}$ .

2. For each couple ( i, j ) ∈ I 2 we ask that any irreducible component of the cycle X$_{i}$ ∩ U$_{j}$ meets the open set W$_{i,j}$ .

Then the conclusion is the same.

Proof. Let Γ be an irreducible component with multiplicity δ in the cycle X$_{i}$ ∩ U$_{j}$ . Let Γ ′ be the irreducible component of X$_{i}$ which contains Γ, and put X$_{i}$ = X ′ i + δ.Γ ′ . Then Γ ′ meets W$_{i,j}$ and there exists a closed analytic subset Y of pure dimension n in | X$_{j}$ | such that its restriction to W$_{i,j}$ is equal to Γ ′ ∩ W$_{i,j}$ : indeed, Y is the union of the irreducible components of X$_{j}$ containing a non empty open set in Γ ′ ∩ W$_{i,j}$ . Note that each of these irreducible components of X$_{j}$ has multiplicity δ in the cycle X$_{j}$ . Then put X$_{j}$ = X ′ j + δ.Y . We see that the cycles X ′ i and X ′ j respectively in U$_{i}$ and U$_{j}$ satisfy again the patching condition X ′ i ∩ W$_{i,j}$ = X ′ j ∩ W$_{i,j}$ .

This allows, for fixed ( i, j ), to make a descending induction on the number (necessarily finite as W$_{i,j}$ is relatively compact) of irreducible components of X$_{i}$ ∩ U$_{j}$ , to show that the condition X$_{i}$ ∩ U$_{j}$ = X$_{j}$ ∩ U$_{i}$ holds. This reduces this lemma to the previous one. □

3.B. Adjusted scales.

The definition of a scale adapted to a cycle is recalled in Definition 1.3.

Definition 3.5. 1. Let Z be a complex space. We shall call adjusted n-scale on Z , written down E := ( U, U $^{′}$, U $^{′′}$, B, B $^{′′}$, j ), the data of a n-scale on Z , E := ( U, B, j ), with additional polydiscs U ′′ ⊂ ⊂ U ′ ⊂ ⊂ U and B ′′ ⊂ ⊂ B . We call E the underlying scale of the adjusted scale E .
2. We shall say that **the adjusted scale E** is adapted to a **n-cycle X** in Z when we have

$$j^{-1}(\bar{U} \times (\bar{B} \setminus B'')) \cap |X| = \emptyset.$$

Note that this implies that the underlying scale E is adapted to X, but this condition is more restrictive.

3. When the adjusted scale E is adapted to the n-cycle X, we shall call **degree of X in E** the degree of X in E.

4. We shall call **center of the adjusted scale**, written down $D(E)$, or more simply, $D(E)$, the open set $j^{-1}(U \times B)$ in Z which is also the center of the scale E.

5. We shall call **domain of isotropy of the adjusted scale**, written down $D'(E)$, the open set $j^{-1}(U' \times B)$ in Z.

6. We shall call **domain of patching of the adjusted scale**, written down $D''(E)$, the open set $j^{-1}(U'' \times B'')$ in Z.

**Remarks.**

1. The open set $D''(E)$ is relatively compact in $D'(E)$.

2. When a n-scale E is given, for any compact set K in $D(E)$, there exists an adjusted n-scale E such that E is the underlying scale of E and with $K \subset D''(E)$. Moreover, if E is adapted to a n-cycle $X_0$ in Z, we may choose E in order that it is adapted to $X_0$.

3. As for $X \in C_n^{\text{loc}}(Z)$ the condition to avoid a given compact subset is open in $C_n^{\text{loc}}(Z)$, when the adjusted scale E is adapted to a cycle $X_0$ there exists an open neighbourhood, written down $\Omega_k(E)$, of $X_0$ in $C_n^{\text{loc}}(Z)$ such that $\Omega_k(E)$ is the subset of all n-cycles X in Z for which E is adapted and $\text{deg}_E(X) = k$ where $k := \text{deg}_E(X_0)$.

Let Z be a reduced complex space and let $E = (U, U', U'', B, B'', j)$ be an adjusted scale on Z. For a given integer k consider the continuous map sending a branched covering in $H(\bar{U}, \text{Sym}^k(B''))$ to its isotropy data on $\bar{U}'$ (for the notations see what follows Lemma 2.5)

$$T: H(\bar{U}, \text{Sym}^k(B'')) \to H(\bar{U}', F \otimes E').$$

The graph $\Sigma_{U,U'}(k)$ of this map is a Banach analytic set$^7$, thanks to [Bar75, Proposition 2, p. 81] (see also [BM]).

The set of couples $(f, T(f))$ in $\Sigma_{U,U'}(k)$ for which the associated branched covering is contained in $j(Z \cap \bar{D}(E))$ is a closed Banach analytic subset of $\Sigma_{U,U'}(k)$ being the pull-back by the projection of the subset of elements in $H(\bar{U}, \text{Sym}^k(B''))$ contained in $j(Z \cap \bar{D}(E))$ which is a closed Banach analytic subset of $H(\bar{U}, \text{Sym}^k(B''))$ by Proposition 4, p. 27 of [Bar75] (see also [BM, Chapter V]).

**Definition 3.6.** We shall denote $\mathcal{G}_k(E)$ this Banach analytic set and we shall call it the **k-th classifying space of the adjusted n-scale E** on Z.

We have then a tautological family of n-cycles in the open set $D(E)$ parametrized by $\mathcal{G}_k(E)$. It is an analytic family of cycles in the open set $D'(E)$, in the sense of [Bar75], and the fact that, for $k \ge 1$, locally on $\mathcal{G}_k(E)$, any irreducible component of a branched covering in this family meets $\bar{U}'' \times \bar{B}''$ implies that we have a f-analytic family of cycles in $D'(E)$.

<footnote>$^7$ ↑ homeomorphic to $H(\bar{U}, \text{Sym}^k(B''))$ via the projection!</footnote>
Be careful that the tautological family of cycles on the open set $D(E)$ parametrized by $G_k(E)$ is not, in general, an analytic family of $n$-cycles ; see the example of [Bar75, p. 83] (and also [BM14, Chapter IV]).

The next lemma is an obvious consequence of loc. cit.

**Lemma 3.7.** Let $\mathbb{E}$ be an adjusted $n$-scale on a reduced complex space $Z$ and let $k$ be an integer. The tautological family of $n$-cycles in the open set $D'(\mathbb{E})$ parametrized by $G_k(\mathbb{E})$ has the following “almost universal” property:

For any analytic family of $n$-cycles $(X_s)_{s \in S}$ in $Z$ parametrized by a Banach analytic set $S$ such that for each $s \in S$ the adjusted scale $\mathbb{E}$ is adapted to $X_s$ with $\deg_E(X_s) = k$, there exists an unique holomorphic map

$$f : S \to G_k(E)$$

such that the pull-back by $f$ of the tautological family is the restriction to the open set $D'(\mathbb{E})$ of the given family.

Of course, conversely, such a holomorphic map gives a $f$-analytic family of $n$-cycles on the open set $D'(\mathbb{E})$.

Note that the pull-back family is in fact defined on the open set $D(E)$ but, as already noticed above, it may not be analytic outside $D'(E)$.

As a consequence of this “almost universal” property, we obtain that for any analytic family $(X_s)_{s \in S}$ of $n$-cycles in $Z$ such that for a point $s_0 \in S$ the adjusted $n$-scale $\mathbb{E}$ is adapted to the cycle $X_{s_0}$ with $\deg_{\mathbb{E}}(X_{s_0}) = k$, there exists an open neighbourhood $S'$ of $s_0$ in $S$ such that the previous lemma applies for the family parametrized by $S'$. So we shall have a holomorphic classifying map $f : S' \to G_k(E)$ in this situation.

We shall generalize now the concept of classifying space to the case of a finite family of adjusted $n$-scales.

**Definition 3.8.** Consider a reduced complex space $Z$ and a finite family of adjusted $n$-scales $(\mathbb{E}_i)_{i \in I}$ on $Z$. Assume that they are adapted to a given finite type $n$-cycle $\hat{X}_0$ in $Z$. Assume that any irreducible component of $\hat{X}_0$ meets the open set $W'' := \bigcup_{i \in I} D''(\mathbb{E}_i)$.

We shall call **patching data for $\hat{X}_0$ associated to the family $(\mathbb{E}_i)_{i \in I}$, written $\mathcal{R}((\mathbb{E}_i)_{i \in I}, F)$** or more simply $\mathcal{R}$ when there is no ambiguity, a finite collection of $n$-scales $(F_{i,j,h})$ for $(i,j) \in I^2$, $i \neq j$, where $h$ belongs to a finite set $H(i,j)$ for each couple $(i,j) \in I^2$, $i \neq j$, such that the following properties hold:

i) $F_{i,j,h}$ is a $n$-scale on the open set $D'(\mathbb{E}_i) \cap D'(\mathbb{E}_j)$.

ii) The $n$-scales $F_{i,j,h}$ are adapted to $\hat{X}_0$.

We shall say that the patching data $\mathcal{R}$ are **complete** when the following condition also holds:

iii) For each $i \neq j$ given, the union of domains of the scales $F_{i,j,h}$, $h \in H(i,j)$, covers the compact subset $\overline{D''(\mathbb{E}_i)} \cap \overline{D''(\mathbb{E}_j)}$ of $D'(\mathbb{E}_i) \cap D'(\mathbb{E}_j)$.

**Notations.**

1. In the sequel, when we shall consider a reduced complex space $Z$ and a finite family of adjusted $n$-scales $(\mathbb{E}_i)_{i \in I}$, adapted to a finite type $n$-cycle $\hat{X}_0$ in $Z$, such that any irreducible component of $\hat{X}_0$ meets the open set

$$W'' := \bigcup_{i \in I} D''(\mathbb{E}_i),$$

we shall say that the family $(\mathbb{E}_i)_{i \in I}$ is **convenient** for $\hat{X}_0$.
2. In this setting we shall use the following definitions :

* $W := \bigcup_{i \in I} D(\mathbb{E}_i)$;

* $W' := \bigcup_{i \in I} D'(\mathbb{E}_i)$;

* $W'' := \bigcup_{i \in I} D''(\mathbb{E}_i)$;

* $K := \bigcup_{i \in I} j_i^{-1}(\bar{U}_i \times (\bar{B}_i \setminus B_i''))$.

* When the family $(\mathbb{E}_i)_{i \in I}$ is convenient for a finite type $n$-cycle $\hat{X}_0$, $K$ will be a compact neighbourhood of $K$ disjoint from $\hat{X}_0$.

3. For $\tilde{X} \in \prod_{i \in I} \mathcal{G}_{k_i}(\mathbb{E}_i)$ we shall denote by $X^i$ the closed cycle in $D(\mathbb{E}_i)$ associated to the $i$-th component of $\tilde{X}$.

**Lemma 3.9.** Let $(\mathbb{E}_i)_{i \in I}$ be a finite family of adjusted $n$-scales on $Z$, convenient for a $n$-cycle $\hat{X}_0$ of finite type in $Z$, and let $\mathcal{R}$ be some corresponding complete patching data. There exists an open neighbourhood $\mathcal{V}$ of the image $\hat{X}_0$ of $\hat{X}_0$ in the product $\prod_{i \in I} \mathcal{G}_{k_i}(\mathbb{E}_i)$ such that for each $\tilde{X} \in \mathcal{V}$ we have the following properties:

1. No $X^i$ meets the compact set $K$.

2. For each $i \in I$, any irreducible component of $X^i$ meeting $\overline{D''(\mathbb{E}_i)} \cap \overline{D''(\mathbb{E}_j)}$ with $j \neq i$, meets the open set $\bigcup_h D(F_{i,j,h})$.

3. For each $(i, j, h)$ the scale $F_{i,j,h}$ is adapted to $X^i$ and $X^j$.

4. For each $(i, j, h)$ we have $\deg_{F_{i,j,h}}(X^i) = \deg_{F_{i,j,h}}(X^j) = \deg_{F_{i,j,h}}(\hat{X}_0) = k_{i,j,h}$.

*Proof.* Conditions 1, 3 and 4 are clearly open. An easy consequence of Condition 1, of the inclusion ii) of Definition 3.8 and of Lemma 3.2 is that Condition 2 is also open. $\square$

For a $n$-scale $E := (U, B, j)$ on $Z$ we shall abbreviate $H(\bar{U}, \mathrm{Sym}^k(B))$ in $G_k(E)$.

When we consider a cycle $X_0$ in an adapted scale $E := (U, B, j)$ and when we dispose of a $n$-scale $F := (V, C, h)$ on $U \times B$, adapted to $X_0$, where $h$ is given by an isomorphism of an open set in $U \times B$ into some open neighbourhood of $\bar{V} \times \bar{C}$ in $\mathbb{C}^n \times \mathbb{C}^p$, we have a well-defined map of an open neighbourhood $\mathcal{U}$ of $X_0$ is $H(\bar{U}, \mathrm{Sym}^k(B))$ into $H(\bar{V}, \mathrm{Sym}^l(C))$, where $l := \deg_F(X_0)$, sending $X \in \mathcal{U}$ to the multiform graph associated to $h_*(X)$ in the scale $F$. This is a consequence of the fact that the condition $X_0 \cap h^{-1}(\bar{V} \times \partial C) = \emptyset$ is open in $H(\bar{U}, \mathrm{Sym}^k(B))$ and that the degree of $X$ near enough $X_0$ in the adapted scale $F$ will be equal to $l$.

Such a map, which will be called a **change of scale**, is not holomorphic in general but becomes holomorphic when we add the isotropy condition:

precisely, if $U' \subset \subset U$ and if $h^{-1}(\bar{V} \times \bar{C}) \subset U' \times B$, then the change of scale map

$$ \Sigma_{U,U'} \to H(\bar{V}, \mathrm{Sym}^l(C)) $$

will be holomorphic (see Theorem 4, p. 66 in [Bar75]).

**Definition 3.10.** Let $(\mathbb{E}_i)_{i \in I}$ a finite family of adjusted $n$-scales on $Z$, convenient for a $n$-cycle $\hat{X}_0$ of finite type in $Z$, and let $\mathcal{R}$ be some corresponding patching data. Let $k_i$ be the degree of $\hat{X}_0$ in the adjusted scale $\mathbb{E}_i$. For each $(i, j, h)$, $i \neq j$ we have a couple of holomorphic maps

$$ \prod_{\alpha \in I} \mathcal{G}_{k_{\alpha}}(\mathbb{E}_{\alpha}) \xrightarrow{\longrightarrow} \mathcal{G}_{k_{i,j,h}}(F_{i,j,h}) $$

obtained by the changes of scales $\mathbb{E}_i \to F_{i,j,h}$ and $\mathbb{E}_j \to F_{i,j,h}$, because, by construction, we have $D(F_{i,j,h}) \subset \subset D'(\mathbb{E}_i) \cap D'(\mathbb{E}_j)$.
We shall denote by S ( R ) the intersection of the kernels of these double maps 8 with the open set V built in Lemma 3.9. It is a Banach analytic set and we shall call it the classifying space associated to ( E$_{i}$ )$_{i}$$_{∈}$$_{I}$ , ˆ X$_{0}$ and R .

Remark that the patching data R are not assumed to be complete in the previous definition.

**Proposition 3.11.** Consider a finite family of adjusted n-scales which is convenient for the n-cycle X$_{0}$ in Z and let R be some complete patching data associated. Keeping the previous notations we have for each ( X $^{i}$)$_{i}$$_{∈}$$_{I}$ ∈ S ( R ) a unique n-cycle X ∈ C f $_{n}$( W $^{′}$) such that X ∩ D $^{′}$( E$_{i}$ ) = X i ∩ D $^{′}$( E$_{i}$ ) , ∀ i ∈ I .

Moreover, this defines a tautological family of cycles in W ′ which is a f -analytic family of cycles satisfying the following “almost universal” property:

For any analytic family of n-cycles ( X$_{s}$ )$_{s}$$_{∈}$$_{S}$ in an open neighbourhood of ¯ W parametrized by a Banach analytic set S 9 such that for s$_{0}$ ∈ S we have X$_{s}$$_{0}$ = X$_{0}$ in a neighbourhood of ¯ W , there exists an open neighbourhood S ′ of s$_{0}$ in S and a unique holomorphic map

$f : S' \to S(\mathcal{R})$

such that the pull-back by f of the tautological family parametrized by S ( R ) is the restriction to the open set W ′ of the family ( X$_{s}$ )$_{s}$$_{∈}$$_{S}$ ′ .

Proof. For each ˜ X ∈ V any X i does not meet K . As R is complete, we may use Lemma 3.4 with U$_{i}$ := D $^{′}$( E$_{i}$ ) and W$_{i,j}$ = ⋃ $_{h}$D ( F$_{i,j,h}$ ) to associate to ˜ X a finite type n-cycle X of the open set W $^{′}$. The f -analyticity of the so defined family is obvious. The “almost universal” property is then clear. □

Note that this proposition implies that the map S ( R ) → C f $_{n}$( W $^{′}$) classifying the tautological family of n-cycles in W ′ is a holomorphic map.

3.C. Shrinkage.

**Definition 3.12.** Let Z be a reduced complex space and let E := ( U, U $^{′}$, U $^{′′}$, B, B $^{′′}$, j ) be an adjusted n-scale on Z . For any real τ > 0 small enough we shall denote by E τ the adjusted n-scale on Z defined as E τ := ( U $^{τ}$, U $^{′τ}$, U $^{′′}$, B, B $^{′′}$, j ) . We shall call E τ the τ -shrinkage of E .

Recall that for a polydisc P of radius R , P τ is the polydisc with same center and radius R − τ . The definitions of M ( α ) $^{τ}$, M ( α ) τ are given in the section 2.B.

Remarks.

1. By definition, the shrinkage of E does not change the embedding j and the polydiscs U $^{′′}$, B, B $^{′′}$.

2. As j is a closed embedding of an open set in Z in an open neighbourhood of the compact set ¯ U × ¯ B , it is clear that for any given adjusted n-scale E on Z , there exists a real ε > 0 (depending on E ) such that for any τ ∈ ] 0 , ε [ , E τ is again an adjusted n-scale on Z .

3. If E is an adjusted scale adapted to the n-cycle X$_{0}$ in Z , for τ small enough (depending on E and X$_{0}$ ), the adjusted n-scale E τ remains adapted to X$_{0}$ and we shall have also deg$_{E}$ $^{τ}$( X$_{0}$ ) = deg$_{E}$( X$_{0}$ ).

<footnote n="8">8 ↑ The kernel of a double map f, g : A → B is the pull-back of the diagonal in B × B by the map ( f, g ) : A → B × B .</footnote>

<footnote n="9">9 ↑ Recall that, by definition, this means that for any n-scale E := ( U, B, j ) on Z , adapted to some X$_{s}$$_{0}$ , s$_{0}$ ∈ S , with deg$_{E}$( X$_{s}$$_{0}$ ) = k , we have an open neighbourhood S$_{0}$ of s$_{0}$ in S and a classifying map for the corresponding family of branched coverings f : S$_{0}$ × U → Sym $^{k}$( B ) which is isotropic.</footnote>
4. If $\mathbb{E}$ is an adjusted scale adapted to the $n$-cycle $X_0$ in $Z$, there exists an open neighbourhood $\mathcal{U}$ of $X_0$ in $C_n^{\text{loc}}(Z)$ and a real $\varepsilon > 0$ such that for any $X \in \mathcal{U}$ and any $\tau \in ]0, \varepsilon[$, the adjusted scale $\mathbb{E}^{\tau}$ remains adapted to $X$ with again $\text{deg}_{\mathbb{E}^{\tau}}(X) = \text{deg}_{\mathbb{E}}(X_0)$.

5. In the situation of the previous proposition 3.11, we may, for $\tau > 0$ small enough, keep the same patching data $\mathcal{R}$ on the finite family $(\mathbb{E}_i^\tau)_{i \in I}$ of adjusted $n$-scales; if it was complete, it remains complete and if it was convenient<fnref n="10" /> for the $n$-cycle $X_0$, it remains convenient for the $n$-cycle $X_0$. Then we have a holomorphic restriction map

$$S(\mathcal{R}) \to S^\tau(\mathcal{R})$$

where $S^\tau(\mathcal{R})$ is the classifying space associated to the family $(\mathbb{E}^\tau)_{i \in I}$, and this map is induced by a finite product of linear (continuous) compact maps. This last point is crucial for the finiteness results.

### 3.D. Excellent family.

In order to avoid that our notations become too heavy we shall introduce the following conventions when $\mathcal{H}$ is a $n$-Hartogs figure in $\mathbb{C}^{n+p}$; we shall put, using the notations introduced above

$$U := M(\alpha) \times V, \quad U' := M(\alpha)^{\varepsilon'} \times V^{\varepsilon'}, \quad U'' := M(\alpha)^{\varepsilon''} \times V^{\varepsilon''}, \quad B'' := B^{\varepsilon'}$$

where $\varepsilon' > 0$ is small enough, and where $0 < \varepsilon'' < \varepsilon'$. The choices of $\varepsilon'$ and $\varepsilon''$ will be precised when they are useful. We shall associate to $\mathcal{H}$ the adjusted $n$-scale on $\Delta$ given by:

$$\mathbb{E}_{\mathcal{H}} := (U, U', U'', B, B'', j).$$

We shall put also

$$\tilde{U} := \mathcal{M}(\alpha) \times V, \quad \tilde{U}' := \mathcal{M}(\alpha)^{\varepsilon'} \times V^{\varepsilon'}, \quad \tilde{U}'' := \mathcal{M}(\alpha)^{\varepsilon''} \times V^{\varepsilon''}$$

the holomorphy envelopes respectively of $U$, $U'$ and $U''$. Then we shall have a family of adjusted $n$-scales on $\Delta'$, written down:

$$\mathbb{E}_{\tilde{\mathcal{H}}}^{\eta} := (\tilde{U}^{\eta}, \tilde{U}'^{\eta}, \tilde{U}''^{\eta}, B, B'', j) \quad \text{with} \quad \tilde{U}^{\eta} := \mathcal{M}(\alpha)^{\eta} \times V^{\eta}, \quad \tilde{U}'^{\eta} := \mathcal{M}(\alpha)^{\varepsilon'+\eta} \times V^{\varepsilon'+\eta},$$

where $\eta$ is a non negative real number, small enough (for $\eta = 0$ we shall simply write $\mathbb{E}_{\tilde{\mathcal{H}}}^{\eta}$). Then we shall have the isotropic classifying spaces

$$\mathcal{G}_k(\mathcal{H}) := \sum_{U,U'}(k) \quad \text{and also} \quad \mathcal{G}_k^{\eta}(\tilde{\mathcal{H}}) := \sum_{\tilde{U}^{\eta}, \tilde{U}'^{\eta}}(k).$$

Proposition 2.6 gives a holomorphic analytic extension map $prlgt^\eta$ such that the following diagram commutes:

<figure><img src="image_2.png" /></figure>

**Definition 3.13.** Let $Z$ be a reduced complex space of pure dimension $n + p$, let $\Delta \subset \subset Z$ be a strongly $(n-2)$-concave open set in $Z$ and $\tilde{X}_0$ a $n$-cycle in an open neighbourhood $\Delta'$ of $\bar{\Delta}$ in $Z$. We shall say that a finite family $(\mathcal{H}_a)_{a \in A}$ of $n$-Hartogs figures relative to the boundary of $\Delta$ is **excellent** for the cycle $\tilde{X}_0$ when the following conditions hold, where we write $\tilde{\mathbb{E}}_a$ and $\mathbb{E}_a$ the adjusted scales respectively on $\Delta'$ and $\Delta$ associated to the $n$-Hartogs figure $\mathcal{H}_a$:

<footnote n="10">¹⁰ ↑ See the notations following Definition 3.8.</footnote>
1. The adjusted scales $\tilde{\mathbb{E}}_a$ and $\mathbb{E}_a$ are adapted to the cycle $\tilde{X}_0$.

2. We may choose the patching domains of the adjusted scales $(\tilde{\mathbb{E}}_a)_{a \in A}$ in order that the union

$$ \tilde{D}''(A) := \bigcup_{a \in A} j_a^{-1}(\tilde{U}''_a \times B''_a) $$

contains the compact set $\partial \Delta$.

3. There exists a finite family of adjusted $n$-scales $(\mathbb{E}_b)_{b \in B}$ on $\Delta$, adapted to $\tilde{X}_0$, such that the finite families $(\mathbb{E}_c)_{c \in A \cup B}$ and $(\tilde{\mathbb{E}}_c)_{c \in A \cup B}$ are convenient for $\tilde{X}_0$, where we put $\tilde{\mathbb{E}}_b = \mathbb{E}_b$ for $b \in B$.

Moreover we ask that the union $D''(B)$ of the patching domains of the $(\mathbb{E}_b)_{b \in B}$ covers the compact set $\Delta \setminus \tilde{D}''(A)$ of $\Delta$; so $\tilde{D}''(A) \cup D''(B)$ in an open set containing $\tilde{\Delta}$.

As a consequence, the union $\tilde{D}'(A)^\eta \cup D'(B)$ of the isotropy domains will cover $\tilde{\Delta}$ for $\eta > 0$ small enough.

**Proposition 3.14. (Existence of excellent families)** Let $Z$ be a reduced complex space of pure dimension $n + p$, $\Delta \subset \subset Z$ be a strongly $(n-2)$-concave open set with smooth boundary and $\tilde{X}_0$ a $n$-cycle in an open neighbourhood $\Delta' \subset Z$ of $\tilde{\Delta}$ such that any irreducible component of $\tilde{X}_0$ meets $\Delta$.

Then there exists a finite family $(\mathcal{H}_a)_{a \in A}$ of $n$-Hartogs figures relative to the boundary of $\Delta$ which is excellent for the cycle $\tilde{X}_0$.

*Proof.* First we use Proposition 2.13 to cover $\partial \Delta$ by a finite family of $n$-Hartogs figures relative to the boundary of $\Delta$ adapted to the cycle $\tilde{X}_0$ such that Conditions 1 and 2 hold. Then we build a finite family of adjusted $n$-scales $(\mathbb{E}_b)_{b \in B}$ on $\Delta$, adapted to $\tilde{X}_0$ in order that Condition 3 holds. $\square$

### 3.E. The extension and finiteness theorem.

The next theorem will be crucial in the proof of Theorem 1.1.

**Theorem 3.15.** Let $Z$ be a reduced complex space of pure dimension $n + p$, where $n \ge 2$, $p \ge 1$. Assume that there exists a $\mathcal{C}^2$ exhaustion $\varphi : Z \to [0, 2]$ which is strongly $(n-2)$-convex on the open set $\varphi^{-1}([0, 1[)$ and let $\Delta := \{x \in Z / \varphi(x) > \alpha\}$ for some $\alpha \in ]0, 1[$ which is not a critical value for $\varphi$. Let $\tilde{X}_0$ a closed $n$-cycle of an open neighbourhood $\Delta'$ of $\tilde{\Delta}$ in $Z$ such that any irreducible component of $\tilde{X}_0$ meets $\Delta$. Then there exists an open neighbourhood $\Delta''$ of $\tilde{\Delta}$ in $\Delta'$ and a $f$-analytic family $(\tilde{X}_\xi)_{\xi \in \Xi}$ of $n$-cycles in $\Delta''$ parametrized by a reduced complex space $\Xi$ (of finite dimension) such that $X_{\xi_0} = \tilde{X}_0 \cap \Delta''$ and such that $\Xi$ is isomorphic to an open neighbourhood of $\tilde{X}_0 \cap \Delta''$ in $C_n^f(\Delta'')$. It satisfies the following universal property:

For any $f$-analytic family $(X_s)_{s \in S}$ of $n$-cycles in $\Delta$ parametrized by a Banach analytic set $S$ and such that its value at some $s_0 \in S$ is equal to $\tilde{X}_0 \cap \Delta$, there exists an open neighbourhood $S'$ of $s_0$ in $S$ and an unique holomorphic map

$$ h : S' \to \Xi $$

satisfying the equality $X_s = \tilde{X}_{h(s)} \cap \Delta$ for each $s \in S'$.

*Proof.* Note that $\Delta$ is a relatively compact open set in $Z$ with a $\mathcal{C}^2$ boundary which is strongly $(n-2)$-concave.

Begin by covering the compact set $\partial \Delta$ by an excellent finite family $(\mathcal{H}_a)_{a \in A}$ of $n$-Hartogs figures for $\Delta$ adapted to the cycle $\tilde{X}_0$. Choose then open sets $\Delta_1 \subset \subset \Delta \subset \subset \Delta'' \subset \subset \Delta'$, such that the following properties hold, where we use the notations introduced above for the finite family of the adjusted scales $(\tilde{\mathbb{E}}_a)_{a \in A}$ associated to $(\mathcal{H}_a)_{a \in A}$:
i) $\bar{\Delta}'' \setminus \Delta_1 \subset \bigcup_{a \in A} W''(\tilde{\mathbb{E}}_a)$.

ii) $\bigcup_{a \in A} W(\tilde{\mathbb{E}}_a) \subset \Delta'$.

iii) $\bigcup_{a \in A} \bar{W}(\mathbb{E}_a) \subset \Delta$.

iv) $K := \bigcup_{a \in A} j_a^{-1}(\bar{U}_a \times (\bar{B}_a \setminus B_a'')) \subset \Delta_1$

It is easy to fulfill these conditions for $\Delta_1$ and $\Delta''$ near enough to $\Delta$, as, by assumption, the subsets $\bar{W}(\mathbb{E}_a)$ and $j_a^{-1}(\bar{U}_a \times \bar{B}_a \setminus B_a'')$ are compact in $\Delta$, and as the union of the open sets $W''(\tilde{\mathbb{E}}_a)$ contains $\partial\Delta$.

Note that Condition iv) allows to choose the compact neighbourhood $K$ of $K$ inside $\Delta_1$.

Choose now a convenient finite family $(\mathbb{E}_b)_{b \in B}$ of adjusted $n$-scales on $\Delta$, adapted to $\tilde{X}_0$ in order that the open set $\bigcup_{b \in B} W''(\mathbb{E}_b)$ contains $\bar{\Delta}_1$.

Put $\tilde{\mathbb{E}}_b = \mathbb{E}_b$ for $b \in B$, and define $C := A \cup B$. The family of adjusted scales $(\mathbb{E}_c)_{c \in C}$ in $\Delta$ is then convenient for $\tilde{X}_0 \cap \Delta_1$, up to choosing the patching domains big enough. Fix some complete patching data $\mathcal{R}$ associated to the family $(\mathbb{E}_c)_{c \in C}$.

The family $(\tilde{\mathbb{E}}_c)_{c \in C}$ of adjusted scales in $\Delta'$ is convenient for $\tilde{X}_0 \cap \Delta''$ if we choose the patching domains big enough. Consider now some complete patching data for this family of the form $\mathcal{R} \cup \mathcal{R}''$, that is to say containing the patching scales already in $\mathcal{R}$. Define the following Banach analytic sets:

1. $S_0$ is the classifying space of the family $(\tilde{\mathbb{E}}_c)_{c \in C}$, the degrees being these of $\tilde{X}_0$ in the various scales adapted to the cycle $\tilde{X}_0$, with the patching conditions defined by $\mathcal{R}$. Note that $\mathcal{R}$ is not complete in general.

2. $S_+$ is the classifying space of the family $(\tilde{\mathbb{E}}_c)_{c \in C}$ with the patching conditions defined by $\mathcal{R} \cup \mathcal{R}'$.

3. $S_-$ is the classifying space of the family $(\mathbb{E}_c)_{c \in C}$, with the (complete) patching conditions defined by $\mathcal{R}$.

Then we get a holomorphic extension map

$$ \alpha : S_- \to S_0 $$

deduced from the extension maps in the $n$-Hartogs figures $(H_a)_{a \in A}$.

By definition $S_+$ is a closed Banach analytic subset of $S_0$ as it is defined in $S_0$ by the patching conditions given by $\mathcal{R}''$. Then put $\Xi := \alpha^{-1}(S_+)$. So we have a holomorphic map $\alpha : \Xi \to S_+$. We want to show the following claim:

**Claim.** There exists a holomorphic map $\beta : S_+ \to \Xi$ satisfying the two properties:

1. We have $\alpha \circ \beta = \mathrm{Id}$ and $\beta \circ \alpha = \mathrm{Id}$ in a neighbourhood of the points defined by $\tilde{X}_0$ respectively in $S_+$ and $\Xi$.

2. The holomorphic map $\beta$ is the composition of a holomorphic map induced by a linear (continuous) compact map and a holomorphic map.

As the open set $\bigcup_{c \in C} W''(\tilde{\mathbb{E}}_c)$ contains $\Delta''$, there is, on $S_+$, a tautological family of $n$-cycles which is $f$-analytic on $\Delta''$. Then the “almost universal” property of $S_-$ gives a holomorphic map $\tilde{\beta} : S_+ \to S_-$ which factorizes via the closed Banach analytic subset $\Xi \subset S_-$. Let us show that the holomorphic map $\beta : S_+ \to \Xi$ deduced from this factorization satisfies the two properties of the claim.

First we have $\alpha \circ \beta = \mathrm{Id}$ and $\beta \circ \alpha = \mathrm{Id}$ respectively in $S_+$ and $\Xi$, by definition of $\Xi$ and $\beta$ (see Proposition 2.6).

To see the second property, consider $\tau > 0$ small enough and let us show that the holomorphic map induced by the linear compact (by Vitali's theorem) restriction $r^\tau : S_+ \to S_+^\tau$ factorizes $\beta$, where
$S_+^\tau$ is the classifying space corresponding to the $\tau$-shrinkage $(\tilde{\mathbb{E}}_c^\tau)_{c \in C}$ of the family $(\tilde{\mathbb{E}}_c)_{c \in C}$ with the patching data $\mathcal{R} \cup \mathcal{R}''$. Indeed, for $\tau$ small enough, the tautological family of $S_+^\tau$ is still f-analytic on $\Delta$ and the "almost universal" property of $S_-$ gives again a holomorphic map $\tilde{\beta}^\tau : S_+^\tau \to S_-$. And we have $\beta = \tilde{\beta}^\tau \circ r^\tau$ proving our assertion. In fact, the map $\tilde{\beta}^\tau$ takes its values in $\Xi$ because the $\tau$-shrinkage does not change the patching data deduced from $\mathcal{R} \cup \mathcal{R}''$ for $\tau$ small enough.

We conclude that the Banach analytic set $\Xi$ has finite dimension thanks to the finiteness lemma of [Bar75, p. 8] (see also [BM, Chapter V]). Moreover, it is isomorphic to $S_+$ which is also of finite dimension.

The holomorphic isomorphism $\beta : S_+ \to \Xi$ factorizes via an open neighbourhood of $X_0 \cap \Delta''$ in $C_n^f(\Delta'')$ because $S_+$ parametrizes a f-analytic family of cycles in $\Delta''$, and because a cycle $X$ near enough to $X_0 \cap \Delta''$ defines an element in $\Xi \subset S_-$ as it satisfies the patching conditions $\mathcal{R} \cup \mathcal{R}''$. This shows that we can identify the Banach analytic set $\Xi$ (which is a reduced finite dimensional complex space) with an open neighbourhood of $X_0 \cap \Delta''$ in $C_n^f(\Delta'')$.

The universal property is obvious. $\square$

Note that it is not restrictive to choose $\Delta'' := \{\varphi > \alpha_1\}$ for some $\alpha_1 \in ]0, \alpha[$, near enough to $\alpha$.

### Remarks.

1. The reduced complex space (of finite dimension) $\Xi$ built in the previous theorem parametrizes a f-analytic family of $n$-cycles in the open set $\Delta''$ which is an open neighbourhood of $\bar{\Delta}$. So, when we have a f-analytic family $(X_s)_{s \in S}$ of $n$-cycles in $\Delta$ parametrized by a Banach analytic set $S$ and such that its value at some $s_0 \in S$ is equal to $\tilde{X}_0 \cap \Delta$ we can extend each cycle $X_s$, $s \in S'$, to a $n$-cycle $\tilde{X}_s$ in $\Delta''$ in order that the family $(\tilde{X}_s)_{s \in S'}$ is f-analytic in $\Delta''$, with the condition that each irreducible component of $\tilde{X}_s$ meets $\Delta$ and with the equality $\tilde{X}_s \cap \Delta = X_s$ for each $s \in S'$.

2. We shall see later on that $\Xi$ is also (isomorphic to) an open neighbourhood of the point $\tilde{X}_0 \cap \Delta$ in $C_n^f(\Delta)$.

# 4. Finiteness of the space of $n$-cycles of a reduced strongly $(n-2)$-concave space ($n \ge 2$).

## 4.A. The global extension theorem.

First we have to complete our terminology.

**Definition 4.1.** We shall say that a reduced complex space $Z$ is **strongly $q$-concave**, where $q \ge 0$ is a natural integer, if there exists a real valued $\mathcal{C}^2$ exhaustion function on $Z$, $\varphi : Z \to ]0, 2]$, which is strongly $q$-convex outside the compact set $K := \varphi^{-1}([1, 2])$.

In the sequel, when we consider a reduced complex space $Z$ which is assumed to be $q$-concave, we shall always assume implicitly that we have chosen such an exhaustion $\varphi$. For instance, any reduced compact complex space is strongly $q$-concave for any $q \ge 0$.

When $Z$ is strongly $q$-concave irreducible and non compact of dimension at least $q+1$, the function $\varphi$ achieves its maximum at a point in $K$. So we shall have $\varphi(Z) =]0, u]$ with $u \ge 1$.

**Theorem 4.2.** Let $n \ge 2$ be an integer. Let $Z$ be a reduced complex space which is strongly $(n-2)$-concave. Let $\alpha \in ]0, 1[$ and let $X$ be a finite type $n$-cycle in the open set $Z_\alpha := \{z \in Z / \varphi(z) > \alpha\}$. Then there exists a unique $n$-cycle $\tilde{X}$ in $C_n^f(Z)$ such that $\tilde{X} \cap Z_\alpha = X$.

The proof will use the following remark.
**Remark.** Consider a closed irreducible analytic subset Γ of dimension n in Z. As the restriction $\varphi|_{\Gamma}$ of the exhaustion $\varphi$ to Γ must reach its maximum (as $\varphi$ is continuous and proper), this maximum cannot be obtained at a point in which $\varphi$ is strongly (n-2)-convex. So we have $\Gamma \cap \varphi^{-1}[1, 2]) \neq \emptyset$.

*Proof.* The uniqueness of $\tilde{X}$ is a consequence of the previous remark: if $\tilde{X}_i$, $i = 1, 2$, are in $C_n^f(Z)$ and satisfy $\tilde{X}_i \cap Z_\alpha = X$, then any irreducible component $\Gamma_1$ of $\tilde{X}_1$ has to meet $Z_\alpha$ and so has to contain an open set in $\tilde{X}_2$. Then it has to be an irreducible component of $\tilde{X}_2$ and its multiplicity in $\tilde{X}_1$ and $\tilde{X}_2$ must coincide. So $\tilde{X}_1 = \tilde{X}_2$.

To show that this cycle exists, consider first the case where $X$ is compact in $Z_\alpha$. Then $\tilde{X} := X$ is a solution. So it enough to consider the case where $X$ is irreducible and non compact. Thanks to [ST71, Theorem 8.3], for each $z \in \partial Z_\alpha$ there exists an open set $U_z$ and an unique closed analytic set $X_z$ in $Z_\alpha \cup U_z$ of pure dimension n such that $X_z \cap Z_\alpha = X$. Choose a finite set of points $z_1, \dots, z_N$ and open sets $U'_i \subset \subset U_i := U_{z_i}$ such that the union of the $U'_i$ covers the compact set $\partial Z_\alpha$. Let $\Omega := Z_\alpha \cup (\bigcup_{i=1}^N U'_i)$ and put

$$X_1 := (X \cup (\bigcup_{i=1}^N X_{z_i})) \cap \Omega.$$

Let us show that $X_1$ is a closed analytic subset in $\Omega$. Consider $z \in \Omega$. If $z$ is in $Z_\alpha$ we have $X_1 = X$ in a neighbourhood of $z$ and the assertion is clear. If not, either $z$ is not in any $\partial U'_i$ and $X_1$ is the union of the $X_{z_i}$ in a neighbourhood of $z$ and the assertion is clear, or $z$ is in $\partial U'_{j_1}, \dots, \partial U'_{j_k}$ for $j_1, \dots, j_k$ in $[1, N]$. As the set $X_{j_h}$ is closed and analytic in $Z_\alpha \cup U_{j_h}$, then $X_1$ is again the union of the $X_{z_i}$ near $z$ in $\Omega$, and the assertion is proved.

So in this situation there exists a real positive $\beta < \alpha$ such that $Z_\beta \subset \Omega$. Let $X_2$ be the irreducible component of $X_1 \cap Z_\beta$ which contains $X$; then $X_2$ is a closed irreducible analytic subset of $Z_\beta$ such that $X_2 \cap Z_\alpha = X$.

Now let

$$\gamma := \inf\{\beta \le \alpha / \exists X_\beta \text{ irreducible } n\text{-cycle of } Z_\beta \text{ such that } X_\beta \cap Z_\alpha = X\}.$$

Then what we obtained above shows that we have $\gamma < \alpha$, and, applying the same arguments to the cycle $X_\gamma$ defined on $Z_\gamma$ via the cover of $Z_\gamma$ by the $Z_\beta, \beta > \gamma$ in which we already built an irreducible $n$-cycle $X_\beta$ extending $X$, we conclude that $\gamma = 0$ and that there exists an (unique) irreducible $n$-cycle $\tilde{X}$ in $Z$ extending $X$. $\square$

### 4.B. Some consequences.

We shall give first some easy consequences of the fact that the reduced complex space Z is strongly n-concave.

**Proposition 4.3.** Let $n \ge 2$ be an integer and let Z be a reduced complex space which is strongly (n-2)-concave. Then the natural map $j: C_n^f(Z) \to C_n^{\text{loc}}(Z)$ is a homeomorphism. Moreover, for each $\alpha \in ]0, 1[$ the restriction map

$$\text{res}_\alpha : C_n^f(Z) \to C_n^f(Z_\alpha)$$

is well defined and is also a homeomorphism.

*Proof.* Let us prove first that any $n$-cycle $X$ in $Z$ has finitely many irreducible components. As this implies the same result for each $Z_\alpha$ for $\alpha \in ]0, 1[$, this will imply the fact that the restriction map $\text{res}_\alpha$ is well defined, and then bijective as a consequence of Theorem 4.2.

As the family of irreducible components of a $n$-cycle is locally finite, only finitely many irreducible components of $X$ can meet the compact set $K := \varphi^{-1}([1, 2])$. But we have seen in the remark following the previous theorem that any irreducible component of $X$ must meet $K$. So $X$ is a finite type cycle.

To show the continuity of $\text{res}_\alpha^{-1}$ it is then enough to prove that $j$ is a homeomorphism which is an easy consequence of the lemma below.
**Lemma 4.4.** Let $Z$ be a reduced complex space and let $(X_\nu)_{\nu \ge 0}$ be a sequence of $n$-cycles in $Z$ converging in $C_n^{\text{loc}}(Z)$ to a cycle $Y$. Assume that there exists a relatively compact open set $\Omega$ in $Z$ such that any irreducible component of each $X_\nu$ and of $Y$ meets $\Omega$. Then all these cycles are of finite type and the sequence converges to $Y$ in $C_n^f(Z)$.

*Proof of Lemma 4.4.* First if $Y = \emptyset$, for $\nu \gg 1$ the cycle $X_\nu$ will be disjoint from the compact set $K := \bar{\Omega}$, and this implies that $X_\nu$ is the empty cycle. So the conclusion holds in this case. If $Y$ is not empty, let $U$ be a relatively compact open set in $Z$ meeting each irreducible component of $Y$. We have to show, by definition of the topology of $C_n^f(Z)$, that for $\nu \gg 1$ each irreducible component of $X_\nu$ meets $U$. If it is not the case, passing to a subsequence, we may assume that for each $\nu$ there exists an irreducible component $\Gamma_\nu$ of $X_\nu$ disjoint from $U$. As, passing again to a subsequence, we may assume that the sequence $(\Gamma_\nu)$ converges in $C_n^{\text{loc}}(Z)$ to a cycle $\Gamma$, we shall have $|\Gamma| \subset |Y|$ and $|\Gamma| \cap U = \emptyset$. To conclude, it is enough to show that $\Gamma$ is not the empty cycle, as any irreducible component of $\Gamma$ is also an irreducible component of $Y$ and then meets $U$ by hypothesis. As each $\Gamma_\nu$ is not empty, it has to meet $K = \bar{\Omega}$. This implies that $\Gamma$ also meets $K$ and so is not empty. This contradicts our assumption. $\square$

*End of the proof of Proposition 4.3.* We have proved that $j$, and then also each $j_\alpha : C_n^f(Z_\alpha) \to C_n^{\text{loc}}(Z_\alpha)$ for $\alpha \in ]0, 1[$, is a holomorphic homeomorphism. To conclude the proof we have to show the continuity of $\text{res}_\alpha^{-1}$, and this reduces to prove that if the sequence $(X_\nu)$ of $C_n^{\text{loc}}(Z)$ is such that the sequence $(X_\nu \cap Z_\alpha)$ converges in $C_n^{\text{loc}}(Z_\alpha)$, then it converges in $C_n^{\text{loc}}(Z)$. Let $Y_\alpha \in C_n^{\text{loc}}(Z_\alpha)$ be the limit of this sequence in $C_n^{\text{loc}}(Z_\alpha)$ and let $Y \in C_n^{\text{loc}}(Z)$ be the cycle extending it. Let $A$ be the set of $\beta \in ]0, \alpha[$ such that the sequence $(X_\nu \cap Z_\beta)$ converges in $C_n^{\text{loc}}(Z_\beta)$ to $Y \cap Z_\beta$. Then $\alpha$ is in $A$ so $A$ is not empty. Put $\gamma := \inf A$. Theorem 3.15 implies that $\gamma = 0$ and we obtain also the convergence in any $C_n^{\text{loc}}(Z_\beta)$, for any $\beta > 0$; this gives the convergence in $C_n^{\text{loc}}(Z)$, as, by definition, a $n$-scale on $Z$ is also a $n$-scale on $Z_\beta$ for $\beta > 0$ small enough. $\square$

### 4.C. An analytic extension criterion.

The aim of this paragraph is to prove the following analytic extension result.

**Theorem 4.5.** Let $Z$ be a complex space and $n$ an integer. Consider a $f$-continuous$^{11}$ family $(X_s)_{s \in S}$ of $n$-cycles of finite type in $Z$ parametrized by a reduced complex space $S$. Fix a point $s_0$ in $S$ and assume that the open set $Z'$ in $Z$ meets all irreducible components of $X_{s_0}$ and such that the family of cycles $(X_s \cap Z')_{s \in S}$ is analytic at $s_0$. Then there exists an open neighbourhood $S_0$ of $s_0$ in $S$ such that the family $(X_s)_{s \in S_0}$ is $f$-analytic.

The hypotheses translated in terms of classifying maps means that we have a continuous map $\varphi : S \to C_n^f(Z)$ such that the composed map $r \circ \varphi$ is holomorphic at $s_0$, where $r : C_n^f(Z) \to C_n^{\text{loc}}(Z')$ is the restriction map.

Then the theorem says that there exists an open neighbourhood $S_0$ of $s_0$ in $S$ such that the map $\varphi$ is holomorphic on $S_0$. Note that, as $r$ is holomorphic$^{12}$, the hypothesis that $\varphi$ is holomorphic at $s_0$ is a necessary condition.

This result is not true in general if we take for $S$ a non smooth Banach analytic set which is not of finite dimension (locally). The reader may find a counter-exemple with an isolated singularity in [BM, Chapter V].

The key point for the proof of the previous theorem is the following analytic extension result.

<footnote>$^{11}$ $\uparrow$ This means that we have a continuous family of finite type $n$-cycles such that its graph is quasi-proper over $S$. This is equivalent to the continuity of the classifying map $\varphi : S \to C_n^f(Z)$ of this family.</footnote>

<footnote>$^{12}$ $\uparrow$ in the sense that for any holomorphic map $\psi : T \to C_n^f(Z)$ of a reduced complex space $T$ the composed map $r \circ \psi$ is holomorphic.</footnote>
**Proposition 4.6.** Let $S$ a reduced complex space and let $\emptyset \neq U_1 \subset U_2$ be two polydiscs in $\mathbb{C}^n$. Let $f: S \times U_2 \to \mathbb{C}$ a continuous function, holomorphic on $\{s\} \times U_2$ for each $s \in S$. Assume moreover that the restriction of $f$ to $S \times U_1$ is holomorphic. Then $f$ is holomorphic on $S \times U_2$.

*Proof of Proposition 4.6.* First consider the case $S$ smooth. As the question is local on $S$ it is enough to consider the case where $S$ is an open set in $\mathbb{C}^m$. Fix an open relatively compact polydisc $P$ in $S$. The function $f$ defines a map $F: U_2 \to \mathcal{C}^0(\bar{P}, \mathbb{C})$, where we write down $\mathcal{C}^0(\bar{P}, \mathbb{C})$ the Banach space of continuous functions on $\bar{P}$, via the formula $F(t)[s] = f(s, t)$ for $t \in U_2$ and $s \in \bar{P}$. First we shall show that the map $F$ is holomorphic.

Let $U \subset \subset U_2$ be a polydisc. For any fix $s \in S$ we have

$$ \frac{\partial f}{\partial t_i}(s, t) = \frac{1}{(2i\pi)^n} \cdot \int_{\partial\partial U} f(s, \tau) \cdot \frac{d\tau_1 \wedge \cdots \wedge d\tau_n}{(\tau_1 - t_1) \cdots (\tau_i - t_i)^2 \cdots (\tau_n - t_n)} \quad \forall t \in U \quad \forall i \in [1, n]. $$

where $t := (t_1, \dots, t_n)$ are coordinates on $\mathbb{C}^n$. This Cauchy formula shows that $F$ is $\mathbb{C}$-differentiable and its differential at the point $t \in U$ is given by

$$ h \mapsto \sum_{i=1}^{n} F_i(t).h_i, \quad h \in \mathbb{C}^n, $$

where $F_i$ is associated to the function

$$ (s, t) \mapsto \frac{\partial f}{\partial t_i}(s, t) \quad i \in [1, n] $$

which is holomorphic for each fixed $s \in S$ thanks to the Cauchy formula above.

Let $H(\bar{P}, \mathbb{C})$ the closed subspace of $\mathcal{C}^0(\bar{P}, \mathbb{C})$ of functions which are holomorphic on $P$. Our hypothesis implies that the restriction of $F$ to the non empty open set $U_1$ takes its values in this subspace. Let us show that this is also true on $U_2$. Assume that there exists $t_0 \in U_2$ such that $F(t_0)$ is not in $H(\bar{P}, \mathbb{C})$. Thanks to the Hahn-Banach theorem we can find a continuous linear form $\lambda$ on $\mathcal{C}^0(\bar{P}, \mathbb{C})$, vanishing on $H(\bar{P}, \mathbb{C})$, and such that $\lambda(F(t_0)) \neq 0$. But the function $t \mapsto \lambda(F(t))$ is holomorphic on $U_2$ and vanishes on $U_1$; this contradicts $\lambda(F(t_0)) \neq 0$. So $F$ takes values in $H(\bar{P}, \mathbb{C})$ and $f$ is holomorphic on $S \times U_2$ when $S$ is a complex manifold.

The case where $S$ is a weakly normal complex space follows immediately.

When $S$ is a general reduced complex space, the function $f$ is meromorphic and continuous on $S \times U_2$ and holomorphic on $S \times U_1$. So the closed analytic subset $Y \subset S \times U_2$ along which $f$ may not be holomorphic has no interior point in each $\{s\} \times U_2$. The analytic extension criterion of [BM14, Chapter IV, Criterion 3.1.7] allows to conclude. $\square$

*Proof of Theorem 4.5.* Let $|G| \subset S \times Z$ be the graph of the $f$-continuous family $(X_s)_{s \in S}$ and let $A$ be the set of points in $(\sigma, \zeta) \in |G|$ admitting an open neighbourhood $S_\sigma \times Z_\zeta$ in $S \times Z$ such that the family of cycles $(X_s \cap Z_\zeta)_{s \in S_\sigma}$ is analytic. Remark that, thanks to our hypothesis, the open set $A$ in $|G|$ meets every irreducible component of $\{s_0\} \times |X_{s_0}|$.

Assume to begin that there exists a smooth point $z_0$ of $|X_{s_0}|$ in the boundary of $A \cap (\{s_0\} \times |X_{s_0}|)$. Choose a $n$-scale $E := (U, B, j)$ on $Z$ which is adapted to $X_{s_0}$ and satisfying :

$$ \deg_E(|X_{s_0}|) = 1, \quad j_*(X_{s_0}) = k.(U \times \{0\}) \\ z_0 \in j^{-1}(U \times B), \quad j(z_0) := (t_0, 0). $$

It is clear that such a $n$-scale exists as $z_0$ is a smooth point in $|X_{s_0}|$. Let $S_1$ be a sufficiently small open neighbourhood of $s_0$ in $S$ and let $f: S_1 \times U \to \text{Sym}^k(B)$ be the (continuous) classifying map for the family $(X_s)_{s \in S_1}$ in the scale $E$. As $j^{-1}(U \times \{0\})$ meets $A$, there exists a non empty polydisc $U_2 \subset U$ such that $U_2 \times \{0\}$ is contained in $A$. Then we may apply Proposition 4.6 to each scalar
component of $f$ in order to obtain that $f$ is holomorphic on $S_1 \times U$. Moreover, as the same argument applies to any linear projection of $U \times B$ to $U$ near enough the vertical one; this implies that $f$ is an isotropic map, up to shrinking slightly $U$. This contradicts the fact that the point $(s_0, z_0)$ is in the boundary of the open set $A \cap (\{s_0\} \times |X_{s_0}|)$ of $|X_{s_0}|$.

If the boundary of $A \cap (\{s_0\} \times |X_{s_0}|)$ is contained in the singular set of $|X_{s_0}|$, we may apply the analytic extension criterion of [BM14, Chapter IV, Criterion 3.1.7], and we obtain directly that $A$ contains $|X_{s_0}|$. So in any case the family of cycles $(X_s)_{s \in S}$ is analytic at $s_0$. As the graph $|G|$ is, by assumption, quasi-proper on $S$, it is enough to use the next proposition (which is proved in [Bar15, Proposition 2.2.3]) to conclude.

**Proposition 4.7.** Let $Z$ and $S$ be reduced complex spaces and let $(X_s)_{s \in S}$ be a $f$-continuous family of $n$-cycles in $Z$. Assume that this family is analytic in $s_0 \in S$. Then there exists an open neighbourhood $S'$ of $s_0$ in $S$ such that the family $(X_s)_{s \in S'}$ is a $f$-analytic family of $n$-cycles in $Z$.

### 4.D. Proof of Theorem 1.2 and its corollary

We shall begin by a lemma which will give the case where the $n$-cycle $X_0$ is compact.

**Lemma 4.8.** Let $Z$ be a strongly $(n-2)$-concave reduced complex space. Then $C_n(Z)$ is open in $C_n^f(Z)$.

*Proof.* Let $X_0$ be a compact cycle in $Z$. There exists $\alpha \in ]0, 1[$ such that $X_0$ is contained in $Z_\alpha = \{x \in Z / \varphi(x) > \alpha\}$. So $X_0$ does not meet the compact set $\varphi^{-1}(\{\alpha\})$. This is an open condition in $C_n^f(Z)$. And as any irreducible $n$-dimensional analytic subset in $Z$ has to meet $K$, if it does not meet $\varphi^{-1}(\{\alpha\})$ it is contained in $Z_\alpha$ (by connectedness). Then any $X \in C_n^f(Z)$ which is near enough $X_0$ is contained in $Z_\alpha$ so is compact. $\square$

Note that under the hypothesis of the previous lemma, $C_n(Z)$ is not closed in $C_n^f(Z)$ in general, as one can see taking $Z := \mathbb{P}_N \setminus \{0\}$ and considering the set of hyperplanes in $\mathbb{P}_N$.

As we already know from [Bar75] that $C_n(Z)$ is a reduced complex space, Theorem 1.2 and its corollary are proved near a compact $n$-cycle in $Z$.

*The case where $X_0$ is not compact.* Of course we are in the case where $Z$ is not compact. Fix $X_0$ a non compact $n$-cycle in $C_n^f(Z)$ and choose an $\alpha \in ]0, 1[$ which is not a critical value of $\varphi$ and of the restriction of $\varphi$ to $|X_0|$; this is possible thanks to Sard's Lemma 3.1 and the fact that $\varphi(|X_0|)$ contains $]0, 1[$.

Consider now the reduced complex space $\Xi$ constructed in Theorem 3.15. It is an open neighbourhood of $X_0 \cap \Delta''$ in $C_n^f(\Delta'')$ and we may assume that $\Delta'' := Z_\beta$ for some $\beta < \alpha$ very near $\alpha$. Due to Proposition 4.3 it is homeomorphic to an open neighbourhood $\mathcal{V}$ of $X_0$ in $C_n^f(Z)$. So the restriction map

$$res_0: \mathcal{V} \to \Xi$$

is holomorphic, bijective and is a homeomorphism. Now the continuity of $res_0^{-1}$ and the finiteness of $\Xi$ allow to apply Theorem 4.5 because we already know that the tautological family of cycles parametrized by $\Xi$ is $f$-analytic on the open set $\Delta''$ which is an open set which meets any irreducible component of each cycle in this family (because $K \subset Z_\beta$). Then $res_0^{-1}$ is holomorphic and so $res_0$ is an isomorphism of Banach analytic sets. $\square$
### 4.E. A compactness criterion for the connected components of the reduced complex space $C_n^f(Z)$ when $Z$ is strongly $(n-2)$-concave

For a reduced complex space $Z$ which is compact, the compactness of the connected components of $C_n(Z)$ is a consequence of the existence of a $\mathscr{C}^1$ $2n$-differential form on $Z$ which is $d$-closed and such that its $(n, n)$ part is positive definite in the Lelong sense. Indeed, this gives that the volume (computed with this $(n, n)$ part) of the $n$-cycles is constant on connected components. The result follows then from Bishop's theorem [Bis64] (see [BM14, Chapter IV] for details).

In the case of a non compact strongly $(n-2)$-concave reduced complex space $Z$ we have the following analogous result :

**Proposition 4.9.** Let $Z$ be a reduced complex space which is strongly $(n-2)$-concave. Assume that there exists on $Z$ a $\mathscr{C}^1$ $2n$-differential form $\omega$ which is $d$-closed with compact support and such that its $(n, n)$ part is positive definite in the Lelong sense in a neighbourhood of $K := \varphi^{-1}([1, 2])$, and everywhere non negative in the Lelong sense. Then the connected components of $C_n^f(Z)$ are compact.

*Proof.* For $\alpha < 1$ near enough to 1 and for any continuous hermitian metric $h$ on $Z$ there exists a constant $C$ such that the following inequality holds:

$$ vol_h(X \cap Z_\alpha) \leq C \cdot \int_X \omega \quad \text{for any cycle } X \in C_n^f(Z). $$

As the function $X \mapsto \int_X \omega$ is locally constant on $C_n^f(Z)$ because $d\omega = 0$ (the direct image of $\omega$ as a current is $d$-closed, so locally constant at smooth points of $C_n^f(Z)$, and this current is a continuous function on $C_n^f(Z)$ thanks to Proposition IV 2.3.1 of loc. cit.), we have a uniform bound for the volume of $X \cap Z_\alpha$ for $X$ in a given connected component of $C_n^f(Z)$. This implies that the closure of the image of this connected component in $C_n^f(Z_\alpha)$ is compact, thanks to Bishop's theorem (see [BM14, Chapter IV, Theorem 2.7.20]). But the restriction map $C_n^f(Z) \to C_n^f(Z_\alpha)$ is a homeomorphism by Proposition 4.3, so the image of a connected component is closed and then compact. $\square$

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