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

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.


Introduction
The aim of this article is to show that in a reduced strongly (n − 2)-concave 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 f n (Z) of finite type cycles with its natural topology. We obtain the following results.
Theorem 1.1. Let n ≥ 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 C 2 exhaustion function ϕ : Z →]0, 2] which is strongly (n − 2)-convex outside the compact set K := ϕ −1 ([1, 2]). For any α ∈]0, 1[ and any ncycle X 0 in an open neighbourhood of the compact set ϕ −1 ([α, 2]) there exists β ∈]0, α[ such that, if Z β := {z ∈ Z / ϕ(z) > β}, the cycle X 0 extends in a unique way to the open set Z β and admits an open neighbourhood U in the space C f n (Z β ) such that the ringed space defined by U and the sheaf of holomorphic functions on U is a (reduced) complex space locally of finite dimension.
Recall that a holomorphic function h : U → C on an open set in C f n (Z β ) is a continuous function on U such that for any holomorphic map f : S → 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 • f is holomorphic.
Theorem 1.2. Consider the same situation as in the previous theorem, and let now X 0 ∈ C f n (Z) be a finite type n-cycle in Z. Then there exists β ∈]0, 1[ and open neighbourhoods V and U respectively of X 0 in C f n (Z) and of X 0 ∩ Z β in C f n (Z β ) such that the restriction map r : V → 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 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]). When X is a n-cycle in Z, the n-scale E is adapted to X when |X| ∩ j −1 (Ū × ∂B) = ∅.
Note that when E is a n-scale adapted to a n-cycle X in Z, the projection of U ×B on U restricted to j * (X ∩j −1 (U ×B)) gives a finite proper map of degree k ≥ 0 and the fibers of this map are classified by a holomorphic map f : U → 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∈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 ∈ 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 ∈ S 0 the scale E is adapted to X s . ii) Assume that k := deg E (X s 0 ). Then for each s ∈ S 0 we have deg E (X s ) = k.
iii) There exists a holomorphic map f : S 0 × U → Sym k (B) such that for each s ∈ S 0 the restriction of f to {s 0 } × 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 which is a closed analytic subset in S × 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∈S and when the projection of its graph pr : |G| → S is quasi-proper 3 we shall say that (X s ) s∈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∈S and when the projection of its graph pr : |G| → 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.
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 f n (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 (reduced complex spaces) → (sets) S → {f-analytic family of n-cycles in Z 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 ∈ X be a point in X where the supremum of the restriction of the exhaustion function ϕ to X is obtained. Then the point x 0 is in the compact set K = ϕ −1 ([1, 2]) because the Levi form of ϕ 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 f n (Z) → C loc n (Z) is an isomorphism of ringed spaces and C f n (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.

Hartogs figures 2.A. Banachization
For the analytic extension via Hartogs figures, the use of the Banach spaces H(Ū , C) of continuous functions, holomorphic inside on a compact polydiscŪ is not well adapted. We shall use the Banach space B(U, C) of bounded holomorphic functions on U with the "sup" norm on U . Of course, H(Ū , C) is a closed Banach subspace in B(U, C).
Proposition 2.1. Let U be a relatively compact polydisc in C n and let S be a Banach analytic set. Let F : S → B(U, C) be a holomorphic map. Then the corresponding function f : for (s, t) in S × U is holomorphic (and locally on S uniformely bounded on U ).
Conversely, if we have a holomorphic function f : S × U → C and an open polydisc U ⊂⊂ U , the associated map F : Proof. The evaluation function ev : B(U, C) × U → 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 × id U and ev. So it is holomorphic.
The converse is consequence of the linear isometric inclusion of H(Ū , C) in B(U , C) as F : S → H(Ū , C) is holomorphic as soon as f is holomorphic (see [Bar75] or [BM]). 4 ↑ The set C loc n (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 f n (Z) → C loc n (Z) is continuous but is not a homeomorphism onto its image. See [Bar15] for the comparison between compact sets in these two spaces.

2.B. n-Hartogs figure on a complex space
For ε > 0 small enough, we define For a polydisc of radius R in C m we shall denote by P ε the polydisc with same center and radius R − ε for 0 < ε < R.
Definition 2.2. Let n ≥ 2 and p ≥ 1 two integers and let ∆ ⊂⊂ ∆ be two open sets in a reduced complex space Z. We shall call H := (M, M, B, j) a n-Hartogs figure in Z relative to the boundary of ∆, the following data • a relatively compact polydisc B in C p such that there exists α ∈ (R * + ) 2 and a relatively compact polydisc V in C n−2 with the following conditions: i) The map j induces a closed embedding of M in M(α) × V × B.
ii) The map j induces a closed embedding of Definition 2.3. Let n ≥ 2 and p ≥ 1 two integers and let ∆ ⊂⊂ ∆ be two open sets in a reduced complex space Z. Let H = (M, M, B, j) be a n-Hartogs figure in Z relative to the boundary of ∆ and let X 0 be a n-cycle in ∆. We shall say that H is adapted to X 0 when the following condition is satisfied : Remarks.
1. Let H be a n-Hartogs figure in Z relative to the boundary of ∆. If the open set ∆ 1 ⊂⊂ ∆ has a boundary ∂∆ 1 near enough to ∂∆, then H is again a n-Hartogs figure relative to the boundary of ∆ 1 . For instance, if ∆ := {ϕ > 0} where ϕ is a continuous proper function on Z, we may choose ∆ 1 := {ϕ > ε} for ε > 0 small enough.
2. Note that the n-scale E H := (M (α) × V, B, j) on ∆ associated to H is adapted to X 0 as soon as the n-Hartogs figure H is adapted to X 0 .
3. IfX 0 is a n-cycle in ∆ such that its restriction to ∆ is equal to X 0 , the n-scale EH := (M(α) × V, B, j) on ∆ is adapted toX 0 if and only if the n-Hartogs figure H is adapted to X 0 .
Note that the n-scale EH is not a n-scale on ∆ although the subsetM(α) ×V × ∂B is contained in ∆.
Definition 2.4. In the situation above we define, for ε > 0 small enough, the n-Hartogs figure H ε on ∆ as follows : where we use the notations It is obvious to see that when H is a n-Hartogs figure relative to the boundary of ∆, then H ε is again a n-Hartogs figure relative to the boundary of ∆ for all ε > 0 small enough.
Moreover, if H is adapted to the n-cycle X 0 of ∆, the same is true for H ε for all ε > 0 small enough.
Lemma 2.5. Let V be a relatively compact open polydisc in C q . The restriction map is a linear isometry of Banach spaces.
Note that the restriction map induces also an isometry for all ε > 0 small enough.
Proof. Let f (v, t 1 , t 2 ) := m∈Z a m (v, t 1 ).t m 2 the Laurent expansion of the holomorphic function f : The holomorphic functions a m , m ∈ Z on the product of V by the disc {|t 1 | < α 1 } are given by the formula As the holomorphy of f on M P (α) × V implies that a m ≡ 0 for each negative m on the open set {|t 1 − α 1 /2| < α 1 /4} × V , we conclude that the functions a m are identically zero for m < 0 and so f is holomorphic on M(α) × V . This shows that the restriction map res is bijective (and also it is linear continuous) between the two Fréchet spaces O(M(α) × V ) and O(M (α) × V ); so it is an isomorphism of Fréchet spaces.
Let us show that if f is in B(M(α)×V, C), then res(f ), which belongs to the space B(M (α)×V, C), has the same "sup" norm. For this purpose fix ε > 0 small enough. AsM(α) ε ×V ε is a compact polydisc in M(α)×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 (α) ε ×V ε , the desired equality follows. Conversely, if g is in B(M (α) × V, C), its analytic extension f to M(α) × V will be bounded on the boundary of M(α) ε × V ε by the sup of g on M C (α) ε × V ε . So we obtain the equality of the "sup" norms for g and f respectively on M (α) × V and M(α) × V .
The Banach analytic set B(U, Sym k (C p )). Recall that if p ≥ 1 and k ≥ 1 are integers, there exists a closed embedding (in fact given by a polynomial map) of Sym k (C p ) := (C p ) k S k into the vector space E(k) := k h=1 S h (C p ) given by the elementary tensorial symetric functions 5 . If U is an open relatively compact polydisc in C n , the subset B(U, Sym k (C p )) is closed and Banach analytic in the Banach space B(U, E(k)). Indeed, if Q : E(k) → C N is a polynomial map such that Q −1 (0) = Sym k (C p ), then the holomorphic map Nevertheless Remark. The obvious map H(Ū , E(k)) → B(U, E(k)) is a closed (linear) isometry and induces a holomorphic inclusion map and for all ε > 0 the restriction is a (linear and continuous) compact map which induces a holomorphic restriction map This remark will allow us to use Lemma 2.5 with Banach analytic sets like H(Ū , Sym k (B)).
Notations. Let k ∈ N and let U ⊂⊂ U ⊂⊂ C n and B ⊂⊂ C p be polydiscs. We shall note Σ U,U (k) the Banach analytic set classifying the couples of an element in H(Ū , Sym k (B)) with its isotropy data onŪ . Recall that for a holomorphic map f : U → Sym k (B) the isotropy data on the polydisc U is the map It corresponds to the collection of holomorphic maps for all i ∈ [1, min(n, p)] and m ∈ [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 , . . . , f k , by the formula These maps are always holomorphic on all U and determine the trace map for the projection π : X f → U of the holomorphic differential forms on U × B. The subset is defined as the graph of the map f → T (f ) := i,m T i m (f ) which is not holomorphic in general. Nevertheless it is a Banach analytic subset and its natural projection 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 × U → Sym k (B) will give an analytic family of cycles in U × B parametrized by S if and only if the corresponding isotropy data (given by the maps T i m (f /S ) on S × U ) are holomorphic on S × 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.
Moreover, this holomorphic map induces a holomorphic map, again for ε > 0 small enough which factorizes the restriction map through the restriction Proof. Lemma 2.5 gives that the map is an isometry of Banach spaces. It is clear that its inverse sends B(M(α) × V, Sym k (C p )) into the Banach analytic subset B(M (α)×V, Sym k (C p )) and that if the map f ∈ B(M(α)×V, Sym k (C p )) takes values in 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 which is isotropic on the product of S with any relatively compact subset in the open set M (α) ε/3 × V ε/3 will have an analytic extension which will be isotropic on any relatively compact open set in M(α) ε/3 × V ε/3 . So it will be isotropic on the closure of the open set M(α) 2ε/3 × V 2ε/3 . Proposition 2.7. Let n ≥ 2 and p ≥ 1 be integers and let U 1 × B 1 the product of two polydiscs with centers 0 respectively in C n and C p . Denote by (t 1 , · · · , t n , x 1 , · · · , x p ) coordinates on U 1 × B 1 . Let ϕ be a real valued function of class C 2 on U 1 × B 1 , such that where the real numbers ρ 2 , σ 1 , · · · , σ p are positive (so ϕ is (n − 2)-convex near (0, 0) and dϕ 0,0 = 0). Let ∆ be the open set {ϕ > 0} in U 1 × B 1 and let ∆ be an open neighbourhood of the compact set ∆ in C n+p . Let X 0 be a closed analytic subset of pure dimension n in ∆ such that each irreducible component of X 0 meets ∆ and such that Then there exists α ∈ (R * + ) 2 and polydiscs V and B ⊂⊂ B 1 with centers 0 respectively in C n−2 and C p such that the following conditions are satisfied : Proof. Choose the polydisc B ⊂⊂ B 1 small enough in order that we have This is possible as we have |X 0 | ∩ {t 1 = · · · = t n = 0} ⊂ {0} and as ϕ is positive on a small enough punctured neighbourhood of the origin in the p-plane {t 1 = · · · = t n = 0} × C p . So we shall have for any small enough open neighbourhood W of the origin in C n . A immediate consequence is that Conditions 1, 3 and 4 will be satisfied as soon as α and V are small enough.

Remarks.
1. We only used Condition ( * ) for X 0 and Inequality ( * * ) for ϕ in a neighbourhood of the origin in the proof above.
2. Sufficient conditions on ϕ ∈ C 2 to satisfy (@@) are: i) The origin is not a critical point of ϕ.
ii) The Levi form of ϕ at 0 has, at most, (n − 2) non positive eigenvalues in the complex tangent hyperplane to the real hypersurface {ϕ(z) = 0}; the existence of real function ϕ ∈ C 2 such that ∆ = {ϕ > 0} and satisfying these two conditions is equivalent to the fact that the open set ∆ has a strongly (n − 2)-concave smooth boundary near the origin (see Definition 2.10 given below). Indeed, if ϕ 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 {ϕ(z) = 0}, its order 2 Taylor expansion at the origin is written, in suitable local holomorphic coordinates (τ, x) where Q is a holomorphic homogeneous degree 2 polynomial and where the real numbers ρ 2 and σ j , j ∈ [1, p] are positive. Define new local holomorphic coordinates Then we obtain (@@).
3. Condition ( * ) implies that X 0 has no local irreducible component at 0 contained in the hyperplane {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 ϕ 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 τ 1 + Q(τ, x) = 0 locally defined near 0 for ϕ given. Then, as soon as the restriction ϕ |X 0 has not 0 as a critical point, Condition ( * ) could be realized when the Levi form of ϕ has at most (n − 2) non positive eigenvalues on the complex tangent hyperplane at the origin of the hypersurface {ϕ = 0} . 4. One may easily see that under our hypothesis, the cycle X 0 meets the open set ∆ when it contains 0. Indeed, the analytic subset is nonempty, has dimension at least 1 and meets the complement of ∆ only at the origin.
Of course, assuming that 0 ∈ X 0 , the proposition shows that, in fact, X 0 contains a branched 5. In the situation of Proposition 2.7, for any continuous family (X s ) s∈S of n-cycles in ∆ parametrized by a Banach analytic set S such that X s 0 = X 0 ∩ ∆, there exists an open neighbourhood S of s 0 in S, such that for each s ∈ S Condition 4 remains true after analytic extension of the cycles (see Proposition 2.6), because, thanks to Condition 3, M(α) ×V × ∂B is a compact subset in ∆.

2.C. q-concave open sets.
Definition 2.8. Let ϕ : U → R be real valued C 2 function on an open set U in C N . We shall say that ϕ 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 ϕ : Z → R a real valued C 2 function on a reduced complex space Z. We shall say that ϕ is strongly q-convex if locally near each point of Z it can be induced by a 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 ϕ is strongly p.s.h.
Definition 2.10. Let Z be a reduced complex space and let ∆ be a relatively compact open set in Z. We shall say that ∆ has a smooth C 2 boundary when for each point z in ∂∆ there exists a local holomorphic embedding j : W → 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 C 2 boundary in U such that We shall say that the open set ∆ ⊂ Z with smooth C 2 boundary is strongly q-concave at a point z ∈ ∂∆ if, in some local holomorphic embedding j : 2. The restriction of the Levi form at j(z) of ϕ to the complex hyperplane tangent at j(z) to the real hypersurface {ϕ(x) = ϕ(j(z))} in U has at most q non positive eigenvalues.
We shall say that ∆ is strongly q-concave if ∆ is strongly q-concave near each point in ∂∆.
Remark. Assume that Z is of pure dimension q + p. If the defining function ϕ of ∆ satisfies Conditions 1 and 2 above, we can compose ϕ with a real strictly increasing (non critical) convex C 2 function (this does not change the level sets {ϕ = constant}), in order that c • ϕ is C 2 strongly q-convex (and non critical) near z. Conversely, if ϕ is a real valued C 2 function which is strongly q-convex and not critical near at a point z ∈ Z, the open set {ϕ(x) > ϕ(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 ≥ 2 and p ≥ 1 be integers, let Z be a reduced complex space of pure dimension n + p and let ∆ := {ϕ > 0} be an open set with C 2 smooth boundary in Z. Let X 0 be a n-cycle in an open neighbourhood of a point z ∈ ∂∆ such that the function ϕ |X 0 is not critical at z.
Assume that ∆ is strongly (n − 2)-concave near z ; then there exists a n-Hartogs figure H := (M, M, B, j) relative to the boundary of ∆, adapted to X 0 , and such that the point z lies in 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 C n+p , with p ≥ p an integer. As we may choose the function ϕ strongly (n − 2)-convex such that dϕ z = 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 ∈ X 0 , as we assumed that ϕ |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 H will be zero.

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 C n+p a C 2 function ϕ : U → R and a non critical zero z 0 on ϕ. So ϕ(z 0 ) = 0 and dϕ z 0 = 0. Let D := {z ∈ U / ϕ(z) < 0} and let H be the complex hyperplane tangent at z 0 to the real hypersurface {ϕ = 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 ∈ ∂D if the restriction to H of the Levi form of ϕ 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 ∈ U / − ϕ(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 −ϕ at z 0 has at most q non positive eigenvalues.
If the signature of the restriction to H of the Levi form of ϕ 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 −ϕ is strongly (p − 1)-convex at the point z 0 .
In order that a C 2 exhaustion function ϕ : Z →]0, 2] on a reduced complex space Z gives relatively compact q-concave subsets Z α := {ϕ(z) > α} for each α ∈]0, 1[ which is not critical for ϕ, we see that it is enough that the Levi form of ϕ at each point in ϕ −1 (]0, 1[) has at most q non positive eigenvalues. That is to say that ϕ 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 C 2 -exhaustion ϕ : Z →]0, 2] which is (n − 2)-strongly convex on the open set ϕ −1 (]0, 1[) . So we need to assume that n ≥ 2.

2.E. Boxed Hartogs figures
Definition 2.12. Let n ≥ 2 and p ≥ 1 be integers and let ∆ ⊂⊂ ∆ be two open sets in a reduced complex space Z. Let H = (M, M, B, j) and H := (H , M , B, j) be two n-Hartogs figures in Z relative to the boundary of ∆ given by the same (local) embedding j and having the same polydisc B ⊂⊂ C p . We shall say that these two n-Hartogs figures are boxed when we can choose α, α , V, V in Definition 2.2 in order to have For instance, if ε > 0 is small enough, the n-Hartogs figures (H, H ε ) are boxed (see Definition 2.4).
Proposition 2.13. Let n ≥ 2 and p ≥ 1 be integers, let Z be a reduced complex space of pure dimension n + p and let ∆ ⊂⊂ Z be an open set with smooth C 2 boundary in Z which is strongly (n − 2)-concave. Assume that ∆ := {ϕ > 0} and let X 0 be a n-cycle in an open neighbourhood ∆ of the compact set∆, such that any irreducible component of X 0 meets ∆. Then there exists a finite family of boxed n-Hartogs figures (H a , H a ) a∈A relative to the boundary of ∆, such that the following conditions hold: 1. The open sets M a for a ∈ A cover the boundary ∂∆. Remark. Let X 0 be any n-cycle in ∆ . Choosing the open set ∆ small enough around the compact set∆, we can assume that the cycle X 0 has only finitely many irreducible components in ∆ and that each of them which is not compact meets ∂∆ (see Remark 3 following Proposition 2.7).
Corollary 2.14. In the situation of Proposition 2.13, if we assume that the open set ∆ containinḡ ∆ is small enough, any irreducible component Γ of the cycle X 0 in ∆ satisfies for all a ∈ A and all η > 0 small enough: is the holomorphic map of analytic extension built in Proposition 2.6.
Remark. In the situation of the previous corollary, choosing ε > 0 small enough, there exists, for each a ∈ A, a holomorphic extension map which lifts the map prlgt a : 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 × M ε a to an analytic family of branched coverings in M η a which is isotropic on S × M η+ε a .
Proof of Proposition 2.13. Corollary 2.11 and the remark following it implies the existence, for each z ∈ ∂∆ of a n-Hartogs figure H z relative to the boundary of ∆, contained in ∆ and satisfying the following properties: As the open sets M z cover the compact set ∂∆ 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 H a := H ε a and choosing ε > 0 small enough.
Proof of Corollary 2.14. Let Γ be an irreducible component of X 0 meeting M a for some a ∈ A. Then Γ meets M a . As Γ does not meetM (α) a ×V a × ∂B a because H a is adapted to X 0 , the intersection Γ ∩M a is the graph of an element γ ∈ H(M (α) a ×V a , Sym ka (B a )) with k a ∈ N * 6 .
The If an irreducible component of X 0 does not meet anyM a it has to be compact and contained in ∆. In this case the desired equality is obvious.

The extension and finiteness theorem 3.A. Some useful lemmas
The version below of Sard's lemma is more or less classical. Proof. Firstly note that a point z ∈ Z is critical for ϕ if, by definition, the differential of ϕ 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 ≤ n − 1 for some integer n ≥ 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 ϕ |S i has measure 0. So the critical set of ϕ |S is again of measure 0. But a critical point of ϕ which belongs to S is a critical point of ϕ |S . So it is enough to show that the set of critical values of ϕ restricted to the complex connected manifold Z \ S has measure 0. This is the classical Sard's lemma. 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 uniformely 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 . ThenΓ meets K and not V . Indeed, if (t 0 , x 0 ) would be inΓ ∩ V , there exists open neighbourhoods U 1 and B 1 of t 0 and x 0 respectively inŪ andB 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(Ū , 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: 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 .
The following variant will be used. 1. For each couple (i, j) ∈ I 2 an open subset W i,j ⊂⊂ U i ∩ U j is given and we ask that 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.
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 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. Remarks.

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 ⊂ 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 ∈ C loc n (Z) the condition to avoid a given compact subset is open in C loc n (Z), when the adjusted scale E is adapted to a cycle X 0 there exists an open neighbourhood, written down Ω k (E), of X 0 in C loc n (Z) such that Ω k (E) is the subset of all n-cycles X in Z for which E is  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∈S of n-cycles in Z such that for a point s 0 ∈ S the adjusted n-scale E is adapted to the cycle X s 0 with deg 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 → 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 (E i ) i∈I on Z. Assume that they are adapted to a given finite type n-cycleX 0 in Z. Assume that any irreducible component ofX 0 meets the open set W := i∈I D (E i ).
We shall call patching data forX 0 associated to the family (E i ) i∈I , written R((E i ) i∈I , F ) or more simply R when there is no ambiguity, a finite collection of n-scales (F i,j,h ) for (i, j) ∈ I 2 , i = j, where h belongs to a finite set H(i, j) for each couple (i, j) ∈ I 2 , i = j, such that the following properties hold: ii) The n-scales F i,j,h are adapted toX 0 .
We shall say that the patching data R are complete when the following condition also holds: iii) For each i = j given, the union of domains of the scales F i,j,h , h ∈ H(i, j), covers the compact Notations.
1. In the sequel, when we shall consider a reduced complex space Z and a finite family of adjusted n-scales (E i ) i∈I , adapted to a finite type n-cycleX 0 in Z, such that any irreducible component ofX 0 meets the open set we shall say that the family (E i ) i∈I is convenient forX 0 .
2. In this setting we shall use the following definitions : • W := i∈I D(E i ) ; • W := i∈I D (E i ) ; • W := i∈I D (E i ) ; • When the family (E i ) i∈I is convenient for a finite type n-cycleX 0 , K will be a compact neighbourhood of K disjoint fromX 0 .
3. ForX ∈ i∈I G k i (E i ) we shall denote by X i the closed cycle in D(E i ) associated to the i-th component ofX.
Lemma 3.9. Let (E i ) i∈I be a finite family of adjusted n-scales on Z, convenient for a n-cycleX 0 of finite type in Z, and let R be some corresponding complete patching data. There exists an open neighbourhood V of the imageX 0 ofX 0 in the product i∈I G k i (E i ) such that for eachX ∈ V we have the following properties: 1. No X i meets the compact set K. For a n-scale E := (U, B, j) on Z we shall abbreviate H(Ū , Sym k (B)) in G k (E).

For each
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 × B, adapted to X 0 , where h is given by an isomorphism of an open set in U × B into some open neighbourhood ofV ×C in C n × C p , we have a well-defined map of an open neighbourhood U of X 0 is H(Ū , Sym k (B)) into H(V , Sym l (C)), where l := deg F (X 0 ), sending X ∈ 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 ∩ h −1 (V × ∂C) = ∅ is open in H(Ū , 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: will be holomorphic (see Theorem 4, p. 66 in [Bar75]).
Definition 3.10. Let (E i ) i∈I a finite family of adjusted n-scales on Z, convenient for a n-cycleX 0 of finite type in Z, and let R be some corresponding patching data. Let k i be the degree ofX 0 in the adjusted scale E i . For each (i, j, h), i = j we have a couple of holomorphic maps obtained by the changes of scales E i → F i,j,h and E j → F i,j,h , because, by construction, we have 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 ,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 Moreover, this defines a tautological family of cycles in W which is a f-analytic family of cycles satisfying the following "almost universal" property: Proof. For eachX ∈ 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 toX 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 .

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 ).

If
E is an adjusted scale adapted to the n-cycle X 0 in Z, there exists an open neighbourhood U of X 0 in C loc n (Z) and a real ε > 0 such that for any X ∈ U and any τ ∈]0, ε[, the adjusted scale E τ remains adapted to X with again deg E τ (X) = deg E (X 0 ). 5. In the situation of the previous proposition 3.11, we may, for τ > 0 small enough, keep the same patching data R on the finite family (E τ i ) i∈I of adjusted n-scales; if it was complete, it remains complete and if it was convenient 10 for the n-cycle X 0 , it remains convenient for the n-cycle X 0 . Then we have a holomorphic restriction map where S τ (R) is the classifying space associated to the family (E τ ) i∈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 H is a n-Hartogs figure in C n+p ; we shall put, using the notations introduced above where ε > 0 is small enough, and where 0 < ε < ε . The choices of ε and ε will be precised when they are useful. We shall associate to H the adjusted n-scale on ∆ given by: We shall put alsoŨ the holomorphy envelopes respectively of U , U and U . Then we shall have a family of adjusted n-scales on ∆ , written down: where η is a non negative real number, small enough (for η = 0 we shall simply write EH). Then we shall have the isotropic classifying spaces G k (H) := Σ U,U (k) and also G η k (H) := ΣŨ η ,Ũ η (k).
Proposition 2.6 gives a holomorphic analytic extension map prlgt η such that the following diagram commutes: Definition 3.13. Let Z be a reduced complex space of pure dimension n + p, let ∆ ⊂⊂ Z be a strongly (n − 2)-concave open set in Z andX 0 a n-cycle in an open neighbourhood ∆ of∆ in Z. We shall say that a finite family (H a ) a∈A of n-Hartogs figures relative to the boundary of ∆ is excellent for the cycleX 0 when the following conditions hold, where we writeẼ a and E a the adjusted scales respectively on ∆ and ∆ associated to the n-Hartogs figure H a : 1. The adjusted scalesẼ a and E a are adapted to the cycleX 0 .
2. We may choose the patching domains of the adjusted scales (Ẽ a ) a∈A in order that the unioñ contains the compact set ∂∆.
3. There exists a finite family of adjusted n-scales (E b ) b∈B on ∆, adapted toX 0 , such that the finite families (E c ) c∈A∪B and (Ẽ c ) c∈A∪B are convenient forX 0 , where we As a consequence, the unionD (A) η ∪ D (B) of the isotropy domains will cover∆ for η > 0 small enough.
Proposition 3.14. (Existence of excellent families) Let Z be a reduced complex space of pure dimension n + p, ∆ ⊂⊂ Z be a strongly (n − 2)-concave open set with smooth boundary andX 0 a n-cycle in an open neighbourhood ∆ ⊂ Z of∆ such that any irreducible component ofX 0 meets ∆.
Then there exists a finite family (H a ) a∈A of n-Hartogs figures relative to the boundary of ∆ which is excellent for the cycleX 0 .
Proof. First we use Proposition 2.13 to cover ∂∆ by a finite family of n-Hartogs figures relative to the boundary of ∆ adapted to the cycleX 0 such that Conditions 1 and 2 hold. Then we build a finite family of adjusted n-scales (E b ) b∈B on ∆, adapted toX 0 in order that Condition 3 holds.

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 ≥ 2, p ≥ 1. Assume that there exists a C 2 exhaustion ϕ : Z →]0, 2] which is strongly (n − 2)-convex on the open set ϕ −1 (]0, 1[) and let ∆ := {x ∈ Z / ϕ(x) > α} for some α ∈]0, 1[ which is not a critical value for ϕ. LetX 0 a closed n-cycle of an open neighbourhood ∆ of∆ in Z such that any irreducible component ofX 0 meets ∆. Then there exists an open neighbourhood ∆ of∆ in ∆ and a f-analytic family (X ξ ) ξ∈Ξ of n-cycles in ∆ parametrized by a reduced complex space Ξ (of finite dimension) such that X ξ 0 =X 0 ∩ ∆ and such that Ξ is isomorphic to an open neighbourhood ofX 0 ∩ ∆ in C f n (∆ ). It satisfies the following universal property: For any f-analytic family (X s ) s∈S of n-cycles in ∆ parametrized by a Banach analytic set S and such that its value at some s 0 ∈ S is equal toX 0 ∩ ∆, there exists an open neighbourhood S of s 0 in S and an unique holomorphic map h : S → Ξ satisfying the equality X s =X h(s) ∩ ∆ for each s ∈ S .
Proof. Note that ∆ is a relatively compact open set in Z with a C 2 boundary which is strongly (n − 2)-concave.
Begin by covering the compact set ∂∆ by an excellent finite family (H a ) a∈A of n-Hartogs figures for ∆ adapted to the cycleX 0 . Choose then open sets ∆ 1 ⊂⊂ ∆ ⊂⊂ ∆ ⊂⊂ ∆ , such that the following properties hold, where we use the notations introduced above for the finite family of the adjusted scales (Ẽ a ) a∈A associated to (H a ) a∈A : It is easy to fulfill these conditions for ∆ 1 and ∆ near enough to ∆, as, by assumption, the subsets W (E a ) and j −1 a (Ū a ×B a \ B a ) are compact in ∆, and as the union of the open sets W (Ẽ a ) contains ∂∆.
Note that Condition iv) allows to choose the compact neighbourhood K of K inside ∆ 1 . Choose now a convenient finite family (E b ) b∈B of adjusted n-scales on ∆, adapted toX 0 in order that the open set b∈B W (E b ) contains∆ 1 .
, and define C := A ∪ B. The family of adjusted scales (E c ) c∈C in ∆ is then convenient forX 0 ∩ ∆ 1 , up to choosing the patching domains big enough. Fix some complete patching data R associated to the family (E c ) c∈C .
The family (Ẽ c ) c∈C of adjusted scales in ∆ is convenient forX 0 ∩ ∆ if we choose the patching domains big enough. Consider now some complete patching data for this family of the form R ∪ R , that is to say containing the patching scales already in R. Define the following Banach analytic sets: 1. S 0 is the classifying space of the family (Ẽ c ) c∈C , the degrees being these ofX 0 in the various scales adapted to the cycleX 0 , with the patching conditions defined by R. Note that R is not complete in general.
2. S + is the classifying space of the family (Ẽ c ) c∈C with the patching conditions defined by R∪R .
3. S − is the classifying space of the family (E c ) c∈C , with the (complete) patching conditions defined by R.
Then we get a holomorphic extension map α : S − → S 0 deduced from the extension maps in the n-Hartogs figures (H a ) a∈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 R . Then put Ξ := α −1 (S + ). So we have a holomorphic map α : Ξ → S + . We want to show the following claim: Claim. There exists a holomorphic map β : S + → Ξ satisfying the two properties: 1. We have α • β = Id and β • α = Id in a neighbourhood of the points defined byX 0 respectively in S + and Ξ.
2. The holomorphic map β is the composition of a holomorphic map induced by a linear (continuous) compact map and a holomorphic map.
As the open set c∈C W (Ẽ c ) contains ∆ , there is, on S + , a tautological family of n-cycles which is f-analytic on ∆ . Then the "almost universal" property of S − gives a holomorphic map β : S + → S − which factorizes via the closed Banach analytic subset Ξ ⊂ S − . Let us show that the holomorphic map β : S + → Ξ deduced from this factorization satisfies the two properties of the claim.
To see the second property, consider τ > 0 small enough and let us show that the holomorphic map induced by the linear compact (by Vitali's theorem) restriction r τ : S + → S τ + factorizes β, where S τ + is the classifying space corresponding to the τ -shrinkage (Ẽ τ c ) c∈C of the family (Ẽ c ) c∈C with the patching data R ∪ R . Indeed, for τ small enough, the tautological family of S τ + is still f-analytic on ∆ and the "almost universal" property of S − gives again a holomorphic mapβ τ : S τ + → S − . And we have β =β τ • r τ proving our assertion. In fact, the mapβ τ takes its values in Ξ because the τ -shrinkage does not change the patching data deduced from R ∪ R for τ small enough.
We conclude that the Banach analytic set Ξ 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 β : S + → Ξ factorizes via an open neighbourhood of X 0 ∩ ∆ in C f n (∆ ) because S + parametrizes a f-analytic family of cycles in ∆ , and because a cycle X near enough to X 0 ∩ ∆ defines an element in Ξ ⊂ S − as it satisfies the patching conditions R ∪ R . This shows that we can identify the Banach analytic set Ξ (which is a reduced finite dimensional complex space) with an open neighbourhood of X 0 ∩ ∆ in C f n (∆ ). The universal property is obvious.

Remarks.
1. The reduced complex space (of finite dimension) Ξ built in the previous theorem parametrizes a f-analytic family of n-cycles in the open set ∆ which is an open neighbourhood of∆. So, when we have a f-analytic family (X s ) s∈S of n-cycles in ∆ parametrized by a Banach analytic set S and such that its value at some s 0 ∈ S is equal toX 0 ∩ ∆ we can extend each cycle X s , s ∈ S , to a n-cycleX s in ∆ in order that the family (X s ) s∈S is f-analytic in ∆ , with the condition that each irreducible component ofX s meets ∆ and with the equalityX s ∩ ∆ = X s for each s ∈ S .
2. We shall see later on that Ξ is also (isomorphic to) an open neighbourhood of the pointX 0 ∩ ∆ in C f n (∆).
4. Finiteness of the space of n-cycles of a reduced strongly (n − 2)concave space (n ≥ 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 ≥ 0 is a natural integer, if there exists a real valued C 2 exhaustion function on Z, ϕ : Z →]0, 2], which is strongly q-convex outside the compact set K := ϕ −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 ϕ. For instance, any reduced compact complex space is strongly q-concave for any q ≥ 0.
When Z is strongly q-concave irreducible and non compact of dimension at least q +1, the function ϕ achieves its maximum at a point in K. So we shall have ϕ(Z) =]0, u] with u ≥ 1.
Theorem 4.2. Let n ≥ 2 be an integer. Let Z be a reduced complex space which is strongly (n − 2)concave. Let α ∈]0, 1[ and let X be a finite type n-cycle in the open set Z α := {z ∈ Z / ϕ(z) > α}. Then there exists a unique n-cycleX in C f n (Z) such thatX ∩ Z α = X.
The proof will use the following remark.
Remark. Consider a closed irreducible analytic subset Γ of dimension n in Z. As the restriction ϕ |Γ of the exhaustion ϕ to Γ must reach its maximum (as ϕ is continuous and proper), this maximum cannot be obtained at a point in which ϕ is strongly (n − 2)-convex. So we have Γ ∩ ϕ −1 [1, 2]) = ∅.
Proof. The uniqueness ofX is a consequence of the previous remark: ifX i , i = 1, 2, are in C f n (Z) and satisfyX i ∩ Z α = X, then any irreducible component Γ 1 ofX 1 has to meet Z α and so has to contain an open set inX 2 . Then it has to be an irreducible component ofX 2 and its multiplicity inX 1 and X 2 must coincide. SoX 1 =X 2 .
To show that this cycle exists, consider first the case where X is compact in Z α . ThenX := 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 ∈ ∂Z α there exists an open set U z and an unique closed analytic set X z in Z α ∪ U z of pure dimension n such that X z ∩ Z α = X. Choose a finite set of points z 1 , . . . , z N and open sets U i ⊂⊂ U i := U z i such that the union of the U i covers the compact set ∂Z α . Let Ω := Z α ∪ N i=1 U i and put Let us show that X 1 is a closed analytic subset in Ω. Consider z ∈ Ω. If z is in Z α we have X 1 = X in a neighbourhood of z and the assertion is clear. If not, either z is not in any ∂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 ∂U j 1 , . . . , ∂U j k for j 1 , . . . , j k in [1, N ]. As the set X j h is closed and analytic in Z α ∪ U j h , then X 1 is again the union of the X z i near z in Ω, and the assertion is proved.
So in this situation there exists a real positive β < α such that Z β ⊂ Ω. Let X 2 be the irreducible component of X 1 ∩ Z β which contains X; then X 2 is a closed irreducible analytic subset of Z β such that X 2 ∩ Z α = X. Now let Then what we obtained above shows that we have γ < α, and, applying the same arguments to the cycle X γ defined on Z γ via the cover of Z γ by the Z β , β > γ in which we already built an irreducible n-cycle X β extending X, we conclude that γ = 0 and that there exists an (unique) irreducible n-cyclẽ X in Z extending X.

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 ≥ 2 be an integer and let Z be a reduced complex space which is strongly (n − 2)-concave. Then the natural map j : C f n (Z) → C loc n (Z) is a homeomorphism. Moreover, for each α ∈]0, 1[ the restriction map res α : C f n (Z) → C f n (Z α ) 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 α for α ∈]0, 1[, this will imply the fact that the restriction map res α 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 := ϕ −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 res −1 α 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 ν ) ν≥0 be a sequence of n-cycles in Z converging in C loc n (Z) to a cycle Y . Assume that there exists a relatively compact open set Ω in Z such that any irreducible component of each X ν and of Y meets Ω. Then all these cycles are of finite type and the sequence converges to Y in C f n (Z).
Proof of Lemma 4.4. First if Y = ∅, for ν 1 the cycle X ν will be disjoint from the compact set K :=Ω, and this implies that X ν 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 f n (Z), that for ν 1 each irreducible component of X ν meets U . If it is not the case, passing to a subsequence, we may assume that for each ν there exists an irreducible component Γ ν of X ν disjoint from U . As, passing again to a subsequence, we may assume that the sequence (Γ ν ) converges in C loc n (Z) to a cycle Γ, we shall have |Γ| ⊂ |Y | and |Γ| ∩ U = ∅. To conclude, it is enough to show that Γ is not the empty cycle, as any irreducible component of Γ is also an irreducible component of Y and then meets U by hypothesis. As each Γ ν is not empty, it has to meet K =Ω. This implies that Γ also meets K and so is not empty. This contradicts our assumption.
End of the proof of Proposition 4.3. We have proved that j, and then also each j α : C f n (Z α ) → C loc n (Z α ) for α ∈]0, 1[, is a holomorphic homeomorphism. To conlude the proof we have to show the continuity of res −1 α , and this reduces to prove that if the sequence (X ν ) of C loc n (Z) is such that the sequence (X ν ∩ Z α ) converges in C loc n (Z α ), then it converges in C loc n (Z). Let Y α ∈ C loc n (Z α ) be the limit of this sequence in C loc n (Z α ) and let Y ∈ C loc n (Z) be the cycle extending it. Let A be the set of β ∈]0, α] such that the sequence (X ν ∩ Z β ) converges in C loc n (Z β ) to Y ∩ Z β . Then α is in A so A is not empty. Put γ := inf A. Theorem 3.15 implies that γ = 0 and we obtain also the convergence in any C loc n (Z β ), for any β > 0; this gives the convergence in C loc n (Z), as, by definition, a n-scale on Z is also a n-scale on Z β for β > 0 small enough.

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∈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 ∩ Z ) s∈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∈S 0 is f-analytic.
The hypotheses translated in terms of classifying maps means that we have a continuous map ϕ : S → C f n (Z) such that the composed map r • ϕ is holomorphic at s 0 , where r : C f n (Z) → C loc n (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 ϕ is holomorphic on S 0 . Note that, as r is holomorphic 12 , the hypothesis that ϕ 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.
11 ↑ 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 ϕ : S → C f n (Z) of this family. 12 ↑ in the sense that for any holomorphic map ψ : T → C f n (Z) of a reduced complex space T the composed map r • ψ is holomorphic. Proposition 4.6. Let S a reduced complex space and let ∅ = U 1 ⊂ U 2 be two polydiscs in C n . Let f : S × U 2 → C a continuous function, holomorphic on {s} × U 2 for each s ∈ S. Assume moreover that the restriction of f to S × U 1 is holomorphic. Then f is holomorphic on S × 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 C m . Fix an open relatively compact polydisc P in S. The function f defines a map F : U 2 → C 0 (P , C), where we write down C 0 (P , C) the Banach space of continuous functions onP , via the formula F (t)[s] = f (s, t) for t ∈ U 2 and s ∈P . First we shall show that the map F is holomorphic.
where t := (t 1 , . . . , t n ) are coordinates on C n . This Cauchy formula shows that F is C-differentiable and its differential at the point t ∈ U is given by where F i is associated to the function which is holomorphic for each fixed s ∈ S thanks to the Cauchy formula above. Let H(P , C) the closed subspace of C 0 (P , 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 ∈ U 2 such that F (t 0 ) is not in H(P , C). Thanks to the Hahn-Banach theorem we can find a continuous linear form λ on C 0 (P , C), vanishing on H(P , C), and such that λ(F (t 0 )) = 0. But the function t → λ(F (t)) is holomorphic on U 2 and vanishes on U 1 ; this contradicts λ(F (t 0 )) = 0. So F takes values in H(P , C) and f is holomorphic on S × 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 × U 2 and holomorphic on S × U 1 . So the closed analytic subset Y ⊂ S × U 2 along which f may not be holomorphic has no interior point in each {s} × U 2 . The analytic extension criterion of [BM14, Chapter IV, Criterion 3.1.7] allows to conclude.
Proof of Theorem 4.5. Let |G| ⊂ S × Z be the graph of the f -continuous family (X s ) s∈S and let A be the set of points in (σ, ζ) ∈ |G| admitting an open neighbourhood S σ × Z ζ in S × Z such that the family of cycles (X s ∩ Z ζ ) s∈Sσ is analytic. Remark that, thanks to our hypothesis, the open set A in |G| meets every irreducible component of {s 0 } × |X s 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 × U → Sym k (B) be the (continuous) classifying map for the family (X s ) s∈S 1 in the scale E. As j −1 (U × {0}) meets A, there exists a non empty polydisc U 2 ⊂ U such that U 2 × {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 × U . Moreover, as the same argument applies to any linear projection of U × 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 ∩ ({s 0 } × |X s 0 |) of |X s 0 |.
If the boundary of A ∩ ({s 0 } × |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∈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∈S be a f-continuous family of n-cycles in Z. Assume that this family is analytic in s 0 ∈ S. Then there exists an open neighbourhood S of s 0 in S such that the family (X s ) s∈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. Proof. Let X 0 be a compact cycle in Z. There exists α ∈]0, 1[ such that X 0 is contained in Z α = {x ∈ Z / ϕ(x) > α}. So X 0 does not meet the compact set ϕ −1 ({α}). This is an open condition in C f n (Z). And as any irreducible n-dimensional analytic subset in Z has to meet K, if it does not meet ϕ −1 ({α}) it is contained in Z α (by connectedness). Then any X ∈ C f n (Z) which is near enough X 0 is contained in Z α so is compact.
Note that under the hypothesis of the previous lemma, C n (Z) is not closed in C f n (Z) in general, as one can see taking Z := P N \ {0} and considering the set of hyperplanes in 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 f n (Z) and choose an α ∈]0, 1[ which is not a critical value of ϕ and of the restriction of ϕ to |X 0 |; this is possible thanks to Sard's Lemma 3.1 and the fact that ϕ(|X 0 |) contains ]0, 1[. Consider now the reduced complex space Ξ constructed in Theorem 3.15. It is an open neighbourhood of X 0 ∩ ∆ in C f n (∆ ) and we may assume that ∆ := Z β for some β < α very near α. Due to Proposition 4.3 it is homeomorphic to an open neighbourhood V of X 0 in C f n (Z). So the restriction map res 0 : V → Ξ is holomorphic, bijective and is a homeomorphism. Now the continuity of res −1 0 and the finiteness of Ξ allow to apply Theorem 4.5 because we already know that the tautological family of cycles parametrized by Ξ is f-analytic on the open set ∆ which is an open set which meets any irreducible component of each cycle in this family (because K ⊂ Z β ). Then res −1 0 is holomorphic and so res 0 is an isomorphism of Banach analytic sets.