Mats Andersson
email: matsa@chalmers.se&hasam@chalmers.se&wulcan@chalmers.se

Hakan Samuelsson

MATS ANDERSSON & H ÅKAN SAMUELSSON KALM Elizabeth Wulcan

Alain Yger

Alain Yger Segre

2015 Hal-01026124

Segre numbers, a generalized King formula , and local intersections

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Let X be a reduced analytic space of pure dimension and let J be a coherent ideal sheaf on X. Given a point x ∈ X, Tworzewski, [START_REF] Tworzewski | Intersection theory in complex analytic geometry[END_REF], Achilles-Manaresi, [START_REF] Achilles | Multiplicities of a bigraded ring and intersection theory Math[END_REF], and Gaffney-Gassler, [START_REF] Gaffney | Segre numbers and hypersurface singularities[END_REF], independently introduced numbers e 0 (J , X, x), . . . , e n (J , X, x) that in a certain sense measure the singularities of J at x and that generalize the classical Hilbert-Samuel multiplicity. Following [START_REF] Gaffney | Segre numbers and hypersurface singularities[END_REF] we will call them Segre numbers; Tworzewski used the term extended index of intersection whereas Achilles-Manaresi used the term multiplicity sequence. The definition in [START_REF] Tworzewski | Intersection theory in complex analytic geometry[END_REF] goes via a local variant of the Stückrad-Vogel construction, [START_REF] Stückrad | An algebraic approach to the intersection theory Queen's Papers in Pure and[END_REF], introduced in [START_REF] Tworzewski | Intersection theory in complex analytic geometry[END_REF][START_REF] Massey | Numerical Control over Complex Analytic Singularities Mem[END_REF], and a closely related, also geometric, procedure is used in [START_REF] Gaffney | Segre numbers and hypersurface singularities[END_REF]. In [START_REF] Achilles | Multiplicities of a bigraded ring and intersection theory Math[END_REF] the definition is purely algebraic; it is based on Hilbert functions of bigraded rings. It is proved in [START_REF] Achilles | Intersection numbers, Segre numbers and generalized Samuel multiplicities[END_REF] that the definitions in [START_REF] Tworzewski | Intersection theory in complex analytic geometry[END_REF][START_REF] Achilles | Multiplicities of a bigraded ring and intersection theory Math[END_REF][START_REF] Gaffney | Segre numbers and hypersurface singularities[END_REF] yield the same numbers; see also [START_REF] Nowak | Analytic improper intersections. II. Deformation to an algebraic bicone and applications[END_REF]. In this paper we give a (semi-)global representation of these numbers as the Lelong numbers of certain positive closed currents, constructed from a tuple of generators of J . This is part of our main result Theorem 1.1, which is a generalizion of King's formula for these currents.

Let us first recall the definition in [START_REF] Tworzewski | Intersection theory in complex analytic geometry[END_REF]. In that paper X is a subvariety of a smooth manifold Y and J is the pullback to X of the sheaf associated with a smooth submanifold A ⊂ Y . However, we find it advantageous to avoid any ambient space and instead consider an arbitrary coherent ideal sheaf J → X. A sequence h = (h 1 , h 2 , . . . , h n ) in the local ideal J x is called a Vogel sequence of J at x if there is a neighborhood U ⊂ X of x where the h j are defined, such that

(1.1) codim (U \ Z) ∩ (|H 1 | ∩ . . . ∩ |H k |) = k or ∞, k = 1, . . . , n;
here Z is the (reduced) zero set of J and |H ℓ | are the supports of the divisors H ℓ defined by the h ℓ . Notice that, possibly after shrinking U , the common zero set {h = 0} in U equals Z ∩ U and that if f 0 , . . . , f m generate J x , then any generic Date: July 4, 2014.

First three authors partially supported by the Swedish Research Council. Third author also partially supported by the NSF. sequence of n linear combinations of the f j is a Vogel sequence at x. Let X 0 = X and let X Z 0 denote the irreducible components of X 0 that are contained in Z and let X X\Z 0 be the remaining components 1 , so that

X 0 = X Z 0 + X X\Z 0
.

By the Vogel condition (1.1), H 1 intersects X X\Z 0

properly. Set (1.2)

X 1 = H 1 • X X\Z 0
and decompose analogously X 1 into the components X Z 1 contained in Z and the remaining components

X X\Z 1 , so that X 1 = X Z 1 + X X\Z 1
. Define inductively

X k+1 = H k+1 • X X\Z k , X Z
k+1 , and

X X\Z k+1 . Then V h := X Z 0 + X Z 1 + • • • + X Z n
is the Vogel cycle 2 associated with the Vogel sequence h. Let V h k denote the components of V h of codimension k, i.e., V h k = X Z k . Tworzewski defines the extended index of intersection as min lex (mult x V h 0 , . . . , mult x V h n ), where the min lex is taken over all Vogel sequences h of J x .

Let us next recall the definition of Segre numbers in [START_REF] Gaffney | Segre numbers and hypersurface singularities[END_REF], where also so-called polar multiplicities are introduced. Let f be a tuple of generators f 0 , . . . , f m of J x and let h be a Vogel sequence of linear combinations h j = α j • f = α 0 j f 0 + • • • + α m j f m ; notice that any Vogel sequence is on this form for some choice of generators and α j . It is proved in [10, Section 2], see also Section 6 below, that the multiplicities mult x V h k and mult x X X\Z k are independent of α j for generic choices of α j and also independent of f , and these numbers are the Segre numbers, e k (x) = e k (J , X, x), and polar multiplicities, m k (x) = m k (J , X, x), respectively. Throughout we will use this definition of the list e(x) = e(J , X, x) = e 0 (x), . . . , e n (x) of Segre numbers. In Section 8 below we prove that the definitions in [START_REF] Tworzewski | Intersection theory in complex analytic geometry[END_REF] and [START_REF] Gaffney | Segre numbers and hypersurface singularities[END_REF] coincide, i.e., (1.3) e(x) = min lex mult x V h 0 , . . . , mult x V h n .

It is not clear to us whether this coincidence has been noticed in the literature before.

In [START_REF] Achilles | Intersection numbers, Segre numbers and generalized Samuel multiplicities[END_REF], both notions are discussed, and (1.3) follows for the restrictive class of sheaves considered in [START_REF] Tworzewski | Intersection theory in complex analytic geometry[END_REF], but it is not explicitly stated. The coincidence also follows from [START_REF] Rams | On the intersection product of analytic cycles Ann[END_REF]Theorem 3.3] in combination with [START_REF] Gaffney | Segre numbers and hypersurface singularities[END_REF]Lemma 2.2]. We remark that both definitions above are local. Indeed, the Vogel condition (1.1) as well as the genericity of α j depends on x, cf., Remark 1.4. Also the algebraic definition in [START_REF] Achilles | Multiplicities of a bigraded ring and intersection theory Math[END_REF] is local.

Let f be a tuple of generators of the ideal sheaf J . For k = 0, 1, 2, . . . , n we consider the closed positive currents M f k introduced in [START_REF] Andersson | Residue currents of holomorphic sections and Lelong currents Arkiv för matematik[END_REF]. The current M f k coincides with 1 Z (dd c log |f | 2 ) k , where 1 Z is the characteristic function for Z and for k ≤ codim {f = 0} it is well-known that the definition (1.4) coincides with the standard one. Notice that (dd c log |f | 2 ) 0 = 1 and hence, M f 0 = 1 Z is the current of integration over the components of X that are contained in Z; in particular, it vanishes unless f ≡ 0 on some irreducible component of X. See Section 4 for other expressions for M f k . Our main result is the following. Theorem 1.1 (Generalized King's formula). Let X be a reduced analytic space of pure dimension n and let J be a coherent ideal sheaf on X generated by a tuple f of holomorphic functions. Let Z be the zero set of J and let Z k j be the distinguished varieties of J of codimension k. Then

(1.5) M f k = 1 Z (dd c log |f | 2 ) k = j β k j [Z k j ] + N f k =: S f k + N f k , k = 0, . . . , n,
where the β k j are positive integers, the N f k are positive closed currents, the Lelong numbers ℓ x (N f k ) are nonnegative integers that only depend on the integral closure class of J at x, and the set where ℓ x (N f k ) ≥ 1 has codimension at least k + 1. The Lelong number of M f k at x is the Segre number e k (J , X, x). The polar multiplicity m k (J , X, x) coincides with the Lelong number at x of the current

1 X\Z (dd c log |f | 2 ) k .
Here [Z k j ] denotes the Lelong current3 , i.e., the current of integration, associated with the variety Z k j . Recall that the integral closure of J x consists of all holomorphic germs φ such that |φ| ≤ C|f | for some C > 0 at x. For the definition of the (Fulton-MacPherson) distinguished varieties Z k j at x associated to J x , see Section 7 below. It turns out that S f = k S f k = jk β k j Z k j is precisely the cycle that appears in all Vogel cycles obtained from generic (enough) Vogel sequences. It is called the fixed part in [START_REF] Gaffney | Segre numbers and hypersurface singularities[END_REF]. The remaining parts of the Vogel cycles vary with the Vogel sequence and are called the moving parts. Notice that (1.5) is the Siu decomposition, [START_REF] Siu | Analyticity of sets associated to Lelong numbers and the extension of closed postive currents Invent[END_REF], of M f k .

Note that, contrary to the previous local definitions of Segre numbers, Theorem 1.1, gives a (semi-)global representation of the Segre numbers

(1.6) e k (J , X, x) = ℓ x (M f k ).
Moreover, the M f k are obtained as limits of explicit expressions in generators of J .

It follows from Lemma 2.2 that M f k = 0 if k < codim Z and that N f codim Z = 0. The case k = codim Z of (1.5) is precisely the classical King formula.
Remark 1.2. If J x is generated by p < n functions f 0 , . . . , f p-1 , we will see that M f k = 0 for k > p and hence e k = 0 for k > p. However, M g k may be non-vanishing if g is another, larger, set of generators. If, in addition, codim Z x = p, i.e., J x is a complete intersection, then e p (J , X, x) is the only non-zero entry in e(J , X, x). This number is the classical intersection number of the proper intersection of the divisors of the p generators f j , see, e.g., [START_REF] Chirka | Complex analytic sets Kluwer[END_REF].

Corollary 1.3. If J is the radical ideal of a variety Z of pure codimension p, then M f p = [Z].
Remark 1.4. Assume that x is a point where n k (J , X, x) ≥ 1 for some k and let V h be a generic Vogel cycle so that mult x V h k = e k (J , X, x). Then V h k = S f k + W where the moving part W is a positive cycle of codimension k, such that mult x W = n k (J , X, x). Since n k (J , X, y) ≥ 1 only on a set of codimension ≥ k + 1, at most points y on V h k we have that e k (J , X, y) = mult y (S f k ) and hence mult y V h k > e k (J , X, y). As soon as there is a moving part at x it is thus impossible to find a Vogel cycle that represents the Segre numbers in a whole neighborhood of x.

A fundamental ingredient in this paper is a current calculus described in Sections 3 to 5. It gives an expedient analytic approach to Vogel cycles; for instance, it becomes a straightforward matter to form mean values of (the Lelong currents of) such cycles. The currents M f k are in fact such mean values, see Section 6; this is the intuitive idea behind Theorem 1.1. The current calculus is fundamental for the proof of Theorem 1.1, which is given in Section 7, and it makes it possible to provide a proof of (1.3) in our slightly more general setting of a general sheaf J than what was considered in [START_REF] Tworzewski | Intersection theory in complex analytic geometry[END_REF], see Section 8. Our current calculus is also useful for concrete computations of Segre numbers, see Section 11. In Section 9 we prove a certain invariance property of Segre numbers. The motivation in [START_REF] Tworzewski | Intersection theory in complex analytic geometry[END_REF] for introducing these numbers was to develop a new intersection theory. In Section 10 we discuss some local aspects of connections to intersection theory. Our technique to form new currents by averaging Vogel cycles will be the starting point in a forthcoming paper where we will study a kind of global intersection products.

Remark 1.5. This paper is a shortened and slightly elaborated version of [START_REF] Andersson | Non-proper intersections and positive currents I[END_REF]; in that paper can be found, additionally, a discussion of global intersections in the sense of Tworzewski and various examples.

Preliminaries

Let us fix some notation. Throughout this paper X is a reduced analytic space of pure dimension n and J is a coherent ideal sheaf on X. Given a tuple f = (f 0 , . . . , f m ) of holomorphic functions on an analytic space we will use J (f ) to denote the sheaf it generates. Similarly if W ⊂ X is an analytic subset we will use J W to denote the radical sheaf. We will denote the local ring of germs of holomorphic functions at x in X by O X,x . We say that a sequence g 1 , . . . , g m of functions on an analytic space X is a geometrically regular sequence

if codim {g 1 = . . . = g k = 0} = k for 1 ≤ k ≤ m. If X is smooth (or Cohen-Macaulay) a sequence is geometrically regular if and only if it is regular.
Though less natural at first sight it is often computationally more convenient to use regularizations based on analytic continuation rather than smooth regularizations as in (1.4). For instance, if h is a holomorphic function on X then

λ → ∂|h| 2λ ∧ ∂ log |h| 2 2πi ,
a priori defined for Re λ ≫ 0, has a current-valued analytic continuation to a neighborhood of 0 and the value at 0 is the integration current associated to the divisor defined by h. In general, if α(λ) is a current-valued function, defined in a neighborhood of 0, we let α(λ)| λ=0 denote the value at λ = 0.

2.1. Positive currents. Let d c = (4πi) -1 (∂ -∂) so that dd c = (2πi) -1 ∂∂. We briefly recall some basic facts about positive currents, referring to [START_REF] Chirka | Complex analytic sets Kluwer[END_REF][START_REF] Demailly | Courants positifs et théorie de l'intersection[END_REF] for details. Let µ be a positive current of bidegree (k, k) defined in some open set Ω ⊂ C N . Then µ has order zero, so that the restriction 1 S µ is well-defined for any Borel set S ⊂ Ω. If in addition µ is closed and S is analytic, then the Skoda-El Mir theorem asserts that 1 S µ is closed as well. If µ is closed then one can define inductively

(dd c log |z -x| 2 ) j+1 ∧µ = dd c log |z -x| 2 dd c ((log |z -x| 2 ) j ∧µ) , (dd c log |z -x| 2 ) N -k ∧µ is a (N, N )-current.
Its mass at x is the Lelong number ℓ x (µ) at x of µ, which depends semi-continuously on µ, in the sense that (2.1)

ℓ x (µ) ≥ lim sup j→∞ ℓ x (µ j ) if µ j → µ. It follows that x → ℓ x (µ) is upper semi-continuous. Lemma 2.1. If µ is a closed positive (k, k)-current in Ω ⊂ C N , then 4 (2.2) ℓ 0 (µ)[0] = lim λ→0 + ∂|z| 2λ ∧ ∂|z| 2 2πi|z| 2 ∧(dd c log |z| 2 ) N -k-1 ∧µ .
If k = N , then the right hand side of (2.2) shall be interpreted as lim λ→0 + 1 -|z| 2λ µ = 1 {0} µ, so Lemma 2.1 is trivially true in this case.

Sketch of proof. If ξ is a test function, then (2.3) (dd c log |z| 2 ) N -k ∧µ ∧ ξ = lim λ→0 + |z| 2λ -1 λ ∧(dd c log |z| 2 ) N -k-1 ∧ µ∧dd c ξ.
After an integration by parts, the right-hand side of (2.3) may be rewritten as

lim λ→0 + ∂|z| 2λ ∧ ∂|z| 2 2πi|z| 2 ∧(dd c log |z| 2 ) N -k-1 ∧µ∧ξ + lim λ→0 + |z| 2λ (dd c log |z| 2 ) N -k ∧µ∧ξ.
The second term is precisely the action of 1 C N \{0} (dd c log |z| 2 ) N -k ∧µ on ξ, and consequently the point mass at 0 of (dd c log |z| 2 ) N -k ∧µ is the same as the point mass at 0 of the first term, which proves (2.2).

2.2.

Currents on an analytic space. 5 Let X be a reduced analytic space of pure dimension n. Given a local embedding i : X ֒→ C N , we let E X be the sheaf of smooth forms on X, obtained from the sheaf of smooth forms in the ambient space, where two forms are identified if their pullbacks to X reg coincide; it is wellknown that this definition does not depend on the particular embedding. We say that µ is a current on X of bidegree (p, q) if it acts on test forms on X of bidegree (n -p, n -q). Such currents µ are naturally identified with currents τ = i * µ of bidegree (N -n + p, N -n + q) in the ambient space such that τ vanish on the kernel of i * . Observe that the d-operator is well-defined on currents on X. If W is a subvariety of X of pure codimension p ≥ 0, then

φ → [W ].φ = Wreg φ
is a closed (p, p)-current on X; this is the current of integration over W .

Recall that a current ν is normal if both ν and dν have order zero. The following lemma follows immediately from the corresponding one in C N . Lemma 2.2. Suppose that µ is a normal current of bidegree (p, p) on X that has support on a subvariety W of codimension k. If k > p then µ = 0. If k = p and µ is closed, then µ = j α j [W j ] for some numbers α j , where W j are the irreducible components of W of codimension p.

It is readily checked that if we have a proper holomorphic mapping ν : X ′ → X between analytic spaces, then the push-forward ν * is well-defined on currents on X ′ .

Assume that µ is a positive closed current on the analytic space X. Fix x ∈ X and let i : X ֒→ C N be a local embedding. We define the Lelong number ℓ x (µ) as ℓ x (i * µ). After a suitable change of coordinates i can be factorized as i = j • i ′ , where i ′ : X → C M is a minimal embedding and j is the natural embedding C M → C M × C N -M . Since the Lelong number is invariant under holomorphic changes of coordinates, all minimal embeddings are equal up to a holomorphic change of variables, and ℓ x (τ ) = ℓ x (j * τ ), it follows that ℓ x (µ) is well-defined. Thus if Z is a subvariety of an analytic space X and we have an embedding X ֒→ C N , then the number ℓ x ([Z]) is independent of whether we consider [Z] as the Lelong current of Z on X or on C N .

Recall that if Z is a variety in C N , then the multiplicity mult x Z of Z at x coincides with the Lelong number ℓ x ([Z]), see [START_REF] Chirka | Complex analytic sets Kluwer[END_REF]Prop. 3.15.1.2]; here mult x Z is defined as in [START_REF] Chirka | Complex analytic sets Kluwer[END_REF]Ch. 2.11.1]. In particular, the Lelong number of the function 1, considered as a current on an analytic space X, at x is precisely mult x X.

The classical Siu decomposition, [START_REF] Siu | Analyticity of sets associated to Lelong numbers and the extension of closed postive currents Invent[END_REF], of positive closed currents extends immediately to currents on our analytic space X. Let µ be a positive closed (p, p)-current on X; then there is a unique decomposition

µ = i β i [W i ] + N,
where W i are irreducible analytic varieties of codimension p, β i ≥ 0, and, for each δ > 0, the set where ℓ x (N ) ≥ δ is analytic and has codimension strictly larger than p.

Cycles and Lelong currents. Given an analytic cycle Z =

α j W j , where W j are varieties, we let [Z] = α j [W j ] be the associated Lelong current. We will often identify analytic cycles with their Lelong currents. We let |Z| denote the support of Z, and we let 1 Z mean 1 |Z| . If H is a Cartier divisor defined by (a germ of) a holomorphic function h, we will (sometimes) use the notation [h] for [H] and 1 h for 1 |H| . Given an analytic cycle Z = α i W i of pure dimension, the multiplicity of Z at x is defined as

α i mult x W i (this definition follows [10, p. 704]). It follows that mult x Z = ℓ x ([Z]). If Z = n k=0 Z k , where Z k is an analytic cycle of codimension k we define (2.4) mult x Z := (mult x Z 0 , . . . , mult x Z n )
Throughout this paper all analytic cycles are effective, unless otherwise stated.

2.4. Proper intersections. Let Y be a complex manifold and let Z 1 , . . . , Z r be (effective) analytic cycles in Y of pure codimensions p j , j = 1, . . . , r, that intersect properly, i.e., the intersection V of their supports has codimension

p 1 + • • • + p r .
There is a well-defined cycle, called the (proper) intersection of the Z j ,

(2.5) Z r • • • Z 1 = m j V j ,
where V j are the irreducible components of V and m j are certain positive integers.

One can obtain these numbers m j by defining the intersection number i(x), algebraically or geometrically, at each fixed point x of V , and prove that i(x) is generically constant on each V j , see, e.g., [START_REF] Chirka | Complex analytic sets Kluwer[END_REF]. However, by means of currents, (2.5) can be obtained in a more direct way: By an appropriate regularization one can define the wedge product

[Z r ]∧ • • • ∧[Z 1 ]
, see, e.g., [START_REF] Chirka | Complex analytic sets Kluwer[END_REF][START_REF] Demailly | Courants positifs et théorie de l'intersection[END_REF], and this current indeed coincides with the Lelong current of Z r • • • Z 1 . In particular, if the Z j are (effective) divisors defined by holomorphic functions h j , then the Lelong current of the intersection can be obtained explicitly as

(2.6) [Z r • • • Z 1 ] = lim ǫ→0 dd c log(|h j | 2 + ǫ).
At each point x there is a well-defined intersection number

ǫ(x) := j m j mult x (V j );
here mult x (V j ) is the multiplicity of the variety V j at x. The number ǫ(x) is precisely equal to the Lelong number

ℓ x ([Z r • • • Z 1 ]) of the positive closed current [Z r • • • Z 1 ].

Multiplying a Lelong current by a Cartier divisor

In this section we will describe how the inductive construction of a Vogel cycle V h can be expediently expressed as certain products of Lelong currents. Notice that the map

[W ] → 1 Z [W ] is linear. Notice also that if Z, Z ′ are analytic cycles in X, then (3.1) 1 Z ′ [Z] = [Z Z ′ ]; recall that Z Z ′ denotes the irreducible components of Z that are contained in Z ′ . To see (3.1) we may assume that Z is irreducible. If |Z| is contained in |Z ′ |, then 1 Z ′ [Z] = [Z]. Otherwise, |Z| ∩ |Z ′ | has higher codimension than |Z|, and thus 1 Z ′ [Z]
vanishes by Lemma 2.2. Notice that 1 Z is 1 on the components of X that are contained in Z and 0 otherwise, i.e., it is the Lelong current of X Z .

If h is a non-vanishing holomorphic function on (each irreducible component of) the analytic space Z, then log |h| 2 is a well-defined (0, 0)-current on Z. This is clear if Z is smooth and follows in general, e.g., by means of a smooth resolution Z → Z, cf., the proof below. Lemma 3.1. Let Z be an analytic cycle in X, h be a holomorphic function, and let u be a nonvanishing smooth function on X. Then

(3.2) λ → ∂|uh| 2λ ∧ ∂ log |uh| 2 2πi ∧[Z],
a priori defined when Re λ is large, has an analytic continuation to a half-plane Re λ > -ǫ, where ǫ > 0. The value at λ = 0 is independent of u.

If h does not vanish identically on any irreducible component of (the support of ) Z, then this value is equal to dd c (log

|h| 2 [Z]).
Notice that v λ := ∂|uh| 2λ ∧ ∂ log |uh| 2 /(2πi) has continuous coefficients when Re λ > 1, so the product in (3.2) is then well-defined.

Proof. First assume that Z = X = C N and h is a monomial h = z a 1 1 • • • z a N N . Then (3.2) is equal to v λ = ∂|uz a 1 1 • • • z a N N | 2λ ∧ 1 2πi N 1 a j dz j z j + ∂|u| 2 |u| 2 .
One can check that the desired analytic continuation exists, and that the value at λ = 0 is the current N 1 a j [z j ] = dd c log |h| 2 ; in particular, it is independent of u. Consider now the general case. By linearity, we may assume that Z is irreducible. If h vanishes identically on Z and Re λ is large, then v λ ∧ [Z] = 0, and thus it trivially extends to λ ∈ C. Assume that h does not vanish identically on Z. Let i : Z ֒→ X be an embedding and let π : Z → Z be a smooth modification of Z such that π * i * h is locally a monomial; such a modification exists due to Hironaka's theorem on resolution of singularities. After a partition of unity we are back to the case above. It follows that π * i * v λ has an analytic continuation to Re λ > -ǫ for some ǫ > 0 and thus

v λ ∧[Z] = i * π * (π * i * v λ ) has the desired analytic continuation. The value at λ = 0 is equal to i * π * (dd c log |π * i * h| 2 ) which proves the second statement, since (log |h| 2 )[Z] = i * π * (log |π * i * h| 2 ).
Let H denote the Cartier divisor defined by h. We define [H] ∧ [Z] as the value of (3.2) at λ = 0. According to the lemma it does not depend on the particular choice of h defining H. It follows from the definition that

(3.3) [H]∧([Z 1 ] + [Z 2 ]) = [H]∧[Z 1 ] + [H]∧[Z 2 ]
and thus [Z] → [H]∧[Z] is a linear operator on Lelong currents, cf., (3.1). However, in general it is not true that ([

H 1 ]+[H 2 ])∧[Z] = [H 1 ]∧[Z]+[H 2 ]∧[Z] or [H 1 ]∧[H 2 ] = [H 2 ] ∧ [H 1 ]. Since (3.
2) is analytic by Lemma 3.1 it follows that [H] ∧ 1 H [Z] = 0, and so, using (3.3), we get

[H] ∧ [Z] = [H] ∧ 1 H [Z] + 1 X\H [Z] = [H] ∧ 1 X\H [Z]. Now, 1 X\H [Z]
is the current of integration over Z X\H , i.e., the components of Z that are not contained in |H|. Since H and Z X\H intersect properly it follows that [START_REF] Andersson | Residue currents of holomorphic sections and Lelong currents Arkiv för matematik[END_REF]. Summarizing, we get the computation rules

[H] ∧ 1 X\H [Z] = [H • Z X\H ], cf. Section 2.
(3.4) [H]∧[Z] = [H]∧1 X\H [Z] = [H • Z X\H ]. Remark 3.2. It is important to emphasize that [H]∧[Z]
is not the same as (the Lelong current associated with) the intersection H • Z in [START_REF] Fulton | Intersection theory[END_REF]. In fact, if Z is irreducible and contained in H, then [H]∧[Z] = 0, whereas in [START_REF] Fulton | Intersection theory[END_REF] the product is a cycle in Z of codimension 1 that is well-defined up to rational equivalence.

Example 3.3. Let H 1 and H 2 be Cartier divisors and let

H = H 1 + H 2 . Then [H 1 ]∧[H] = [H 1 ]∧[H 2 ] but [H]∧[H 1 ] = 0. Moreover [H 1 ] ∧ 1 H 1 [H] = [H 1 ]∧[H 1 ] = 0 but 1 H 1 [H 1 ]∧[H] = 1 H 1 [H 1 ]∧[H 2 ] = [H 1 ]∧[H 2 ].
We can construct Vogel cycles, cf., Section 1, by inductively applying operators 1 Z and [H]∧. Proposition 3.4. Let X be an analytic space of dimension n and let h = (h 1 , . . . , h n ) be a Vogel sequence of an ideal J with variety Z at x ∈ X, with corresponding divisors H 1 , . . . , H n . Then on X,

(3.5) [X 0 ] = 1, [X ℓ ] = [H ℓ ]∧ • • • ∧[H 1 ] , ℓ = 1, . . . , n and
(3.6) [X Z 0 ] = 1 Z , [X Z ℓ ] = 1 Z [H ℓ ]∧ • • • ∧[H 1 ], ℓ = 1, . . . , n. In particular, (3.7) [V h ] = 1 Z + 1 Z [H 1 ] + 1 Z [H 2 ]∧[H 1 ] + • • • + 1 Z [H n ]∧ • • • ∧[H 1 ].
If we consider X as embedded in some larger analytic space Y , then we have instead

[X 0 ] = [X], [X ℓ ] = [H ℓ ]∧ • • • ∧[H 1 ]∧[X], ℓ = 1, . . . , n and [X Z 0 ] = 1 Z [X], [X Z ℓ ] = 1 Z [H ℓ ]∧ • • • ∧[H 1 ]∧[X], ℓ = 1, . . . , n Proof.
In view of (3.1), (3.6) follows from (3.5). Using (3.3), we have, in view of (1.2), that

[X 1 ] = [H 1 ]∧[X X\Z 0 ] = [H 1 ]∧([X 0 ] -[X Z 0 ]) = [H 1 ] since [H 1 ]∧[X Z 0 ] = [H 1 ]1 Z = 0.
One obtains (3.5) by induction.

Bochner-Martinelli currents

Let f = (f 0 , . . . , f m ) be a tuple of holomorphic functions on X that generates J and let Z be the zero set of J . For Re λ ≫ 0, let

M f,λ 0 := 1 -|f | 2λ M f,λ k := ∂|f | 2λ ∧ ∂ log |f | 2 2πi ∧(dd c log |f | 2 ) k-1 if k ≥ 1,
and

(4.1) M f,λ := ∞ k=0 M f,λ k , where |f | 2 = m j=0 |f j | 2 .
The sum in (4.1) is finite for degree reasons, and when Re λ ≫ 0, M f,λ is locally integrable. We will show that λ → M f,λ k has a currentvalued analytic continuation to Re λ > -ǫ, for some ǫ > 0. We denote the value of M f,λ k at λ = 0 by M f k and we write

M f := k M f k .
The current M f and its components M f k will be referred to as Bochner-Martinelli currents, cf. Remark 4.2 below.

A computation yields that

M f,λ k = λ i 2π ∂|f | 2 ∧ ∂|f | 2 |f | 4-2λ ∧(dd c log |f | 2 ) k-1
which is positive when λ > 0, and thus M f k is a positive current. Note that M f 0 is the current of integration over the components of X, on which f ≡ 0. In particular, if f does not vanish identically on any component of X, then M f 0 = 0. Let π : X → X be a normal modification such that the pull-back ideal sheaf J •O X is principal; for instance one can take the normalization of the blow-up of X along J . Then π * f = f 0 f ′ where f 0 is a section of the holomorphic line bundle L → X corresponding to the exceptional divisor D f of π : X → X, i.e., the divisor defined by J • O X , and f ′ is a nonvanishing tuple of sections of L -1 . Let L be equipped with the metric defined by

|f 0 | L = |π * f | = |f 0 f ′ |, and let (4.2) ω f := dd c log |f ′ | 2 ;
here the right hand side is computed locally for any local trivialization of

L -1 . Then -ω f is the first Chern form of (L, | • | L ), and clearly ω f ≥ 0. Since log |π * f | 2 = log |f 0 | 2 + log |f ′ | 2 it follows from the Poincare-Lelong formula that (4.3) dd c log |π * f | 2 = [D f ] + ω f .
In particular, π * (dd c log |f | 2 ) = ω f outside π -1 {f = 0}. Therefore, for Re λ ≫ 0,

π * M f,λ 0 = 1 -|f 0 f ′ | 2λ (4.4) π * M f,λ k = (2πi) -1 ∂|f 0 f ′ | 2λ ∧∂ log |f 0 f ′ | 2 ∧ω k-1 f , k ≥ 1. (4.5) Now Lemma 3.1 asserts that λ → π * M f,λ k has an analytic continuation to Re λ > -ǫ and since M f,λ k = π * π * M f,λ k for Re λ ≫ 0, it follows that λ → M f,λ k has the desired analytic continuation. Moreover M f 0 = M f,λ 0 | λ=0 = π * (π * M f,λ 0 | λ=0 ) = π * (1 D f ) = 1 {f =0} . (4.6) M f k = M f,λ k | λ=0 = π * (π * M f,λ k | λ=0 ) = π * ([D f ]∧ω k-1 f ), k ≥ 1. (4.7)
Following for example [START_REF] Andersson | Residue currents of holomorphic sections and Lelong currents Arkiv för matematik[END_REF] one can check that for k ≥ 1, (4.8)

M f k = 1 Z (dd c log |f | 2 ) k and 1 X\Z (dd c log |f | 2 ) k = π * (ω k f ). It is not hard to see that M f,λ k is locally integrable for Re λ > 0 and that M f,λ k → M f k as measures when λ → 0 + .
M f k = lim ǫ→0 ǫ(dd c |f | 2 ) k (|f | 2 + ǫ) k+1 .
Remark 4.2. Given a tuple f as above, associated residue currents of Bochner-Martinelli type were introduced in [START_REF] Passare | Residue currents of the Bochner-Martinelli type[END_REF]. Let E be a trivial vector bundle with basis elements e 0 , . . . , e m and consider f = f 0 e 0 + • • • + f m e m as a section of the dual bundle E * with basis elements e * j . Following [START_REF] Andersson | Residue currents and ideals of holomorphic functions Bull[END_REF] one can define a residue current

R f = R f 0 + • • • + R f n , where R f k is a current of bidegree (0, k)
with values in the exterior product Λ k E, such that the coefficients in R f are precisely the currents in [START_REF] Passare | Residue currents of the Bochner-Martinelli type[END_REF]. It is proved in [START_REF] Andersson | Residue currents of holomorphic sections and Lelong currents Arkiv för matematik[END_REF] that

M f k = R f • (df ) k /(2πi) k k!
, where • denotes the natural contraction. For more details, see, e.g., [START_REF] Andersson | Residue currents of holomorphic sections and Lelong currents Arkiv för matematik[END_REF].

Products of Bochner-Martinelli currents

Given tuples f 1 , . . . , f r of holomorphic functions in X, we will give meaning to the product (5.1)

M fr ∧ • • • ∧M f 1 of Bochner-Martinelli currents. The construction is recursive. Assume that M f ℓ ∧ • • • ∧M f 1 is defined; it follows from the proof of Proposition 5.2 that (5.2) λ → M f ℓ+1 ,λ ∧M f ℓ ∧ • • • ∧M f 1 is holomorphic for Re λ > -ǫ, where ǫ > 0. Set (5.3) M f ℓ+1 ∧M f ℓ . . . ∧M f 1 := M f ℓ+1 ,λ ∧M f ℓ ∧ . . . ∧M f 1 λ=0 . We define the products M fr kr ∧ • • • ∧M f 1 k 1 in the analogous way so that (5.4) M fr ∧ • • • ∧M f 1 = kr,...,k 1 ≥0 M fr kr ∧ • • • ∧M f 1 k 1 .
Notice that if the f j are single functions, then M f j = 1 f j + [f j ] and

(5.5)

M fr ∧ • • • ∧ M f 1 = (1 fr + [f r ]) ∧ • • • ∧ (1 f 1 + [f 1 ]); cf. Section 3. Proposition 5.1. If h = (h 1 , . . . , h n ) is a Vogel sequence of some ideal with zero set Z at x, then M hn ∧ • • • ∧ M h 1 = [V h ].
Proof. In light of Lemma 2.2,

1 h ℓ • • • 1 h k+1 [h k ]∧ • • • ∧[h 1 ] = 1 Z [h k ]∧ • • • ∧[h 1 ]. Thus, by (3.4), [h ℓ+1 ]∧1 h ℓ • • • 1 h k+1 [h k ]∧ • • • ∧[h 1 ] = 0.
Hence, in view of (5.5),

(5.6)

M hn ∧ • • • ∧M h 1 = n k=0 1 hn • • • 1 h k+1 [h k ]∧ • • • ∧[h 1 ] = n k=0 1 Z [h k ]∧ • • • ∧[h 1 ]; here we have used that [h n ]∧ • • • ∧[h 1 ]
has support on Z. Now, Proposition 3.4 asserts that the right hand side of (5.6

) is equal to [V h ].
In contrast to the definition of products of residue currents of Bochner-Martinelli type introduced in [START_REF] Wulcan | Products of residue currents of Cauchy-Fantappiè-Leray type Ark[END_REF], the recursively defined products (5.1) are not commutative in general, not even if the tuples just consist of one single function. For instance, in C 2

x,y we have that M xy ∧M y = 0, whereas M y ∧M xy = [0], cf., Example 3.3. Various approaches to recursively defined products of residue currents are investigated in [START_REF] Lärkäng | Various approaches to products of residue currents[END_REF]. Proposition 5.2. Let f 1 , . . . , f r be tuples of holomorphic functions in X, with common zero set Z = {f 1 = . . . = f r = 0}. Then the current M fr ∧ • • • ∧M f 1 , defined by (5.3), is positive and has support on Z.

Let π : X → X be a normal modification such that the sheaves J (f ℓ ) • O X are principal for ℓ = 1, . . . r. As in Section 4, let D f ℓ and ω f ℓ be the corresponding divisors and negative Chern forms, respectively. Then

(5.7) M fr kr ∧ . . . ∧M f 1 k 1 = π * [D fr ]∧ • • • ∧[D f 1 ]∧ω kr-1 fr ∧ • • • ∧ω k 1 -1 f 1 ,
where, if k ℓ = 0, the factor [D f ℓ ] shall be replaced by 1 D f j and the factor ω k ℓ -1 f ℓ shall be removed.

Assume that g 1 , . . . g r are tuples of holomorphic functions in X such that |g ℓ | ∼ |f ℓ | for ℓ = 1, . . . , r. Then there is a normal current T with support on Z such that (5.8)

dd c T = M fr kr ∧ • • • ∧M f 1 k 1 -M gr kr ∧ • • • ∧M g 1 k 1 .
Proof. Iteratively using Lemma 3.1, the computation rules (3.4), and (4.4)-(4.7) we see that the desired analytic continuation of (5.2) exists and that (5.7) holds. It follows that M fr kr ∧ . . .

∧M f 1 k 1 has its support contained in π(|D fr | ∩ • • • ∩ |D f 1 |) = Z. Moreover M fr kr ∧ . . . ∧M f 1
k 1 is the push-forward of a product of positive (1, 1)-currents and positive forms, and hence it is positive.

To prove the last part, it suffices to change one of the

f ℓ to g ℓ with |g ℓ | ∼ |f ℓ |. First notice that then M f ℓ 0 = 1 f ℓ = 1 g ℓ = M g ℓ 0 .
Let us then assume that k ℓ ≥ 1, and that the modification π is chosen so that also J (g ℓ )•O X is principal. By Remark 4.1, there is a smooth global function γ on X such that ω f ℓ -ω g ℓ = dd c γ and thus we can find a smooth global form w such that dd c w = ω k ℓ -1

f ℓ -ω k ℓ -1 g ℓ . Let T := π * τ r ∧ • • • ∧τ ℓ+1 ∧[D f ℓ ]∧w∧τ ℓ-1 ∧ • • • ∧τ 1 ), where τ j = 1 D f j if k j = 0 and τ j = [D f j ] ∧ ω k j -1 f j otherwise. Then T satisfies (5.8). Note that τ r ∧ • • • ∧τ ℓ+1 ∧[D f ℓ ]∧w∧τ ℓ-1 ∧ • • • ∧τ 1 is normal,
and since normality is preserved under push-forward, so is T .

We also define products of Bochner-Martinelli currents and Lelong currents. If f 1 , . . . , f r are tuples of holomorphic functions in X and Z is an analytic subset of X, we define recursively M f 1 ∧[Z] := M f 1 ,λ ∧[Z] λ=0 , and

M f k+1 ∧ • • • ∧M f 1 ∧[Z] := M f k+1 ,λ ∧M f k ∧ • • • ∧M f 1 ∧[Z] λ=0 .
By arguments as in the proof of Proposition 5.2 one can prove that the desired analytic continuations exist, and thus

M fr ∧ • • • ∧M f 1 ∧[Z] is well-defined. It is readily checked that if i : Z ֒→ X, then, for any k 1 , ..., k r ∈ N, (5.9) M fr kr ∧ • • • ∧M f 1 k 1 ∧[Z] = i * [M i * fr kr ∧ • • • ∧M i * f 1 kr ]. Moreover, if Z = Z ′ + Z ′′ , Z ′′ ⊂ {f j = 0}
, and k j > 0, then one checks that (5.10)

M fr kr ∧ • • • ∧M f 1 k 1 ∧[Z] = M fr kr ∧ • • • ∧M f 1 k 1 ∧[Z ′ ], cf.
(the first equality of) (3.4).

For future reference, note that if f is a tuple of holomorphic functions on the analytic space X then (5.11)

M f = M f 1 X = j M f 1 X j ,
where X j are the irreducible components of X.

Proposition 5.3. Let f 1 , . . . , f r be tuples of holomorphic functions in X and let ξ be a tuple of holomorphic functions such that {ξ = 0} = {x}, where x ∈ X. Then (5.12)

M ξ ∧M fr kr ∧ • • • ∧M f 1 k 1 = M ξ n-k ∧M fr kr ∧ • • • ∧M f 1 k 1 = α[x], where k = k 1 + • • • + k r and α is a non-negative integer. If ξ generates the maximal ideal at x ∈ X, then α = ℓ x M fr kr ∧ • • • ∧M f 1 k 1 .
Proof. By Proposition 5.2, M ξ n-k ∧M fr kr ∧ • • • ∧M f 1 k 1 is positive and has support at x, and thus by Lemma 2.2 it is of the form α[x] for some non-negative α. Let π : X → X be a normal modification such that J (f ℓ ) • O X and J (ξ) • O X are principal. Let us use the notation from Section 4. Then, from (5.7), we see that α is an intersection number and hence an integer. Now assume that ξ generates the maximal ideal at x and that i : X ֒→ C N is a local embedding such that i(x) = 0, so that i * [x] = [0]. By the second part of Proposition 5.2 we may assume that f j = i * F j and ξ = i * z for some tuples F j and the standard coordinate system z = (z 1 , . . . , z N ) in C N . Then (5.13) i

* (M ξ n-k ∧M fr kr ∧ • • • ∧M f 1 k 1 ) = M z n-k ∧M Fr kr ∧ • • • ∧M F 1 k 1 ∧[X], cf.
(5.9). By Lemma 2.1, the right hand side of (5.13) is precisely the Lelong number of

M Fr kr ∧ • • • ∧M F 1 k 1 ∧[X] at 0 in C N times [0]. Proposition 5.4. The Lelong number at x of M fr kr ∧ • • • ∧M f 1 k 1 is unchanged if we replace f j by g j such that |f j | ∼ |g j |.
Proof. It follows from the second part of Proposition 5.2, applied to f 1 , . . . , f r , ξ since then T , which has bidegree (n -1, n -1), must vanish by Lemma 2.2.

One can replace all the evaluations in the definition of the product by one single evaluation in the following way; for the proof see [START_REF] Andersson | One parameter regularizations of products of residue currents[END_REF]. Proposition 5.5. Assume that µ j are strictly positive integers such that

µ 1 > µ 2 > . . . > µ r . Then λ → M fr,λ µr kr ∧ • • • ∧M f 1 ,λ µ 1 k 1 is holomorphic in a neighborhood of the half-axis [0, ∞) in C and (5.14) M fr kr ∧ • • • ∧M f 1 k 1 = M fr,λ µr kr ∧ • • • ∧M f 1 ,λ µ 1 k 1 λ=0
. In view of Proposition 5.1 we get Corollary 5.6. If h 1 , . . . , h n is a Vogel sequence of some ideal at some point x and µ j are as in Proposition 5.5, then the Lelong current of the associated Vogel cycle is given as the value at λ = 0 of the function

λ → n k=1 M h k ,λ µ k = n k=1 1 -|h k | 2λ µ k + ∂|h k | 2λ µ k ∧∂ log |h k | 2 /2πi

Bochner-Martinelli currents and Vogel cycles

For a tuple f = (f 0 , . . . , f m ) of holomorphic functions and β = [β 0 : . . . :

β m ] ∈ P m we write β • f := β 0 f 0 + • • • + β m f m . Note that M β•f
only depends on β ∈ P m and not on the choice of homogeneous coordinates. Our first result in this section relates Lelong numbers of Bochner-Martinelli currents to multiplicities of Vogel cycles. Theorem 6.1. Let f = (f 0 , . . . , f m ) be a tuple of holomorphic functions in X, pick x ∈ X, and let Z = {f = 0}. Then for k ≥ 0, and a generic choice of α = (α 1 , . . . , α k ) ∈ (P m ) k , (6.1)

ℓ x 1 Z [α k • f ]∧ • • • ∧[α 1 • f ] = ℓ x (M f k ) .
Here the current on the left hand side of (6.1) should be interpreted as

1 Z if k = 0.
Assume that f = (f 0 , . . . , f m ) generates the ideal sheaf J . Then α 1 • f, . . . , α n • f is a Vogel sequence for a generic choice of α = (α 1 , . . . , α n ) ∈ (P m ) n . By Theorem 6.1 and Proposition 3.4, (6.2)

mult x V α•f k = ℓ x (M f k )
and by Proposition 5.4 the right hand side only depends on J and not on the particular choice of generators f ; in fact, it only depends on the integral closure of J . Thus this gives an independent proof of Gaffney-Gassler's result, [10, Section 2], that the multiplicities of Vogel cycles V h are independent of h for generic h, which guarantees that the Segre numbers are well-defined. Also, (1.6) immediately follows from (6.2).

Proof. Choose a normal modification π : X → X such that J (f ) • O X is principal; we will use the notation from Section 4. Assume moreover that the pullback of the maximal ideal at x is principal, and let D ξ and ω ξ be the corresponding divisor and Chern form, obtained from a tuple ξ that defines the maximal ideal at x.

Let W be any irreducible subvariety of X. Since f ′ is nonvanishing on W it follows that for β outside a hypersurface in P m , the section β • f ′ is not vanishing identically on W . By induction it follows that there is a Zariski-open dense subset A ⊂ (P m ) n such that for each α = (α 1 , . . . , α n ) ∈ A, the sequence

α 1 • f ′ , . . . , α n • f ′ is a geometrically regular sequence on each component of X, |D f |, |D ξ |, and on the support of [D ξ ]∧[D f ]. Since π * (α ℓ • f ) = f 0 α ℓ • f ′ , we have that [α ℓ • f ] = π * [D f ] + [α ℓ • f ′ ] and if α ∈ A, in light of (3.4), thus [α 2 • f ]∧[α 1 • f ] = π * [D f ]∧[α 1 • f ′ ] + [α 2 • f ′ ]∧[α 1 • f ′ ] . By induction, (6.3) [α k • f ]∧ • • • ∧[α 1 • f ] = π * [D f ]∧[α k-1 • f ′ ]∧ • • • ∧[α 1 • f ′ ] + [α k • f ′ ]∧ • • • ∧[α 1 • f ′ ] ,
and so (6.4)

1 Z [α k • f ]∧ • • • ∧[α 1 • f ] = π * [D f ]∧[α k-1 • f ′ ]∧ • • • ∧[α 1 • f ′ ] .
Here we have used that

1 D f [α k • f ′ ]∧ • • • ∧[α 1 • f ′ ]
vanishes by Lemma 2.2, and that (6.5)

1 Z (π * τ ) = π * (1 D f τ ).
For k = 0, 1, (6.1) follows from (4.6), (4.7) and (6.4); in fact, the currents in (6.1) coincide in these cases. Let us now assume that k ≥ 2. We claim that there is a normal current A k such that (6.6)

dd c A k = [D ξ ]∧ω n-k-1 ξ ∧[D f ]∧ ω k-1 f -[α k-1 • f ′ ]∧ • • • ∧[α 1 • f ′ ] . For ℓ = 1, . . . , k, log |α ℓ • f ′ | 2 defines a singular metric on L -1 with first Chern form [α ℓ • f ′ ], cf., (4.2), and thus [α ℓ • f ′ ] is dd c -cohomologous to ω f . More precisely, c ℓ := log(|f ′ | 2 /|α ℓ • f ′ | 2 ) is a global current on X and ω f -[α ℓ • f ′ ] = dd c c ℓ . Now, let A k := [D ξ ]∧ω n-k-1 ξ ∧[D f ]∧ k-1 ℓ=1 ω k-ℓ-1 f ∧c ℓ ∧[α ℓ-1 • f ′ ]∧ • • • ∧[α 1 • f ′ ] .
Then A k is normal. Since α ℓ • f ′ does not vanish identically on any irreducible component of (the support of

) [D ξ ]∧[D f ]∧[α ℓ-1 • f ′ ]∧ • • • ∧[α 1 • f ′ ]
it follows from Lemma 3.1 and the discussion after the proof of that lemma that (6.6) holds. From (the proof of) Proposition 5.3 and (6.4) we get (6.7)

dd c π * (A k ) = ℓ x (M f k ) -ℓ x (1 Z [α k • f ]∧ • • • ∧[α 1 • f ]) [x]
. On the other hand, π * A k is a normal (n-1, n-1)-current, and so since it has support at x, it vanishes according to Lemma 2.2.

Our next result concerns mean values of Bochner-Martinelli currents. In particular, it says that M f can be represented a mean value of Vogel cycles. Theorem 6.2. Assume that f = (f 0 , . . . , f m ) is a tuple of holomorphic functions on X. Then (6.8)

M f k = α∈(P m ) k 1 Z [α k • f ]∧ • • • ∧[α 1 • f ]
where Z = {f = 0}. Moreover, if ν ≥ min(m + 1, n + 1), then (6.9)

M f = α=(α 1 ,...,αν )∈(P m ) ν M αν •f ∧ • • • ∧M α 1 •f .
For the proof we will use the following lemma which is a simple variant of Crofton's formula that should be well-known so we omit the proof, see also [START_REF] Andersson | Non-proper intersections and positive currents I[END_REF]. Lemma 6.3. If φ is a non-vanishing holomorphic (m + 1)-tuple on X, then, in the sense of currents,

β∈P m [β • φ]dσ(β) = dd c log |φ| 2 ,
where dσ is the normalized Fubini-Study metric.

Proof of Theorem 6.2. We use the same notation as in the proof of Theorem 6.1. In view of Lemma 6.3 and (4.2) we have that (6.10)

β∈P m [β • f ′ ]dσ(β) = ω f .
Since all currents are positive we can apply Fubini's theorem and get (6.8) from (6.4) by repeated use of (6.10), cf., (4.7).

We now prove (6.9). By (4.6) and (4.7),

M α ℓ •f = M α ℓ •f 0 + M α ℓ •f 1 = 1 α ℓ •f + [α ℓ • f ].
As in the proof of Proposition 5.1 we get, for generic (α 1 , . . . , α ν ) ∈ (P m ) ν , that

M αν •f ∧ • • • ∧M α 1 •f = ν j=0 1 Z [α j • f ] ∧ • • • ∧ [α 1 • f ] + 1 X\Z [α ν • f ] ∧ • • • ∧ [α 1 • f ].
Moreover, it follows from (6.3) and (6.4) that

1 X\Z [α ν • f ] ∧ • • • ∧ [α 1 • f ] = π * [α ν • f ′ ] ∧ • • • ∧ [α 1 • f ′ ] .
Hence, using (6.8) and Lemma 6.3, we conclude that α=(α 1 ,...,αν )∈(P m ) ν

M αν •f ∧ • • • ∧M α 1 •f = M f + π * (dd c log |f ′ | 2 ) ν = M f ; indeed, (dd c log |f ′ | 2 ) ν = 0 since ν ≥ min(m + 1, n + 1).
By arguments as in the proof of Theorem 6.2 one can check that (6.11)

α∈(P m ) k 1 X\Z [α k • f ]∧ • • • ∧[α 1 • f ] = 1 X\Z (dd c log |f | 2 ) k .
7. Proof of the generalized King formula (Theorem 1.1)

Let X and J be as in Theorem 1.1 and let Z be the variety of J . The (Fulton-MacPherson) distinguished varieties associated with J are defined in the following way, cf., [START_REF] Fulton | Intersection theory[END_REF]: Let ν : X + → X be the normalization of the blow-up of X along J and let E be the exceptional divisor of ν. Then Z j ⊂ X is a distinguished variety if it is the image under ν of an irreducible component of E. Let Z k j be the distinguished varieties of codimension k. Also, we define the irreducible components of X contained in Z to be distinguished varieties (of codimension 0).

Let us first consider the case k = 0. By (5.11) we may assume that X is irreducible. Then either J = (0) or Z is a proper subvariety of X. In the first case M f 0 = M 0 0 = 1 X and if h is a Vogel sequence of J , then necessarily h = (0, . . . , 0) and so V h = V h 0 = X. In the second case M f 0 = 0 and if h is a Vogel sequence of J , then V h 0 = X Z 0 = 0, since X ⊂ Z. It follows that Theorem 1.1 holds for k = 0. Next, consider the case k ≥ 1. Let π : X → X be a normal modification such that J • O X principal. We use the notation from Section 4, so that

M f k = π * ([D]∧ω k-1 f ),
where D = D f . Moreover, we let D k denote the components of D that are mapped to sets of codimension k in X. Note that

D = D p + . . . + D n , if p = codim Z. If ℓ > k, then π * ([D ℓ ]∧ω k-1 f
) is a positive closed (k, k)-current with support on a variety of codimension ℓ > k, and hence it must vanish in view of Lemma 2.2. Thus (7.1)

M f k = S f k + N f k , where (7.2) S f k = π * [D k ]∧ω k-1 f , N f k = π * ℓ<k [D ℓ ]∧ω k-1 f .
Note that M f k = 0 for k < p and N f p = 0. We claim that (7.1) is the Siu decomposition of M f k , cf., Section 2.2. By Lemma 2.2, S f k is the Lelong current of a cycle of codimension k, so it is enough to show that N f k does not carry any mass on varieties of codimension k. Let W ⊂ X be such a variety. By (6.5),

(7.3) 1 W π * ([D ℓ ]∧ω k-1 f ) = j π * (1 π -1 W [D ℓ j ]∧ω k-1 f ),
where D ℓ j are the irreducible components of D ℓ . Then, since ℓ < k, π -1 (W ) does not contain any component D ℓ j , thus each term in the right hand side of (7.3) vanishes, and thus the claim follows.

Since (7.1) is the Siu decomposition of M f k , it follows that S f k is independent of π : X → X. If we take π to be the normalization of the blow-up of J , we see that the Z k j in (1.5) has to be among the distinguished varieties of J . By Proposition 5.3 (for r = 1), the Lelong number of M f k is an integer at each point, and since the Lelong number of N f k generically vanishes on each Z k j , we conclude that the β k j and ℓ x (N f k ) are integers. That ℓ x (N f k ) is an integer can also be seen directly by copying the proof of Proposition 5.3. Moreover, cf., Proposition 5.4, β k j and ℓ x (N f k ) only depend on the integral closure of J at x.

We shall now see that the coefficients β k j of the distinguished varieties are, in fact, ≥ 1, following the proof of Corollary 5.4.19, in [START_REF] Lazarsfeld | Positivity in Algebraic Geometry I. Classical setting: line bundles and linear series Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]. The blow-up π J : Bl J X → X of X along J can be seen as the subvariety of X × P m t defined by the equations t j f k -t k f j = 0, where 0 ≤ j < k ≤ m. Moreover, the line bundle associated with the exceptional divisor is the pullback of O P m t (-1) from P m to Bl J X, so ω t = dd c log |t| 2 represents minus its first Chern class. This form is strictly positive on the fibers of π J , and since the normalization X + → Bl J X is a finite map, the pullback ω of ω t to X + remains strictly positive on the fibers of ν : X + → X as well. Let E j be one of the irreducible component of the exceptional divisor of ν. We conclude that ν

* ([E j ]∧ω k-1 ) is a positive integer times [Z k j ]
, where Z k j := ν(E j ). On the other hand, this current is unaffected if we replace ω by ω f since these two forms are first Chern forms of the same line bundle. It follows that β k j ≥ 1. We saw in the discussion after Theorem 6.1 that ℓ x (M f k ) is equal to the k:th Segre number of J at x. Next, we show that the fixed Vogel components of J are precisely the S f k . Fix a point x ∈ X. As in proof of Theorem 6.1 we can construct, for k ≥ 1 and a generic α ∈ (P m ) n , a normal current A k with support on |D k | such that

dd c A k = [D k ]∧([α k-1 • f ′ ]∧ • • • ∧[α 1 • f ′ ] -ω k-1 f ). Now π * A k is a normal (k -1, k -1)
-current with support on j Z k j , and thus it vanishes by Lemma 2.2. It follows that π

* [D k ]∧[α k-1 • f ′ ]∧ • • • ∧[α 1 • f ′ ]) = S f
k and hence S f k occurs in a generic Vogel cycle at x, meaning that S f k is a fixed Vogel cycle. On the other hand, the cycles

(7.4) π * ( ℓ<k [D ℓ ]∧[α k-1 • f ′ ]∧ • • • ∧[α 1 • f ′ ])
must be moving. Indeed, by (the proof of) Theorem 6.2, taking mean values of (7.4) over all α ∈ (P m ) k , we get the current N f k , which carries no mass on any variety of codimension k, as seen above.

By arguments as in the proof of Theorem 6.1 one shows that for a generic choice of α ∈ (P m ) k , (7.5)

ℓ x (1 X\Z [α k • f ]∧ • • • ∧[α 1 • f ]) = ℓ x (1 X\Z (dd c log |f | 2 ) k )
cf. (6.11). However, it follows from Proposition 3.4 that the left hand side of (7.5) is equal to m k (x). This concludes the proof of Theorem 1.1.

Remark 7.1. One can see more directly that only the distinguished varieties occur in S f k if S f k is defined by (7.2) from an arbitrary normal modification π : X → X. To begin with, π factors over ν, i.e., there exists a modification ν :

X → X + such that π = ν • ν. If ω + is the form associated with J • O X + in X + , then ν * ω + = ω f . Let D k j be an irreducible component of the divisor D k . Since |D k j | ⊂ π -1 (Z), it follows that ν(|D k j |) is contained in one of the components E j of E in X + . If ν(|D k j |) has codimension ≥ 1 in E j , then ν * ([D k j ]∧ω k-1 f ) = ( ν * [D k j ])∧ω k-1 + vanishes by Lemma 2.2. Hence π * ([D k j ]∧ω k-1 ) = ν * ν * ([D k j ]∧ω k-1 ) vanishes unless ν(|D k j |) = E j , in which case π(|D k j |) is a distinguished variety.
Assume that f 0 , . . . , f p-1 is a regular sequence. From the theory for proper intersections we know that

[f p-1 ]∧ • • • ∧[f 0 ] = β j [Z j ]
where Z j are the irreducible components of Z = {f = 0}, and that the intersection only depends on the ideal generated by the f j , cf., Remark 1.2. In particular, the right hand side is unaffected if we replace f j by α j • f for generic α j . From (6.8) we conclude that

(7.6) M f p = [f p-1 ]∧ • • • ∧[f 0 ].
Proof of Corollary 1.3. If Z is smooth, then locally there are coordinates (z, w) so that Z = {w 1 = • • • = w p = 0}. In view of (7.6) we have that

M w p = [w p ]∧ • • • ∧[w 1 ] = [Z],
and hence ℓ x (M w p ) = 1 for x ∈ Z. If Z is reduced and f generates the ideal J Z , therefore ℓ x (M f p ) = 1 for all smooth points x ∈ Z in view of Proposition 5.4. From Theorem 1.1 we know that M f p = β j [Z j ]. Since the smooth points are dense, we conclude that β j = 1 for each j. is a Vogel sequence of J for a generic choice of α, we can choose (α j ) j∈N in (P m ) n such that (α j ) j∈N → α and such that α j • f is a Vogel sequence of J for each j, and moreover, by Theorem 6.1, such that mult x V α j •f = e(x). It then follows that, for k ≤ r,

(8.3) ℓ x (1 Z [α k • f ]∧ • • • ∧[α 1 • f ]) ≤ e k (x) = ℓ x (1 Z [α j k • f ]∧ • • • ∧[α j 1 • f ]). We claim that (8.4) lim j→∞ [α j k • f ]∧ • • • ∧[α j 1 • f ] = [α k • f ]∧ • • • ∧[α 1 • f ] for k ≤ r.
(1 X\Z [α j k • f ]∧ • • • ∧[α j 1 • f ]) = 1 X\Z [α k • f ]∧ • • • ∧[α 1 • f ].
(1 Z [α j r • f ]∧ • • • ∧[α j 1 • f ]) = 1 Z [α r • f ]∧ • • • ∧[α 1 • f ].
By semicontinuity, (2.1), the Lelong number of the right hand side of (8.6) is greater than or equal to the Lelong number of

1 Z [α j r • f ]∧ • • • ∧[α j 1 • f ]. Thus, mult x V α•f
r ≥ e r (x), which gives a contradiction. Hence min lex mult x V α•f = e(x).

Given a positive closed current v, we define ℓ x (v) := (ℓ x0 , ..., ℓ xn ), where ℓ xk denotes the Lelong number at x of the component of v of bidegree (k, k). If v and w are positive and closed, we let v ≤ x w mean that ℓ x (v) ≤ lex ℓ x (w), and v = x w means that ℓ x (v) = ℓ x (w). Observe that if h is a Vogel sequence of an ideal J x , then the zero sets of h and J x coincide. If f 1 , . . . , f n is Vogel sequence of an ideal J x , then in view of Theorems 1.1 and 8.1 and Proposition 5.1, M f ≤ x M fn ∧ . . . ∧M f 1 . In fact we have: Proposition 8.3. Let f 1 , . . . , f s be a sequence of elements in O X,x and let f = (f 1 , . . . , f s ). Then (8.7)

M f ≤ x M fs ∧ . . . ∧M f 1 .
Proof. Let Z := {f = 0}. In order to prove (8.7), we proceed by induction on the number s of functions. Clearly (8.7) holds for s = 1, so assume that it holds for s -1 instead of s. Let f := (f 2 , . . . , f s ). By (5.11) we may assume that X is irreducible and that f 1 does not vanish identically on X, so that

M f 1 = M f 1 1 = [f 1 ]; otherwise M f 1 = M f 1 0 = 1 X and M f = M f
and we are back in the case s -1. Let [W ] := [f 1 ], and let i W j : W j ֒→ X be the irreducible components of W X\Z . Theorem 6.1 asserts that for a generic choice of α ∈ (P s-2 ) n-1 , α • f is a Vogel sequence 6 of J (i * W j f ) and

M α n-1 • f ∧ • • • ∧M α 1 • f = x M f on each W j , so that M α n-1 • f ∧ • • • ∧M α 1 • f ∧[W X\Z ] = x M f ∧[W X\Z ],
cf. the discussion after Lemma 2.2. By the induction hypothesis

M f ∧[W X\Z ] ≤ x M fs ∧ • • • ∧M f 2 ∧[W X\Z ].
In view of (5.9) and (5.10), since [f 1 ] = [W Z ] + [W X\Z ] and f vanishes on Z, we get (8.8)

M α n-1 • f ∧ • • • ∧M α 1 • f ∧M f 1 ≤ x M fs ∧ • • • ∧M f 2 ∧M f 1 .
For a generic choice of α, the sequence f 1 , α 1 • f , . . . , α n-1 • f is a Vogel sequence of J (f ). Thus, by Theorem 8.1, (8.9)

M f ≤ x M α n-1 • f ∧ • • • ∧M α 1 • f ∧M f 1 .
Combining (8.8) and (8.9), we get (8.7).

An invariance property

We have the following invariance property of Segre numbers. In the setting when J is the pullback to X of the radical sheaf of a smooth manifold in some ambient space this was proved in [START_REF] Rams | On the intersection product of analytic cycles Ann[END_REF]Section 5].

Proposition 9.1. Let J be an ideal sheaf on an analytic space X, let J ′ be the pullback of J to X × C w under the projection (x, w) → x, and let i : X ֒→ X × C w be the embedding x → (x, 0). Then

(9.1) e k (J ′ , X × {0}, i(x)) = e k (J , X , x) and (9.
2) e k+1 (J ′ + (w), X × C w , i(x)) = e k (J , X, x).

For the proof we will use the following invariance of Bochner-Martinelli currents.

Lemma 9.2. Let f be a tuple of holomorphic functions on X and let i : X ֒→ X ×C w be the embedding x → (x, 0). Then M (f,w) 0 = 0 and

(9.3) M (f,w) k+1 = i * M f k , k ≥ 0. Moreover, if W ⊂ X is an analytic variety, (9.4) M f ⊗1 k ∧[W × {0}] = i * (M f k ∧[W ]
). If we consider X as embedded in some larger analytic space X ′ and i :

X ′ → X ′ × C w , x → (x, 0), then (9.3) reads M (f,w) k+1 ∧[X × C w ] = i * (M f k ∧[X]). In particular, if f = 0, (9.5) M w 1 ∧[X × C w ] = i * [X] = [X × {0}]. Proof.
Let z be local coordinates on X. Since (z, w) → (f (z), w) does not vanish identically on X × C w , it follows that M (f,w) 0 = 0. Let us now prove (9.3). First consider the case when k = 0. By (5.11) we may assume that X is irreducible. Then either f ≡ 0 on X or the zero set of f has at least codimension 1 in X. In the first case

M (f,w) 1 = M w 1 = [w] = i * 1 = i * M 0 0 = i * M f 0 .
In the latter case the zero set of (f, w) has at least codimension 2 on X × C w , and and so both sides of (9.3) vanish by Lemma 2.2. Thus (9.3) holds for k = 0. Next let π : X → X be a smooth modification such that J • O X is principal and moreover f 0 is locally a monomial; use the notation from Section 4. Observe that then π ⊗ id w : X × C w → X × C w is a smooth modification with the same properties. It follows that it is enough to prove (9.3) in case X is smooth, J = (f 0 ) is principal and f 0 is (in local coordinates) a monomial.

In light of Section 4 we thus have to show that

(9.6) (2πi) -1 ∂(|f | 2 + |w| 2 ) λ ∧∂ log(|f | 2 + |w| 2 )∧(dd c log(|f | 2 + |w| 2 )) k is equal to [f 0 ]∧(dd c log |f ′ |) k-1 ∧[w] = i * M f k when λ = 0. Indeed, at λ = 0, (9.6) is equal to M (f,w)
k+1 . Note that (9.6) is locally integrable for Re λ > 0. Moreover, if Re λ < 1, it is integrable in the w-direction and thus acts on forms that are just bounded in the w-direction. Since M (f,w) k+1 is of order zero and suppM (f,w) k+1 ⊂ {w = 0}, it follows that to check the action of M (f,w) k+1 on test forms, it is enough to consider forms ξ(z, w) = ξ(z), where ξ(z) is any test form in X. However, after the (generically 1 -1) change of variables

f 0 ω = w, so that |f | 2 + |w| 2 = |f 0 | 2 (|f ′ | 2 + |ω| 2 ), the action of (9.6) on ξ is equal to (2πi) -1 z,ω ∂|f 0 | 2λ (|f ′ | 2 + |ω| 2 ) λ ∧∂ log |f 0 | 2 ∧(dd c log(|f ′ | 2 + |ω| 2 )) k ∧ ξ(z).
Taking λ = 0, we get (9.7)

z [f 0 ]∧ ξ(z)∧ ω (dd c log(|f ′ | 2 + |ω| 2 )) k .
One can check that the inner integral in (9.7) is equal to (dd c log |f ′ | 2 ) k-1 , which proves (9.3). Finally we prove (9.4). Let j : W ֒→ X. Then, using (5.9),

M f ⊗1 ∧[W × {0}] = i * j * M j * i * f ⊗1 = i * j * M j * f = i * M f ∧[W ].
Proof of Proposition 9.1. Since the pullback of J ′ to X ≃ X × {0} is just J , (9.1) should be clear. More formally: Let f be a tuple that defines the ideal sheaf J in ,X,x), where we have used (9.4) for the second equality. This proves (9.1).

X. Then f ⊗ 1 defines J ′ in X × C w and e k (J ′ , X × {0}, x) = ℓ x (M f ⊗1 k ∧[X × {0}]) = ℓ x (M f k ∧[X]) = e k (J
To see (9.2) notice that f, w defines J + (w) in X × C w . Thus, using (5.9) and (9.3) we have

e k+1 (J ′ + (w), X × C w , (x, 0)) = ℓ x (M (f,w) k+1 ∧[X × C w ]) = ℓ x (M f k ∧[X]) = e k (J , X, x).

Local intersection numbers

Tworzewski's original motivation for introducing the extended index of intersection was to understand intersection theory in the nonproper case. Let Z 1 , . . . , Z r be subvarieties of a smooth manifold Y that do not necessarily intersect properly.

A standard procedure to define an intersection product Z 1 • • • Z r is to give some reasonable meaning to the intersection

(10.1) ∆ • Z 1 × • • • × Z r , where i : Y ≃ ∆ → Y × • • • × Y is the diagonal in Y × • • • × Y .
In this way one is reduced to the case of two varieties one of which is smooth. Now assume that A, Z are subvarieties of Y , that A is smooth, and (initially) that Z has pure dimension. Let J A denote the radical sheaf of A, and also, for simplicity, the pullback of J A to Z. Following Tworzewski we define local intersection numbers

(10.2) g ℓ (A, Z, x) := e dim Z-ℓ (J A , Z, x) at x. If Z = j α j Z j ,
where the Z j are pure dimensional, we set g ℓ (A, Z, x) := j α j g ℓ (A, Z j , x). The change of indices is made so that ℓ corresponds to the generic multiplicity of components of dimension ℓ of Vogel cycles. We will use the notation A • Z for these lists of local intersection numbers, i.e., (10.3)

A • Z(x) = (g dim Z (A, Z, x), . . . , g 1 (A, Z, x), g 0 (A, Z, x))
The local multiplicities of intersection of general varieties Z 1 , . . . , Z r are then

(10.4) ǫ ℓ (Z 1 , . . . , Z r ; x) = g ℓ (∆, Z 1 × • • • × Z r , (x, . . . , x)),
cf. [START_REF] Tworzewski | Intersection theory in complex analytic geometry[END_REF]Section 6]. We will write

(10.5) Z 1 ⋄ • • • ⋄ Z r (x) = (ǫ ν (Z 1 , . . . , Z r ; x), . . . , ǫ 1 (Z 1 , . . . , Z r ; x), ǫ 0 (Z 1 , . . . , Z r ; x)),
where ν is the dimension of the set-theoretical intersection

Z 1 ∩ • • • ∩ Z r . Note that the product Z 1 ⋄ • • • ⋄ Z r by definition is commutative. It is also inde- pendent of the manifold Y in the following sense: If ι : Y → Y is an embedding of Y in a larger manifold Y , then (10.6) Z 1 ⋄ • • • ⋄ Z r (x) = ι(Z 1 ) ⋄ • • • ⋄ ι(Z r ) ι(x) .
This follows from Proposition 9.1, see also [START_REF] Rams | On the intersection product of analytic cycles Ann[END_REF]Section 5]. The next result, which relates the two local intersections (10.3) and (10.5), we have not found in the literature. Proposition 10.1. Assume that A, Z are subvarities of a manifold Y , and that A is smooth. Then

A ⋄ Z = A • Z.
In particular, if A and B are smooth submanifolds of Y , since A⋄B is commutative, it follows that A • B = B • A.

Proof. Fix x in Y . We may assume, without loss of generality, that Y = C N . Choose local coordinates z = (z ′ , z ′′ ) on C N so that A = {z ′ = 0}, and local coordinates (z, w) on C N × C N . We will show that (10.7) ǫ j (A, Z A ⋄ Z(x) = (mult x Z, 0, . . . , 0).

; x) = e dim A+dim Z-j (J ∆ , Z × A, x) = ℓ x (M z-w dim A+dim Z-j ∧[Z × A]
In the nonproper case the classical intersection product Z 1 • • • Z r in the sense of Fulton, [START_REF] Fulton | Intersection theory[END_REF], cannot represent the local intersection multiplicities in any reasonable sense. Indeed, in general

Z 1 • • • Z r is a cycle of codimension codim Z 1 +• • •+codim Z r , determined modulo rational equivalence on Z 1 ∩ • • • ∩ Z r .
Tworzewski, [START_REF] Tworzewski | Intersection theory in complex analytic geometry[END_REF]Sections 5,[START_REF] Andersson | One parameter regularizations of products of residue currents[END_REF], proves that there is unique cycle Z 1 • • • • • Z r , that he calls the intersection product, such that In a forthcoming paper we will introduce, in the case Y = P n , a global current that represents, at each point, the local intersection multiplicities, and respects Bezout's formula, in a reasonable sense. It is obtained as the mean value of various Vogel sequences, based on global variants of the ideas in Section 6 above.

Examples

Let us start by some computations of Bochner-Martinelli currents and Segre numbers. Our first example illustrates that the currents M f in general depend on the set of generators f although the Lelong numbers only depend on (the integral closure of) the ideal generated by f , cf. Remark 1.2. Here we have used that with the generic parametrization C ∋ t → [-t, 1] ∈ P 1 , we have

[α]∈P 1 [α 0 + α 1 y]dσ(α) = t∈C [y -t]∧ dt∧d t π(1 + |t| 2 ) 2 = dy∧dȳ π(1 + |y| 2 ) 2 .
Next we will discuss a simple example where a moving component occurs.

Example 11.2. Consider the tuple f = t 3 (t 1 , t 2 , t 3 ) = t 3 t in X = C 3 t , with zero set Z = {t 3 = 0}. We will compute the Segre numbers e k (0) = e k (J (f ), C 3 , 0), k = 0, 1, 2, 3. Let α • f be a Vogel sequence of J (f ) at 0 of the form α 1 • f, . . . , α 3 • f . Let us compute the corresponding Vogel cycle V α•f . First note that X X\Z 0 = X 0 = X. Thus, by Proposition 3.4,

[X 1 ] = M t 3 (α 1 •t) 1 = [t 3 ] + [α 1 • t] = [X Z 1 ] + [X X\Z 1
].

Furthermore, using (3.3) and (3.4), we get

[X 2 ] = M t 3 (α 2 •t) 1 ∧M t 3 (α 1 •t) 1 = [t 3 ]∧[α 1 • t] + [α 2 • t]∧[α 1 • t] = [X Z 2 ] + [X X\Z 2
] and

[X 3 ] = M t 3 (α 3 •t) 1 ∧M t 3 (α 2 •t) 1 ∧M t(α 1 •t) 1 = ([t 3 ] + [α 3 • t])∧[α 2 • t]∧[α 1 • t] = 2[0] = [X Z 3 ], for a generic α. Hence [V h ] = [V h 1 ] + [V h 2 ] + [V h 3 ] = [t 3 ] + [t 3 ]∧[α 1 • t] + 2[
0] and, in particular, e 0 (0) = 0, e 1 (0) = 1, e 2 (0) = 1, e 3 (0) = 2.

Observe that V h 1 and V h 3 are fixed, whereas V h 2 is moving. A computation, using Theorem 6.2 and Lemma 6.3, yields

M f 0 = 0, M f 1 = [t 3 ], M f 2 = [t 3 ]∧dd c log(|t 1 | 2 + |t 2 | 2 ), M f 3 = 2[0].
The following simple lemma is useful for computations.

Lemma 11.3. Let X and X ′ be two analytic spaces of dimension n, let τ : X ′ → X be a holomorphic map, and let f be a tuple of holomorphic functions on X. Assume that τ is proper, surjective, and generically r to 1. Then (11.1)

rM f k = τ * M τ * f k .

(1. 4 )

 4 (dd c log |f | 2 ) k := lim ǫ→0 dd c log(|f | 2 + ǫ) k ;

Remark 4 . 1 .

 41 For future reference, let g be a tuple of holomorphic functions such that |g| ∼ |f |, i.e., there exists C ∈ R such that |f |/C ≤ |g| ≤ C|f |, and let π : X → X be a normal modification such that both J (f ) • O X and J (g) • O X are principal. Then |f 0 f ′ | ∼ |g 0 g ′ | and since f ′ and g ′ are non-vanishing it follows that f 0 and g 0 define the same divisor on X. Therefore the corresponding negative Chern forms ω f and ω g are dd c -cohomologous, i.e., there is a global smooth function γ such that dd c γ = ω f -ω g .By combining [4, Proposition 3.2] and[START_REF] Samuelsson | Regularizations of products of residue and principal value currents[END_REF] Corollary 4] it follows that (4.9)

  ), cf. (5.9), coincides with(10.8) g j (A, Z, x) = e dim Z-j (J A , Z, x) = ℓ x (M z ′ dim Z-j ∧[Z]). Note that M z-w k ∧[Z × A] = M (z ′ ,z ′′ -w ′′ ) k ∧[Z × {w ′ = 0}]. Let (z ′ , z ′′ , w ′ , η ′′ ), where η ′′ = z ′′ -w ′′ , be new coordinates on C N × C N . Then (9.3) implies that M (z ′ ,η ′′ ) k+dim A ∧[Z×{w ′ = 0}] = i * M z ′ k ∧[Z×{w ′ = 0}], where i : C 2N -dim A z ′ ,z ′′ ,w ′ ֒→ C 2N z ′ ,z ′′ ,w ′ ,η ′′ . Moreover, by (9.4), M z ′ k ∧[Z × {w ′ = 0}] = j * M z ′ k ∧[Z], where j : C N z ′ ,z ′′ ֒→ C 2N -dim A z ′ ,z ′′ ,w ′ . Hence M z-w dim A+dim Z-j ∧[Z × A] = i * j * M z ′ dim Z-j ∧[Z]and thus (10.7) is equal to (10.8).Example 10.2. If Z j intersect properly, then also (10.1) is a proper intersection and it is well-known that Z 1 • • • Z r coincides with the intersection (10.1) (after identifying Y ≃ ∆). It follows thatǫ ℓ (Z 1 , . . . , Z r ; x) = mult x (Z 1 • • • Z r ) for x ∈ |Z 1 • • • Z r | and ℓ = dim(Z 1 • • • Z r ),and 0 otherwise, cf.[START_REF] Tworzewski | Intersection theory in complex analytic geometry[END_REF] Theorem 6.5]. In particular, if Z is a subvariety of the smooth manifold A, in view of (10.6),(10.9)

k

  mult x (Z 1 • • • • • Z r ) k = ℓ ǫ ℓ (Z 1 , . . . , Z r ; x)for each point x, where the index k denotes the component of dimension k. By definition this product respects the (sum of the) local intersection multiplicities, but it does not respect Bezout's formula in general. For instance, the self-intersection in P 2 of any smooth curve C is just C itself, and thus deg (C • C) = (deg C)2 unless C is a line.

Example 11 . 1 . 2 =

 1112 Let us consider the primary ideal (x) in C 2 x,y . We know from Corollary 1.3 and Remark 1.2, respectively, that M x 1 = [x] and that M x 2 = 0. Let us now consider the pair (x, xy) of generators for the same ideal. We first consider the Vogel cycles obtained from generic linear combinations of these generators. SinceZ = {x = 0} and [α 0 x + α 1 xy] = [x(α 0 + α 1 y)] = [x] + [α 0 + α 1 y] we have that 1 Z [α 0 x + α 1 xy] = [x]as expected, since [x] must be a fixed component in any Vogel cycle. A simple computation yields that1 Z [β 0 x + β 1 xy]∧[α 0 x + α 1 xy] = [x]∧[α 0 + α 1 y]for generic choices of α and β, cf. Section 3. Thus the component of the Vogel cycle of codimension 2 is non-vanishing for generic α, β. Taking mean values over P 1 we get, cf., Theorem 6.2,M x,xy [x]∧ dy∧dȳ π(1 + |y| 2 ) 2 .

  Since α j • f and α • f are Vogel sequences, the currents in (8.5) intersect properly with [α j k+1 • f ] and [α k+1 • f ], respectively. In light of [7, Chapter 2, Corollary 12.3.4] or [25, Theorem 3.6], (8.4) holds for k + 1, and the claim follows by induction.

	Proposition 8.2 and (8.3) imply that
	(8.6)	lim j→∞

We will often identify a cycle with its corresponding Lelong current.

By slight abuse of notation we will write [0], instead of the formally more correct [{0}], to denote point evaluation at 0.

For a more detailed exposition of currents on an analytic space we refer to[START_REF] Herrera | Residues and principal values on complex spaces Math[END_REF] Section 4.2].

Acknowledgement: We thank Terry Gaffney for fruitful discussions. We also thank the referee for important comments on a previous version. This work was partially carried out while the authors visited the Mittag-Leffler Institute.

The minimality property

Recall that the lexicographical order on R N is a total order, defined by (x 1 , . . . , x N ) ≤ lex (y 1 , . . . , y N ) if there is an 1 ≤ ℓ ≤ N such that x i = y i for i ≤ ℓ and x ℓ < y ℓ . We let min lex denote the minimum with respect to the lexicographical order. We will now give a proof that Tworzewski's extended index of intersection coincides with the list of Segre numbers. More precisely we will prove: Theorem 8.1. Let J be a coherent ideal sheaf on X and let e(x) be the list of associated Segre numbers at x. Then (1.3) holds, where the min lex is taken over all Vogel sequences h of ideals with the same integral closure as J x .

Moreover, if f is a tuple of generators of J (or any ideal with the same integral clousure as J ) then it suffices to take the min lex over all Vogel sequences of the form

As mentioned in the introduction, Theorem 8.1 is known in the case when J is the pullback to X of the radical sheaf of a smooth manifold A in some ambient space. For the proof of Theorem 8.1 we will need the following result; if Z is smooth this is precisely Theorem 3.4 in [START_REF] Tworzewski | Intersection theory in complex analytic geometry[END_REF]. Proposition 8.2. Assume that (W j ) j∈N and W are subvarieties of X of pure dimension such that lim j→∞ [W j ] = [W ] as currents on X. Let Z be a fixed subvariety of X, let x be a fixed point in Z, and assume that

for all j. Then there is a neighborhood U of x in X, in which lim j→∞ (1

Proof. Since the currents [W j ] are positive and locally uniformly bounded, so are the currents 1 Z [W j ]. Thus, there is a subsequence (1 Z [W j k ]) k∈N converging to a positive closed current with support on W ∩ Z. By Lemma 2.2 this current is the integration current [V ] for some effective cycle V (with possibly real coefficients).

By (8.1) and semicontinuity, (2.1), we have that

), and combined with (8.2) and the fact that V and W are effective cycles, it follows that

The last statement follows by complementarity.

Proof of Theorem 8.1. Since each Vogel sequence h can be realized as α • f for some choice of f and α, it is easy to check that the first statement follows from the second one. Let f be a tuple of generators of J . By definition, e(x) = mult x V α•f for almost all α, and thus it is enough to prove that e(x)

Suppose that e(x) ≤ lex min lex mult x V α•f . Then there is an r and a Vogel sequence

Moreover, if ξ is a tuple that defines the maximal ideal at x ∈ X, then the Segre numbers at x associated with J = J (f ) on X are given by

for test forms ψ. Taking analytic continuations to λ = 0, we get (11.1). In view of Proposition 5.5 we have

In particular, it follows from Lemma 11.3 that

Example 11.4. Let r, s be relatively prime integers and consider the cusp

Since we have the parametrization τ : t → (t s , t r ) of X, using (11.3) we get

This multiplicity is of course well-known, and can be computed in various other ways.

We will now proceed with some computations of local intersection numbers.

x , where m ≥ 1, and let A = {x 2 = x 3 = 0}. Since A is smooth and contained in X, and X is smooth outside the origin in C 3 , we must have that A ⋄ X(x) = (mult x A, 0, . . . , 0) for x = 0, cf., (10.9).

We shall now compute the local intersection numbers at 0. To this end we consider a generic Vogel sequence of J A on X at the origin and compute the corresponding Vogel cycle. Let H 1 be a generic hyperplane that contains A, defined by

2 and it has order m at the origin. We conclude that

As an illustration, let us also compute ǫ 0 (A, X, 0) = e 2 (J A , X, 0) as the Lelong number of a certain Bochner-Martinelli current. Notice that τ : (t

is the mean value of all

, by (3.3) and (3.4), we thus have that

The following example is related to Example 11.2 above.

Example 11.6. The mapping γ :

3 ) is proper and injective, so that X := γ(C 3 ) is a subvariety of C 6 . Let A = {z 3 = z 4 = z 5 = z 6 = 0}. Then A is smooth and contained in X and, since X is smooth outside 0, it follows from (10.9) that A ⋄ X(x) = (mult x A, 0, . . . , 0) for x = 0. We want to determine the local intersection numbers ǫ 3-k (A, X, 0) = e k (J A , X, 0) at 0. Since J A has codimension 1 in X, e 0 (0) = 0. Moreover, by King's formula, M z 3 ,z 4 ,z 5 ,z 6 1 is a Lelong current on X, and at a point x = 0 we know that the Lelong number is 1 if x ∈ A and 0 otherwise. We conclude that M z 3 ,z 4 ,z 5 ,z 6 1 = [A], and hence e 1 (1) = mult 0 A = 1. By Lemma 11.3,

where we have used that the ideal γ * z is generated by t 1 , t 2 , t 2 3 , that the ideal γ * J A is generated by t 3 (t 1 , t 2 , t 3 ), and that e k (A, X, 0) only depends on the ideals. In light of Example 11.2 thus, e 2 (A, X, 0) =

3 ) 1

where t ′ = (t 1 , t 2 ). To see the last equality, in view of Theorem 6.1, one can replace dd c log |t| 2 by a generic hyperplane [α • t]. In a similar way one concludes that e 3 (A, X, 0) = 2.

Institut de Mathématiques, Université Bordeaux 1, 33405, Talence, France E-mail address: Alain.Yger@math.u-bordeaux1.fr


G Salomon

L Fouché

S Lepoutre

A Aspect

T Bourdel

All optical cooling of 39 K to Bose Einstein condensation

We report the all-optical production of Bose Einstein condensates (BEC) of 39 K atoms. We directly load 3 × 10 7 atoms in a large volume optical dipole trap from gray molasses on the D1 transition. We then apply a small magnetic quadrupole field to polarize the sample before transferring the atoms in a tightly confining optical trap. Evaporative cooling is finally performed close to a Feshbach resonance to enhance the scattering length. Our setup allows to cross the BEC threshold with 3 × 10 5 atoms every 7s. As an illustration of the interest of the tunability of the interactions we study the expansion of Bose-Einstein condensates in the 1D to 3D crossover.

I. INTRODUCTION

As versatile, extremely well controlled, and isolated systems, ultracold atoms are widely used for quantum simulation [1]. A key feature in ultracold atoms experiments is that interactions are characterized by a unique parameter, i.e. the scattering length, which can be tuned using magnetic Feshbach resonances [2]. In the case of fermions, changing the interaction strength has allowed the study of the BEC-BCS crossover [3][4][5] as well as the thermodynamical properties of fermionic systems at unitarity [6][7][8]. The properties of Bose-Einstein condensates can also be greatly modified [9][10][11][12][13][14][15]. More recently, strongly interacting Bose gases have received special interest both from experimental and theoretical points of view [16][17][18][19][20][21].

Due to the existence of both fermionic and bosonic isotopes as well as broad Feshbach resonances at low magnetic field, potassium is an attractive alkali [22]. However the most common isotope -the bosonic 39 K -suffers from two major limitations. The first one is a narrow hyperfine structure that hinders sub-Doppler cooling using conventional techniques. It has recently been overcome by implementing either bright molasses close to resonance [23] or gray molasses on the D1 transition [24,25]. The second limitation is due to a small and negative background scattering length at zero magnetic field, which leads to a minimum of the collision cross section at relatively low energy [23]. As a result, evaporative cooling in a magnetic trap has proved to be inefficient. To circumvent these problems a solution is to use other species to sympathetically cool potassium to quantum degeneracy [13,26]. This however demands a second laser system. More recently, pioneering work combining a quadrupole trap and direct transfer to a deep optical dipole trap has led to the Bose Einstein condensation of 39 K in a single species experiment [27].

In this paper, we report the first production of a Bose Einstein condensate of 39 K using an all optical method [28,29], i.e. by direct loading of a dipole trap from gray molasses without the aid of another atomic coolant, nor pure magnetic trapping. Optical cooling and trapping allows us to use the magnetic field to adjust the scatter-ing length to an optimal value. This allows for a rapid production of degenerate ultracold gases with tunable interactions.

The paper is organized as follows. The direct loading of a large volume dipole trap from gray molasses is described in section II. The procedure to obtain a polarized and dense sample is then discussed in section III. The evaporation close to a Feshbach resonance is described in section IV. Finally, the tunability of the interaction is used to study the expansion of BEC in the 1D to 3D crossover in section V.

II. DIPOLE TRAP LOADING

The beginning of the experiment is described in [24]. 39 K atoms are collected from a vapor in a two dimensional (2D) magneto-optical trap (MOT), which loads a 3D-MOT in a second ultra high vacuum chamber. The 3D-MOT saturates at 5 × 10 9 atoms after a 2.5 s loading stage. The cloud is further compressed in a hybrid compressed MOT (CMOT) configuration combining light on the D1 and the D2 transitions. A gray molasses working on the D1 transition [24,30,31] then cools the gas down to T = 6 µK.

The atoms are loaded in a large dipole trap derived from a 30 W multimode IPG fiber laser ELR-30-1550-LP operating at 1550 nm. The 21 W beam is focused down to a waist radius at 1/e 2 of ω 1 = 150 µm and shined on the atoms at the beginning of the molasses phase [29,32,33]. It creates a U 1 = k B × 40 µK deep optical potential (k B is the Boltzmann constant). The highest trapped atom number of 3 × 10 7 is obtained for the laser parameters leading to the coldest gray molasses. This is expected as the induced light shifts of the order of a few megahertz are small compared to the laser detunings and moreover do not affect the Raman condition between the hyperfine ground states in the gray molasses [24].

As compared to a sudden loading at the end of the molasses phase, the superposition of the optical trap with the gray molasses leads to a doubling of the number of captured atoms. We cannot collect more atoms using longer superposition times than ∼ 10 ms, an effect which we interpret as a density dependent efficiency of the gray I. Experimental parameters during the steps preceding evaporation. We report the atom number N , the temperature T , the density n, the magnetic field gradient in the strong direction γ, and the Feshbach field B0.

molasses limited by photon reabsorption (above ∼ 2 × 10 11 cm -3 ) [34]. Our transferred atom number of 3 × 10 7 roughly matches an estimation given by the number of atoms being at a position where their kinetic energy does not exceed the local trap depth [35]. This is consistent with our optimization of the captured atom number by increasing the trap waist up to 150 µm. For even larger waists, gravity plays in important role and hinders an efficient trapping.

III. SAMPLE POLARIZATION AND COMPRESSION

During the gray molasses the atoms are distributed among the different F = 1 and F = 2 ground states. In the very last 0.5 ms of the gray molasses, the repumping light is switched off in order to optically pump the atoms in the F = 1 sublevels, forming a stable mixture. Due to the large Rayleigh length of the optical dipole trap of 4.5 cm, the atoms are not confined along the beam direction. The initial (rms) size of the cloud is in this direction is 1 mm, i.e. the size of the gray molasses. In order to perform an efficient evaporation at a high collision rate, we thus need to confine and compress the cloud to a higher density. This can for example be done by directly adding another crossing optical dipole trap [29]. In the case of potassium, this is not sufficient because of the scattering length a bg ≈ -30×a 0 where a 0 is the Bohr radius. Indeed this low and negative value implies the existence of a Ramsauer collision rate minimum at a relatively low collision energy of k B × 400 µK [27]. When compressing the gas, the temperature increases towards this energy and prevents the evaporation from being efficient.

To circumvent this problem, we take advantage of 39 K Feshbach resonances [22] to tune the scattering length. We thus need to polarize the gas in a single spin state because there is no magnetic field region where the scattering lengths for the different F = 1 sublevels are simultaneously favorable to initiate an efficient evaporation. Experimentally the optical pumping efficiency is limited to 60% due to a too high density at the end of the gray molasses. Moreover optical pumping leads to residual heating and to a factor 4 decrease in the atom number captured in the optical trap [36].

We thus adopt another strategy that is filtering out unwanted atomic levels. In practice, we apply, in addition to the optical trap, a small magnetic quadrupole field of 14 G.cm -1 using the MOT coils. In this configuration, only the atoms in the |F = 1, m F = -1 state experience magnetic confinement along the beam direction, while the others are lost due to a repulsive magnetic potential (see fig. 1). This configuration allows us, after 500 ms, to keep one third of the initially trapped atoms in a pure spin state.

A second 13W dipole trap beam with a 21 µm waist radius is then crossed at the centre of the large trap with an angle of 56 degrees [29]. The atoms from the first beam are transferred into this second highly confining trap on a time scale of 1 s. After 2s the magnetic quadrupole is switched off and we are left with N = 4×10 6 atoms in the |F = 1, m F = -1 state at 220 µK in this tightly confining optical trap (second beam). Since the temperature is then higher than the trap depth of the large trap, the latter has no important effect in the cloud trapping. At this stage, the trapping frequencies can thus be approximated by the ones of the confining trap, ω /2π = 145(7) Hz in the longitudinal direction and ω ⊥ /2π = 8.7(0.5) kHz in the radial one. The resulting phase space density D = N ω 2

⊥ ω (h/k B T ) 3 is 4.5 × 10 -4 . Note that, even in this highly confining dipole trap at 1550 nm, we do not observe any light induced losses. This is in contrast to the situation observed with a multimode laser at 1070 nm [27].

IV. EVAPORATIVE COOLING IN THE OPTICAL TRAP

In order to perform efficient evaporative cooling to quantum degeneracy, we first adjust the magnetic field to 550 G in order to tune the scattering length to a = 130×a 0 [22]. It increases the collision rate to 15 ms -1 . The forced evaporation proceeds as follows (see fig. 2). In the first part of the evaporation the power of the tightly confining trap is decreased by a factor ∼ 25 until the trap depths of the two beams are of the same order. In the second part of the evaporation the power of both beams is decreased simultaneously. The condensation transition is crossed with 3 × 10 5 atoms after 2 s of evaporation.

In order to evaluate the efficiency of our evaporation, we plot in figure 3 the phase space density D and the collision rate Γ = 2/π n 0 (8πa 2 ) k B T /m, where n 0 = N ω 2

⊥ ω (m/2πk B T ) 3/2 is the central density, and m the atomic mass [START_REF] Dalibard | Lectures notes on cold atoms at ENS[END_REF]. The temperature and the number of atoms in the gas are measured through the size of the cloud after a time of flight of 20 ms. The trap frequencies are deduced from the independently measured properties of the two dipole trap beams. In our analysis, we assume that the longitudinal trap frequency is either given by the longitudinal frequency of the tightly confining beam or given by the radial confinement of the large beam depending if the atoms predominantly occupies the cross region (see Fig. 3) .

To quantify the efficiency of the evaporation process we use the derivative ̺ = -dlogD/dlogN . We measure an average efficiency ̺ ∼ 3 (see fig. 3). The collision rate always remains larger than some hundreds per second which is high enough for an efficient evaporation. The increase of the collision rate around 10 6 atoms corresponds to the transfer of the atoms in the crossed region [38]. It appears as a sudden jump because of our approximation of an abrupt increase in the longitudinal frequency of the trap.

Our evaporation in an optical trap is both efficient and fast. It takes 2 s to reach the condensation threshold while losing a factor ∼10 in atom number. The chosen scattering length is optimal and changing it to a higher or lower value during the evaporation does not improve our efficiency. We could also evaporate with similar efficiency when using the same scattering length close to the two other low field Feshbach resonances at 33 G and 162 G. ∼ U1 where U1 is the depth of the large beam. The potential is changed by reducing the power of the confining beam and the atoms enter the crossed region when kBT < ∼ U1. The large beam then determines the longitudinal frequency. The crossover between the two trapping regimes is shown by a dashed line. The powers of both beams are reduced together during the remaining part of the evaporation.

V. CONDENSATE WITH TUNABLE INTERACTIONS

In this section, we use our ability to tune the scattering length to study Bose-Einstein condensates in the 1D to 3D crossover. More precisely, we study the expansion of condensates released from an elongated trap at different magnetic fields. At the end of the evaporation producing Bose-Einstein condensates of N c ∼ 2×10 4 atoms, the trapping frequencies are ω /2π = 24(1) Hz in the longitudinal direction and ω ⊥ /2π = 305 (10) Hz in the radial one. The magnetic field is first linearly ramped from 550 G to values between 400 G and 555 G, which corresponds to scattering lengths between -30 a 0 and 300 a 0 . After an additional 200 ms equilibration time, we then switch off the optical trap and let the gas freely expands. After 10 ms the interaction energy is already converted into kinetic energy and we can switch off the magnetic field without modifying the expansion [START_REF]If we switch the trap and the magnetic field simultaneously[END_REF]. After t = 34.5 ms of expansion, the condensates are finally imaged at zero field by fluorescence using the MOT beams at resonance during 50 µs. We observe the onset of three body losses when the scattering length is set to values higher than 300 a 0 . At magnetic fields lower than 505 G, i.e. for a < 0, the collapse of the Bose-Einstein condensate is observed through its disappearance [START_REF] Sackett | [END_REF].

For 10 < a < 300 a 0 , we study the radial size of the condensate after expansion. Figure 4 shows this size as a function of the parameter N c a which characterizes the FIG. 3. Phase space density (orange circles) and two body collision rate Γ (blue triangles) during the evaporation process as functions of the atom number. The corresponding average efficiency is ̺ ∼ 3. The dashed line represents the stage at which the atoms enter the crossed region of the traps as explained in fig. 2 strength of the interactions. We observe that due to the increase of the interaction energy, the size after expansion grows when N c a increases. In practice, we have chosen to fit the shape of the condensate with a Thomas-Fermi profile [41][42][43]. Because µ is not much larger than hω ⊥ , we are in the 1D-3D crossover and the Thomas-Fermi theory is not applicable. We have thus developed a model to adjust our data. It is based on an accurate interpolated value of the local chemical potential in the 1D-3D crossover in a radially trapped cloud µ = hω ⊥ √ 1 + 4n 1D a, where n 1D is the 1D density along the longitudinal direction [44]. This formula interpolates between µ = hω ⊥ , in the 1D regime, and µ = h√ 4n 1D a, which corresponds to the 3D Thomas-Fermi regime. In a cylindrical section of length l, the number of atoms is n 1D l and the energy is

E = l n 1D 0 µ(n)dn = lhω ⊥ (1 + 4n 1D a) 3/2 /6a. (1)
Thanks to the virial theorem in 2D [45], half of this energy is potential and the other half is the sum of the kinetic and interaction energies, which is the release energy in expansion E rel . Using the local density approximation in the longitudinal direction, we can then calculate the radial release energy per particle. In order to compare with the data, we finally compute the corresponding Thomas-Fermi radius after expansion R = 7E rel /m×t. The theoretical curve matches well our data. Note that the latter are fitted with a single free parameter that is the atom number calibration or in other words the detection efficiency. The obtained scaling factor is consistent with the atom number calibration obtained via the study of the condensed fraction as a function of T /T 0 c around the transition. We estimate a 30% uncertainty on the measured number of atoms.

VI. CONCLUSION

We have demonstrated a new route to produce 39 K BECs in a single species experiment. It is based on an all optical method that allows for a fast production of degenerate quantum gases. The atoms are directly loaded in a dipole trap from D1 gray molasses. After filtering out unwanted Zeeman substates evaporation is performed close to a Feshbach resonance with a high efficiency. It results in producing BECs with tunable interactions every 7 s in our setup. As an example we have finally studied the expansion of a BEC in the 1D to 3D crossover while tuning the scattering length finding good agreement with theory. In future works, the atom number in the BEC can be increased by using more power in the dipole trap thus allowing the use of larger waists. Our method will facilitate future studies of degenerate Bose gases with tunable interactions and will find applications in both interferometry as well as quantum simulation.

5 TABLE

 5 ) n (cm -3 ) γ (G.cm -1 ) B0(G) the beginning of the evaporation 4 × 10 6 220 7.5 × 10 13 0 550.

FIG. 1 .

 1 FIG. 1. Sample polarization procedure in a quadrupole field. Each Zeeman sublevel (represented with different colors) experiences a different longitudinal potential. It is a confining potential only for the mF = -1 (in light orange). The two others are repulsive and given by the second order Zeeman shift for the mF = 0 (in blue), and by the first order Zeeman shift for the mF = +1 (in black).

FIG. 2 .

 2 FIG. 2. Power reduction during the evaporation process of the large beam (orange circles) and of the tightly confining one (blue triangles). During the first part of the evaporation the atoms mainly see the second (confining) trap. Indeed kBT > ∼ U1 where U1 is the depth of the large beam. The potential is changed by reducing the power of the confining beam and the atoms enter the crossed region when kBT < ∼ U1. The large beam then determines the longitudinal frequency. The crossover between the two trapping regimes is shown by a dashed line. The powers of both beams are reduced together during the remaining part of the evaporation.

FIG. 4 .

 4 FIG.4. Experimental radial Thomas Fermi size after 34.5 ms of expansion as a function of Nca, where Nc is the number of condensed atoms. The theoretical curve (continuous blue line) corresponds to the size given by an approximated theory in the 1D to 3D crossover. The blue dashed line corresponds to the Thomas-Fermi model in 3D which does not match our data.

ACKNOWLEDGMENTS

We acknowledge P. Bouyer and P. Wang for important experimental contributions, and F. Moron and A. Villing for technical assistance. This research was supported by CNRS, Ministère de l'Enseignement Supérieur et de la Recherche, Direction Générale de l'Armement, ANR-12-BS04-0022-01, RTRA: Triangle de la physique, iSense, ERC senior grant Quantatop, DIM Nano'K from region Ile-de-France. LCFIO is member of IFRAF.