On non-proper intersections and local intersection numbers

Given equidimensional (generalized) cycles μ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _1$$\end{document} and μ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _2$$\end{document} on a complex manifold Y we introduce a product μ1⋄Yμ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _1\diamond _{Y} \mu _2$$\end{document} that is a generalized cycle whose multiplicities at each point are the local intersection numbers at the point. If Y is projective, then given a very ample line bundle L→Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L\rightarrow Y$$\end{document} we define a product μ1∙Lμ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _1{\bullet _L}\mu _2$$\end{document} whose multiplicities at each point also coincide with the local intersection numbers. In addition, provided that μ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _1$$\end{document} and μ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _2$$\end{document} are effective, this product satisfies a Bézout inequality. If i:Y→PN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i:Y\rightarrow {\mathbb P}^N$$\end{document} is an embedding such that i∗O(1)=L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i^*\mathcal O(1)=L$$\end{document}, then μ1∙Lμ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _1{\bullet _L}\mu _2$$\end{document} can be expressed as a mean value of Stückrad–Vogel cycles on PN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb P}^N$$\end{document}. There are quite explicit relations between ⋄Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\diamond }_Y$$\end{document} and ∙L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bullet _L}$$\end{document}.


Introduction
Let Y be a complex manifold of dimension n. A cycle on Y is a locally finite linear combination, over Z, of subvarieties of Y . Assume that μ 1 and μ 2 are equidimensional cycles on Y . If they intersect properly, i.e., the expected dimension ρ = dim μ 1 + dim μ 2 − n is equal to the dimension of their set-theoretical intersection V = |μ 1 | ∩ |μ 2 |, then there is a well-defined intersection cycle where V j are the irreducible components of V and m j are integers. If μ 1 and μ 2 do not intersect properly, i.e., dim V > ρ, following [7], the product μ 1 · Y μ 2 is represented by a cycle of dimension ρ on V that is determined up to rational equivalence, i.e., a Chow class on V . In case Y = P n there is a construction of a product μ 1 · SV μ 2 due to Stückrad and Vogel, [11,13], that is represented by a cycle on V with components of various degrees. This cycle, which we call a SV-cycle, is obtained by a quite explicit procedure, which however  [9], see also [6], one can obtain the Chow class μ 1 · Y μ 2 from a generic representative of μ 1 · SV μ 2 .
In the 1990s Tworzewski, [12], introduced local intersection numbers (μ 1 , μ 2 , x) at each point x, 0 ≤ ≤ dim V , which reflect the complexity of the intersection at dimension . In particular, if the intersection is proper, then (μ 1 , μ 2 , x) is the multiplicity of μ 1 · Y μ 2 at x when = dim V and 0 otherwise. In this case thus all these numbers are represented by the global cycle μ 1 · Y μ 2 . In general however there is no single cycle whose multiplicities of its components of various dimensions are precisely the local intersection numbers for all points of Y , that is, which represents all these local intersection numbers.
Since both the local and global intersections are defined within algebraic geometry, it is natural from such a point of view to look for a way to unify these theories. However since this cannot be done by cycles it is natural to look for slightly more general geometric objects that may have the desired local multiplicities and at the same time in a reasonable sense represent the global intersection products.
To this end, in [4], together with Eriksson and Yger, we introduced, for any reduced analytic space X , the group B k (X ) of generalized cycles of dimension k, modulo a certain equivalence relation, that contains the group Z k (X ) of cycles of dimension k as a subgroup. The generalized cycles classes in B k (X ) share many properties with (usual) cycles. For instance, each μ ∈ B k (X ) has a well-defined (integer) multiplicity mult x μ at each point x ∈ X and a Zariski support |μ|. Each generalized cycle class is a unique sum of irreducible generalized cycle classes. Moreover, the generalized cycle classes μ in B k (X ) that have Zariski support |μ| on a subvariety Z ⊂ X are naturally identified with B k (Z ); see Sect. 2.1 below for precise definitions and statements. One can think of generalized cycle classes as mean values of cycles. We let B(X ) = ⊕ m 0 B k (X ) if m = dim X . For μ 1 , μ 2 ∈ B(P n ) we defined with Eriksson and Yger, [5], an element μ 1 • μ 2 ∈ B(P n ) that is equal to μ 1 · P n μ 2 if the intersection is proper, and whose multiplicities at each point coincide with the local intersection numbers. If μ j have pure dimensions and the expected dimension ρ := dim μ 1 + dim μ 2 − n (1 .1) is non-negative, then we have the Bézout equality Roughly speaking, μ 1 • μ 2 is defined as a mean value of SV-cycles μ 1 · SV μ 2 in case μ j are cycles.
In this paper we introduce two global intersection products that both respect all the local intersection numbers. The first one is defined on an arbitrary complex manifold. The second one generalizes the •-product and satisfies a Bézout inequality, but it is only defined on projective manifolds.
Here is our first main theorem.
with the following properties: where ( ) denotes the component of dimension .
For the definitions of the cohomology groups H * , * (V ), see Sect. 2. Part (ii) means that Y solves our representation problem. However, (iv) suggests that already the component of dimension ρ is as 'big' as the Chow class in a cohomological sense. In particular, if Y = P n this implies that which by Bézout's equality and (1.2) is equal to deg (μ 1 • μ 2 ) if ρ ≥ 0. In general, the degree of the full generalized cycle class is much larger. For effective generalized cycle classes, in particular for cycles, we have the estimate, see Sect. 4, The constant, which is the best possible, blows up when the intersection is far from being proper, i.e., when dim V − ρ is large. It is thus natural to look for an extension of the •-product, in order to get a representation of the local intersection numbers that is not 'too big'. To do this we restrict to a projective manifold Y . Let L → Y be a very ample line bundle. By definition then there is an embedding and extend to general μ ∈ B(Y ) by linearity. Here is our second main result.

Theorem 1.2
Assume that Y is a projective manifold and let L → Y be a very ample line bundle.
is commutative and Z-bilinear. It depends on the choice of L but not on the embedding i.
where ( ) denotes the component of dimension . (iv) If the μ 1 and μ 2 are effective, then μ 1 • L μ 2 is effective and (1.8) (v) If μ 1 , μ 2 are cycles that intersect properly, then where · · · are terms with lower dimension and vanishing multiplicities.
In view of (iii) thus μ 1 • L μ 2 has the 'right' multiplicities at each point, whereas (iv) says that we have control of the total mass of μ 1 • L μ 2 . In case Y = P n and L = O(1), then • L coincides with •. In this case the dots in (1.9) vanish. However, in general they do not, see Example 7.3 in Sect. 7.
The plan of this paper is as follows. In Sect. 2 we recall necessary material from [4,5]. The Y -product is defined in Sect. 3 and Theorem 1.1 is proved. The relation to the •-product on P n is discussed in Sect. 4. In Sect. 5 we prove Theorem 1.2 and provide formulas that relate Y and • L . In Sect. 6 we provide some further properties of these products, and in the final section, Sect. 7, we give various explicit examples.

Preliminaries
Throughout this section X is a reduced analytic space of dimension n. We let Z k (X ) denote the Z-module of k-cycles on X . Given μ ∈ Z k (X ) there is the associated closed current [μ], the Lelong current, of bidegree (n −k, n −k). We will often identify μ and its Lelong current. If nothing else is stated the definitions and results in this section are from [4, Sections 3 and 4].

Generalized cycles
The group GZ k (X ) of generalized cycles of dimension k was introduced in [4]. It is the Z-module generated by (closed) (n − k, n − k)-currents of the form τ * α, where τ : W → X is a proper mapping and α =ĉ 1 (L 1 )∧ . . . ∧ĉ 1 (L r ), (2.1) where L j → W are Hermitian 1 line bundles, andĉ 1 (L j ) are the associated first Chern forms. We let GZ(X ) = ⊕ n 0 GZ k (X ). Here W can be any complex variety but by virtue of Hironaka's theorem we may assume that W is a connected manifold. It is clear that generalized cycles are closed currents of order 0. Moreover, their Lelong numbers (multiplicities, see below) are integers. This means that GZ(X ) is a quite restricted class of closed currents. We are basically interested in a certain quotient space (i) We have a natural inclusion Z k (X ) → B k (X ) for each k and hence an inclusion has a well-defined Zariski support |μ|; it is the smallest Zariski closed set such that μ has a representative in GZ k (X ) that vanishes in its complement. (iii) Given μ ∈ B(X ) also its restriction 1 V μ to the subvariety V ⊂ X is an element in B(X ). (iv) If f : X → X is a proper mapping, then the push-forward f * induces a mapping f * : B k (X ) → B k (X ) that coincides with the usual push-forward on cycles.
(v) If i : X → X is an embedding, then i * : B k (X ) → B k (X ) is injective, and the image is precisely the elements in B k (X ) with Zariski support on i(X ). (vi) If E → X is a vector bundle, then we have natural mappings c k (E) : B * (X ) → B * −k (X ). The image of μ is represented byĉ k (E)∧μ, whereμ ∈ GZ(X ) represents μ andĉ k (E) is the Chern form associated with a (smooth) Hermitian metric on E.
In the recent paper [14] Yger introduces the related notion of algebraic generalized cycle as a generalization of (complex) algebraic cycle.

Irreducibility
A generalized cycle class μ ∈ B(X ) is irreducible if its Zariski support |μ| is an irreducible subvariety and μ has a representativeμ with Zariski support |μ| such that 1 Wμ = 0 for each subvariety W ⊂ X that does not contain |μ|. This condition onμ is equivalent to thatμ is a (finite) sum of elements of the form τ * α, where α is a form as in (2.1) on W , and τ : W → |μ| is surjective. Notice that these various terms can have different dimensions. Each element in GZ(X ), and in B(X ), has a unique decomposition in irreducible components with different Zariski supports. Each irreducible element has in turn a unique decomposition in components of various dimensions. There is a unique decomposition where μ f ix is an ordinary cycle, whose irreducible components are called the fixed components of μ, and μ mov , whose irreducible components are the moving components. Each moving component has strictly lower dimension than its Zariski support.

Multiplicities
If μ is a cycle, then the multiplicity mult x μ at x ∈ X is precisely the Lelong number at x of the associated Lelong current. If X is not smooth, then mult is an embedding and X is smooth. There is a suitable definition of Lelong number that extends to all generalized cycles and it turns out to depend only of their classes in B(X ), see [4,Section 6]. In this way we have for each μ ∈ B k (X ) well-defined multiplicities mult x μ at all points x ∈ X , and these numbers are integers. They are local in the following sense: If U ⊂ X is an open subset, then we have natural restriction mappings r U : If i : X → X , where X is smooth and μ ∈ B(X ), then Assume that μ = γ ∧μ ,where μ, μ ∈ B(U ) and γ is smooth and has positive degree. Then mult x μ = 0.

Effective generalized cycle classes
In [5, Section 2.4] was introduced the notion of effective generalized cycle class μ ∈ B(X ) generalizing the notion of effective cycle. It means precisely that μ has a representativê μ ∈ GZ(X ) that is a positive current. Effective generalized cycle classes have non-negative multiplicities at each point.

The cohomology groups H * , * (X)
We define H * , * (X ) as the vector space of closed ( * , * )-currents of order 0 modulo the subspace generated by all dτ for currents τ of order 0 such that also dτ has order 0, cf. [4,Section 10].
If f : X → X is proper and n = dim X , then we have natural mappings f * : Each μ ∈ Z k (X ) defines an element in the Chow group A k (X ) and the mapping r k induces a mapping A k (X ) → H n−k,n−k (X ).

The B-Segre class
Assume that J is a coherent ideal sheaf on X with zero set Z . Also assume that J is generated by a holomorphic section σ of a Hermitian vector bundle E → X . That is, J is locally generated by the tuple of holomorphic functions obtained when σ is expressed in a local frame of E. Such a section σ exists if X is projective. For any μ ∈ B(X ), following [4, Section 5], let whereμ is a representative of the class μ. The existence of the limit is highly non-trivial and relies on a resolution of singularities.
If J is locally a complete intersection, that is, defines a regular embedding, then one can define the Segre classes S(J , μ) without a section σ as above. We show this in Sect. 3 below in the case that J is the ideal sheaf of a submanifold V ⊂ X .

Segre numbers
Given a coherent ideal sheaf J → X with zero set Z , and μ ∈ B(X ), there are, at each point x, non-negative integers e k (J , X , x) for k = 0, 1, . . . , dim Z , called the Segre numbers. They were introduced independently by Tworzewski, [12], and Gaffney-Gassler, [8], as the multiplicity of the component of codimension k of a generic local SV-cycle in J x . A purely algebraic definition was introduced in [1] and the equivalence to the geometric definition was proved in [2]. If Z is a point, then the Segre number is precisely the Hilbert-Samuel multiplicity. In [3] was introduced an analytic definition.
Given μ ∈ B(X ) we have the integers that are called the Segre numbers of J on μ in [4, Section 2.6]. If σ is a section of a Hermitian vector bundle that defines J andμ is a representative of μ, then Locally we can choose σ and the (smooth) Hermitian metric so that log |σ | 2 is plurisubharmonic. If follows from (2.8) and (2.5), and the Skoda-El Mir theorem, that the Segre numbers

Local intersection numbers
Let X be smooth, assume that μ 1 , μ 2 ∈ B(X ) have pure dimensions, and let d = dim μ 1 + dim μ 2 . Furthermore, let J be the sheaf that defines the diagonal in X × X and let j : X → X × X be the natural parametrization. We define the local intersection numbers saying that (μ 1 , μ 2 , x) is the local intersection number at dimension . These numbers are biholomorphic invariants, and if we have an embedding i : X → X in a larger manifold X , then it follows from (2.6), (2.7) and (2.9) that Let X be a complex manifold, i : V → X a submanifold, and J V the corresponding ideal sheaf. Recall that if there is a holomorphic section σ of a vector bundle E → X such that σ generates J V , then there is an embedding N V X → E| V ; see, e.g. [4,Lemma 7.3].

(i) If U ⊂ X is open and σ is a holomorphic section of a Hermitian vector bundle E → U such that σ generates J V in U and the embedding N V U → E| V ∩U is an embedding of Hermitian vector bundles, then S k
Proof The proof is based on some ideas in [10]. Letμ ∈ GZ(X ) and assume thatμ = τ * α, where τ : W → X is a proper holomorphic mapping and α is a product of first Chern forms of Hermitian line bundles on W . We can assume that τ * J V is principal and that W is smooth. Consider the commutative diagram where D is the divisor of τ * J V . Let L → W be the line bundle corresponding to D; for future reference we recall that L| D is the normal bundle of D. We will show below that the Hermitian metric on N V X induces a metric on L| D . Let ω =ĉ 1 (L| * D ) be the first Chern form of the dual bundle. Then is in GZ(X ). This will be our definition of S k (J V ,μ). However, a priori this definition depends on the representation τ * α ofμ.
Let us now describe the induced metric on L| D . Let κ = codim V . We recall the following ad hoc definition; cf. [4,Section 7]. A section ξ of N V X is a choice of a κ-tuple ξ(s) locally on V for each local holomorphic κ-tuple s generating J V such that ξ(Ms) = Mξ(s) on V for any locally defined holomorphic matrix M invertible in a neighborhood of V .
Assume that U , σ , and E are as in (i). Assume also that s is a holomorphic κ-tuple generating J V in an open set U . In view of the definition of a section of N V X above, if we consider s as a section of the trivial rank κ bundle F → U , then we can identify F| V ∩U with N V U . Notice that this identification induces a Hermitian metric on F| V ∩U ; we extend it to a Hermitian metric on F in an arbitrary way. Since both s and σ generate Let U = τ −1 (U ∩U ). In D ∩U we get that a := τ * A| D∩U embeds τ * N V U in τ * E| D . Moreover, we have a similar situation in τ −1 (U ) and in τ −1 (U ) as we had in U ∩ U since the ideal sheaf τ * J V , which defines D, is generated by τ * s in τ −1 (U ) and τ * σ in τ −1 (U ). In the same way as above, since L| D is the normal bundle of D, we thus get embeddings Using the ad hoc definition of a section of a normal bundle it is straightforward to check that the latter embedding restricted to U is the composition of It follows that the metrics induced on L| D by the embeddings in τ * F and τ * E, respectively, coincide on L| D∩U . In particular, if σ is a holomorphic κ-tuple generating J V in U , so that E| V ∩U can be identified with N V U , it follows that the metric on N V X induces a metric on L| D .
With this metric on L| D , M σ k ∧μ equals the left-hand side of (3.1) in U by [4,Eq. (5.9)]. In view of (2.5), M σ k ∧μ is independent of the representation τ * α ofμ. It follows that (3.1) is independent of the representation τ * α ofμ, and we take (3.1) as our definition of S k (J V ,μ). Then M σ k ∧μ = S k (J V ,μ) in U and (i) is proved. We now note that (ii) follows. Indeed, in view of [4,Section 3], the image of (3.1) in B(X ) is 0 ifμ is 0 in B(X ) and, moreover, it is independent of the Hermitian metric on L.
Let N (Y × Y ) → be the normal bundle.
For the proof of Theorem 1.1 we need the following lemma. Recall that a coherent ideal sheaf J → X , with zero set Z , on a reduced space X of pure dimension defines a regular embedding of codimension κ if codim Z = κ and locally J is generated by κ functions. Then there is a well-defined normal bundle N J X over Z . See, e.g., [4,Section 7]. Lemma 3.3 Let X be a reduced space and let ι : X → X be a reduced subspace. Assume that the coherent sheaf J → X defines a regular embedding of codimension κ in X , and that J = ι * J defines a regular embedding of codimension κ in X . Then N J X = ι * N J X .
Let Z and Z denote the zero sets of J and J , respectively.
Proof By assumption, locally we have a set of generators s = (s 1 , . . . , s κ ) for J . If s is another such κ-tuple, then (on the overlap) there is an invertible holomorphic κ × κ matrix a(s, s ) such that s = a(s, s )s. The matrices so obtained form the transition matrices on Z for the bundle N J X . Now the lemma follows by noting that ι * s and ι * s are minimal sets of generators for J = ι * J and hence ι * a(s, s ) are transition matrices for N J X → Z .
It follows from (2.7) and (2.9) that where · · · are smooth forms of positive bidegree, it follows from the dimension principle that where · · · are smooth forms of positive degree times generalized cycle classes. Now (ii) follows from (3.3), (3.4) and the comment after (2.4).
We now prove (iii). Assume that μ j are cycles that intersect properly. Then intersects X := μ 1 ×μ 2 properly so that if ι : X → Y ×Y , then J := ι * J defines a regular embedding in X . In view of Lemma 3.3, (2.6) and (2.2) we have, using the notation S(J , 1 X ) = S(J , X ), (3.5) where the last equality is precisely [4, Theorem 1.4]. Here [Z J ] is the Lelong current of the fundamental cycle associated with J . Its Zariski support is precisely Z but there is a certain multiplicity of each irreducible component of Z . Since the right hand side of (3.5) has the expected dimension ρ, cf. (1.1), (3.5) implies that Since μ j intersect properly, this product is equal to μ 1 · Y μ 2 by [5, Proposition 5.8 (i)]. Thus Now (iii) follows from (3.5)-(3.7). We know from [5, Theorem 1.3] that the image of μ 1 · B(Y ) μ 2 coincides with the image of the Chow class μ 1 · Y μ 2 . Thus (iv) follows from (3.7). This concludes the proof.
For future reference we include the following simple proposition.

Proposition 3.4 Assume that we have an embedding i
Proof Notice now that if and are the diagonals in Y × Y and Y × Y , respectively, then We claim that If J is generated by a holomorphic section of a Hermitian vector bundle over Y × Y , then this follows from (2.6) since (i × i) * (μ 1 × μ 2 ) = i * μ 1 × i * μ 2 . The general case follows since (2.6), with f replaced by i × i, J = J , and μ = μ 1 × μ 2 , still holds in view of Proposition 3.1. Identifying B( ) and B( ) with B(Y ) and B(Y ), respectively, we have Therefore, by (3.2), (3.10), and (3.11),

The and •-products on P n
We first recall the definition of the •-product on P n . Let η 0 , . . . , η n be sections of L = O(1) P 2n+1 that define the join diagonal J in P 2n+1 , cf. [5,Section 6]. Let J J be the sheaf that defines J . Let η k be holomorphic functions that represent η k in a given local frame for L. Then dd c log |η| 2 • := dd c log(|η 0 | 2 + · · · + |η n | 2 ) is a well-defined global current. For μ ∈ B(P 2n+1 ) we define V k ( J , L, μ) as the classes in whereμ is a generalized cycle that represents μ. It is proved in [5,Section 4] that the Monge-Ampère products in (4.1) are well-defined and that V k ( J , L, μ) is independent of the choice of representativeμ and sections η 1 , . . . η n defining J . If k > n + 1 in (4.1), then If μ 1 , μ 2 ∈ B(P n ), then there is a natural class μ 1 × J μ 2 , see [5,Section 6], in B(P 2n+1 ), generalizing the usual join when μ 1 , μ 2 are cycles, and dim(μ 1 × J μ 2 ) = dim μ 1 +dim μ 2 +1 if μ 1 and μ 2 have pure dimensions. Let j : is the class in B(P n ) defined by Let ω P n be the first Chern class of O(1) → P n , for instance represented by the Fubini-Study metric form. If i : W → P n is a linear subspace, with the induced metric, then ω W = i * ω P n . We will often write ω without subscript. Recall thatĉ(T P n ) = (1 + ω) n+1 .

Proposition 4.1 Let = P n and let
We have the relations (4.5) and Since k ≤ n + 1, ≥ ρ. Moreover, each term in the sum (4.2) has support on V . Hence the sum runs from max(ρ, 0) to dim V .
Proof With the notation in [5,Section 7] we have that μ 1 μ 2 = i !! (μ 1 × μ 2 ). It follows from [5, Proposition 7.1] that By the second van Gastel type equality in [4,Corollary 9.9] we get (4.7) Since k ≤ n + 1 it follows that ≥ ρ, cf. (4.4), and since all terms in the sum have Zariski support on V , each term with larger than dim V must vanish in view of the dimension principle. Hence (4.7) is precisely (4.5).
The equality (4.6) follows from (4.5) and Lemma 4.2 below. Notice that although the sum in (4.5) happens to begin at = max(ρ, 0) it will give rise to terms of lower dimension so (4.6) must start at k = 0.

Lemma 4.2 Assume that A = ≥0
A is a graded C-algebra and ω : A → A maps A +1 → A , ≥ 0, and A 0 → 0. Moreover, let r be a fixed integer. Assume that a = a 0 + a 1 + · · · , where a are elements in A , and let b k be the elements in A k so that (4.8) This lemma is probably well-known but we sketch a proof. (1 + ω) +r a .

Sketch of proof
Since (z + ω) +r a = (1 + ω/z) +r z +r a it follows that i.e., T ω a(z) is obtained by formally replacing each occurrence of z inâ(z) by z + ω. It is now clear that T −ω • T ω = I d which proves the lemma.
Proof of (1.4) From (4.5) we have that (4.10) since ω j ∧(μ 1 • μ 2 ) = 0 for degree reasons when j > . We get the estimate In view of the proof we have equality in (1.4) if ρ ≥ 0 and in addition only the term with = dim V occurs. Thus only the term corresponding to = dim V = k occurs in (4.5). If in addition ρ ≥ 0, i.e., 2k ≥ n, then each term in the expansion of (1+ω) −ρ gives a contribution and therefore, so that the estimate (1.4) is sharp.

The • L -product on a projective manifold Y
We shall now see that if Y is projective and L → Y is a very ample line bundle, then there is an associated product • L with the desired local multiplicities and a Bézout inequality for effective generalized cycle classes. By definition 'very ample' means that there is an embedding where the right hand side is the •-product in P M . We shall see that • L only depends on L and not on the embedding i.
Proof By Proposition 3.4 and (4.5) we have Notice that since which is the same as (5.5). Now (5.6) follows by Lemma 4.2.
Notice that there may occur negative powers of 1 − ω L and 1 + ω L in the sums (5.5) and (5.6).
Recall that if μ ∈ B k (Y ), then, cf. (1.6), Part (iv) follows from the analogous statement for • on P M . In fact, first notice that μ is effective if and only if i * μ is. Then observe that if μ has pure dimension k, then If μ 1 and μ 2 are effective, then Let us now consider (v). If μ 1 and μ 2 are cycles that intersect properly on Y , then by . All other terms from (1 − ω L ) k−d−1 , or for k < ρ, will give contributions of strictly lower dimension, and they have vanishing multiplicities, see Sect. 2.3.
We have the following consequence of the proof.

Some further properties
In this section we still assume that Y is a projective manifold. Assume that μ 0 , μ 1 ∈ B(Y ) and that γ is a smooth (closed) form in an open subset U ⊂ Y . We say that μ 1 = γ ∧μ 0 in U if there are generalized cycles μ 0 and μ 1 representing μ 0 and μ 1 , respectively, such that μ 1 = γ ∧μ 0 in U . We have the following version of Proposition 8.4 in [5].
in U . If L → Y is a very ample line bundle, then Proof Fix suitable representatives μ 0 , μ 1 , μ 2 in GZ(Y ) and a section η that defines the diagonal in Y × Y . Moreover, letĉ(N (Y × Y )) be a fixed representative of the Chern class c(N (Y × Y )). As usual, let j : Y → Y × Y be the natural parametrization of . Then j * (μ 1 Y μ 2 ) is represented, cf. Sect. 2.6, by the generalized cycle in U × U . By (6.3) and (6.4), in U × U . Now (6.1) follows.
Assume that γ has pure degree ν. Then by (6.1), Let d be as in Proposition 5.1 and letd = dim μ 0 + dim μ 2 ; then d =d − ν. By (5.6), since the terms with r < ν in the last sum vanish when multiplied by γ .

Examples
We first recall the so-called Segre embedding Notice that i * ω = ω L = ω x + ω y , where ω, ω x and ω y are (representatives of) c 1 (O(1) P (m+1)(n+1)−1 ), c 1 (O(1) P m x ), and c 1 (O(1) P n y ), respectively. We now consider the self-intersection of an exceptional divisor. and get the embedding i = σ • j : Y → P 5 . We claim that and E Y E = E − ω L ∧E. Since ω x = 0 on E we have, cf. (7.1), Let us next look at an example where Y is embedded into P M for a minimal M, and where the terms · · · of lower dimension in (1.9) do not vanish. We want to compute Y • L Y . Since ω L = i * ω = ω x + ω y , cf. (7.1), it follows that ω 2 L = (ω x + ω y ) 2 = 2ω x ∧ω y and thus Let us check (7.5) by means of (5.5) in Proposition 5.1 and (7.4). Notice that d = dim Y + dim Y = 4,ρ = d − 3 = 1, and V = Y so that dim V = 2. Moreover, L . Assuming (7.5), the right hand side of (5.5) equals Hence our guess is correct. Clearly one can just as well start with (7.4) and apply (5.6). By similar computations one then gets (7.5), as expected.