On non-proper intersections and local intersection numbers

Given pure-dimensional (generalized) cycles $\mu_1$ and $\mu_2$ on a complex manifold $Y$ we introduce a product $\mu_1\diamond_{Y} \mu_2$ 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 $L\to Y$ we define a product $\mu_1\bl \mu_2$ whose multiplicities at each point also coincide with the local intersection numbers. In addition, provided that $\mu_1$ and $\mu_2$ are effective, this product satisfies a B\'ezout inequality. If $i\colon Y\to \Pk^N$ is an embedding such that $i^*\Ok(1)=L$, then $\mu_1\bl \mu_2$ can be expressed as a mean value of St\"uckrad-Vogel cycles on $\Pk^N$. There are quite explicit relations between $\di_Y$ and $\bl$.


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 [9], 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 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 involves various choices. By van Gastel's formula, [11], one can obtain the Chow class µ 1 · Y µ 2 from a generic representative of µ 1 · SV µ 2 . In the '90s Tworzewski, [14], introduced local intersection numbers ǫ ℓ (µ 1 , µ 2 , x) at each point x, 0 ≤ ℓ ≤ dim V , which reflect the complexity of the intersection at dimension ℓ. In particular, if the intersection is proper, then ǫ ℓ (µ 1 , µ 2 , x) is the multiplicity of µ 1 · Y µ 2 at x when ℓ = 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 Section 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 (1.1) ρ := dim µ 1 + dim µ 2 − n 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 ( ) ℓ denotes the component of dimension ℓ.
For the definitions of the cohomology groups H * , * (V ), see Section 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 deg (µ 1 ⋄ P n µ 2 ) ρ = deg (µ 1 · P n µ 2 ), 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 Section 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 i : and extend to general µ ∈ B(Y ) by linearity. Here is our second main result.
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 ( ) ℓ denotes the component of dimension ℓ.
(iv) If the µ 1 and µ 2 are effective, then µ 1 • L µ 2 is effective and (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 Section 7.
The plan of this paper is as follows. In Section 2 we recall necessary material from [4,5]. The ⋄ Y -product is defined in Section 3 and Theorem 1.1 is proved. The relation to the •-product on P n is discussed in Section 4. In Section 5 we prove Theorem 1.2 and provide formulas that relate ⋄ Y and • L . In Section 6 we provide some further properties of these products, and in the final section, Section 7, we give various explicit examples.
Acknowledgment We would like to thank Bo Berndtsson, Martin Raum and Jan Stevens for valuable discussions on questions in this paper. We would also like to thank the referee for important comments and suggestions.

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].
2.1. 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 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 B(X) = ⊕ n 0 B k (X), where B k (X) are quotient spaces of GZ k (X). For precise definitions and proofs of the properties listed below, see [4,Sections 3 and 4]. (i) We have a natural inclusion Z k (X) → B k (X) for each k and hence an inclusion Z(X) = ⊕ n 0 Z k (X) → B(X) = ⊕ n 0 B k (X). (ii) Each µ ∈ B k (X) 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 [16] A. Yger introduces the related notion of algebraic generalized cycle as a generalization of (complex) algebraic cycle.
2.2. 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 on µ 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.

2.3.
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 x µ = mult i(x) i * µ if i : X → X ′ 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 : B k (X) → B k (U ), and mult x µ = mult x r U µ.

2.4.
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 representativeμ ∈ GZ(X) that is a positive current. Effective generalized cycle classes have non-negative multiplicities at each point.
If f : X → X ′ is proper and n ′ = dim X ′ , then we have natural mappings f * : If X is smooth, then H * , * (X) is naturally isomorphic to the usual cohomology groups H * , * (X, C).
For each k there is a natural mapping r k : Z k (X) → H n−k,n−k (X) that takes µ ∈ Z k (X) to its Lelong current [µ]. This mapping extends to a mapping B k (X) → H n−k,n−k (X). 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).
2.6. 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 Section 3 below in the case that J is the ideal sheaf of a submanifold V ⊂ X.

2.7.
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, [14], and Gaffney-Gassler, [10], 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 e k (J , µ, x) are non-negative if µ is effective. We have that e k (J , X, x) = e k (J , 1 X , x), see [4].
2.8. 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 ǫ ℓ (µ 1 , µ 2 , x) is the local intersection number at dimension ℓ. 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 for each x ∈ X.
we need a definition of S(J ∆ , µ 1 × µ 2 ) when J ∆ is not necessarily generated by a global holomorphic section of a Hermitian vector bundle 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 ֒→ E| V ; see, e.g., [4,Lemma 7.3].
Proposition 3.1. Assume that the normal bundle N V X → V is equipped with a Hermitian metric. For anyμ ∈ GZ(X) and k = 0, 1, 2, . . ., there is a generalized cycle S k (J V ,μ) ∈ GZ dimμ−k (X) with the following properties. (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 Proof. The proof is based on some ideas in [12]. Letμ ∈ GZ(X) and assume that µ = 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] 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 J V in U ∩ U ′ there is a holomorphic A ∈ Hom(F, E) in U ∩ U ′ such that σ = As. In V ∩ U ∩ U ′ , the embedding N V U ֒→ E| V ∩U , which by assumption is an embedding of Hermitian bundles, is then realized by A| V ; cf. [4,Lemma 7.3].
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.
If we identify ∆ and Y , then 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.
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), 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
For future reference we include the following simple proposition.
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 ℓ = 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 =
ℓ≥0 A ℓ is a graded C-algebra and ω : A → A maps A ℓ+1 → A ℓ , ℓ ≥ 0, and A 0 → 0. Moreover, let r be a fixed integer. Assume that a = a 0 + a 1 + · · · , where a ℓ are elements in A ℓ , and let b k be the elements in A k so that This lemma is probably well-known but we sketch a proof.
Sketch of proof. We can identify a ∈ A with the A-valued meromorphic function z →â(z) = ℓ≥0 z ℓ+r a ℓ .
Since (z + ω) ℓ+r a ℓ = (1 + ω/z) ℓ+r z ℓ+r a ℓ 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 = Id which proves the lemma.
Proof of (1.4). From (4.5) we have that since ω j ∧(µ 1 • µ 2 ) ℓ = 0 for degree reasons when j > ℓ. 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 ℓ = dim V occurs.
Example 4.3. Let µ 1 and µ 2 be the same k-plane V in P n . Then V • V = V , see [5,Section 1]. Thus only the term corresponding to ℓ = 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) ℓ−ρ gives a contribution and therefore, since ℓ = dim V , 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.
Given an embedding (5.3) let us select a maximal linearly independent subset s 0 , . . . , s N ′ of the s k . Notice that then N ′ ≤ N . Let i ′ : Y → P N ′ be the embedding defined by these sections. Then, there is a linear subspace ι : V → P M such that i = ι • i ′ . In view of [5, Proposition 6.7], i and i ′ give rise to the same product on Y .
Thus we can assume that our embedding (5.1) is defined by (5.3), where s 0 , . . . , s M is a linearly independent set in H 0 (Y, L). In view of [5, Example 6.4] the product • L only depends on the subspace of H 0 (Y, L) spanned by the given sections.
Proof. By Proposition 3.4 and (4.5) we have Notice that since N i(Y ) P M = T P M /T (i(Y )), 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), Proof of Theorem 1.2. Parts (i) and (ii) follow from Theorem 1.1 and (5.6). Part (iii) follows from the corresponding statement for • = • P M and (2.4). Alternatively, it follows from (5.6) and Theorem 1.1 (ii). 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 i * (µ 1 • L µ 2 ) = i * µ 1 • i * µ 2 is effective, see [5, Let us now consider (v). If µ 1 and µ 2 are cycles that intersect properly on Y , then by Theorem 1.1, Now k = ρ together with the term 1 from (1 − ω L ) k−d−1 gives us µ 1 · Y µ 2 , cf. (5.7). 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 Section 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]. Proposition 6.1. Assume that µ 0 , µ 1 , µ 2 ∈ B(Y ), γ is smooth in the open set U ⊂ Y , and µ 1 = γ ∧ µ 0 in U . Then 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. Section 2.6, by the generalized cycle 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.
We have the following version of Proposition 8.3 in [5].
If a is a point in Y , then and Proof. We can assume that µ = τ * α, where τ : W → Y is proper and α is a product of components of Chern forms.
In particular we have Proposition 6.3. If µ 1 , µ 2 ∈ B(Y ) are represented byμ 1 ,μ 2 , and η is a section of a Hermitian bundle E → Y × Y that defines J ∆ , then j * (µ 1 ⋄ Y µ 2 ) is represented by the limits One gets a formula for µ 1 ⋄ Y µ 2 by taking π * of (6.14), where π : Y × Y → Y is the projection onto the first (or the second) factor, since π • j = Id Y . One can get similar formulas for µ 1 • L µ 2 by combining (6.14) and (5.6).
Example 7.1. Consider the blowup Y = Bl p P 2 of P 2 at the point p = [1 : 0 : 0], and let both µ 1 and µ 2 be the exceptional divisor E. We have the embedding Let L → Y be the pullback to Y of this line bundle, which we for simplicity denote in the same way so that ω L = ω x + ω y .
We now compose with the Segre embedding and get the embedding i = σ • j : Y → P 5 . We claim that In fact, the image of E in P 2 × P 1 is {[1 : 0 : 0]} × P 1 y so the image in P 5 is the line {[y 0 : y 1 : 0 : 0 : 0 : 0]}. Therefore i * E •i * E = i * E, see the remark after [5, Theorem 1.1], and thus (7.2) holds. Next we compute E⋄ Y E. In view of (7.2) only the term with ℓ = 1 occurs in ( 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. be the Segre embedding. Note that Y · Y Y = Y since it is a proper intersection. It follows from Theorem 1.1 that 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 holds, cf. Proposition 5.2. Thus Y • L Y must have degree 2·2 = 4. On the other hand, · · · in (1.9) can only contain a term µ of dimensionρ = 1, since all components of i * Y • i * Y must have dimension at least the expected dimensionρ, cf. (1.5) and Section 4. Thus deg L µ = 2. For symmetry reasons it is natural to guess that 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, c(T Y ) = c(T P 1 x ) ∧ c(T P 1 y ) = (1 + ω x ) 2 ∧ (1 + ω y ) 2 = (1 + 2ω x ) ∧ (1 + 2ω y ) = 1 + 2(ω x + ω y ) + 4ω x ∧ ω y = 1 + 2ω L + 2ω 2 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.