Nonproper intersection products and generalized cycles

In this article we develop intersection theory in terms of the $\mathcal{B}$-group of a reduced analytic space. This group was introduced in a previous work as an analogue of the Chow group; it is generated by currents that are direct images of Chern forms and it contains all usual cycles. However, contrary to Chow classes, the $\mathcal{B}$-classes have well-defined multiplicities at each point. We focus on a $\mathcal{B}$-analogue of the intersection theory based on the St\"uckrad-Vogel procedure and the join construction in projective space. Our approach provides global $\mathcal{B}$-classes which satisfy a B\'ezout theorem and have the expected local intersection numbers. An essential feature is that we take averages, over various auxiliary choices, by integration. We also introduce $\mathcal{B}$-analogues of more classical constructions of intersections using the Gysin map of the diagonal. These constructions are connected via a $\mathcal{B}$-variant of van Gastel's formulas. Furthermore, we prove that our intersections coincide with the classical ones on cohomology level.


Introduction
Let Y be a smooth manifold of dimension n. Assume that µ 1 , . . . , µ r are cycles on Y of pure codimensions κ 1 , . . . , κ r , respectively, let κ := κ 1 + · · · + κ r , and let V be the set-theoretic intersection V = |µ 1 | ∩ · · · ∩ |µ r |. If µ j intersect properly, that is, if codim V = κ, then there is a well-defined intersection cycle where V j are the irreducible components of V and m j are integers. In the nonproper case there is no canonical intersection cycle. However, following Fulton-MacPherson, see [9], there is an intersection product µ 1 · Y · · · · Y µ r , which is an element in the Chow group A n−κ (V ); that is, the product is represented by a cycle on V of dimension n − κ that is determined up to rational equivalence. For instance, the self-intersection of a line L in P n is obtained by intersecting L with a perturbation of L. If n = 2 one gets an arbitrary point on L, whereas if n ≥ 3 the intersection is empty. In case Y = P n there is an intersection product due to Stückrad and Vogel, [12,14], that in general consists of components of various dimensions. For instance the selfintersection of a line is actually the line itself independently of n. However, in general a nonproper intersection has so-called moving components, that are only determined up to rational equivalence. There is a relation to the classical (Fulton-MacPherson) intersection product via van Gastel's formulas, [11], see also [8].
Date: September 2, 2019. The first, third and fourth author were partially supported by the Swedish Research Council.
The main objective of this paper is to introduce a product of cycles in P n that at each point carries the local intersection numbers and at the same time have reasonable global properties, such as respecting the Bézout formula. To this end we must extend the class of cycles, and our construction is based on the Z-module GZ(X) of generalized cycles on a (reduced) analytic space X introduced in [5]. It is the smallest class of currents on analytic spaces that is closed under multiplication by components of Chern forms and under direct images under proper holomorphic mappings. It turns out that generalized cycles inherit a lot of geometric properties and preferably can be thought of as geometric objects. Actually we are primarily interested in a certain natural quotient group B(X) of GZ(X). Each µ in GZ(X) has a well-defined Zariski support |µ| ⊂ X that only depends on its class in B(X). For a subvariety V ֒→ X there is a natural identification of B(V ) with elements in B(X) that have Zariski support on V . The group of cycles Z(X) is naturally embedded in B(X). Given µ ∈ B(X) also its restriction 1 V µ to the subvariety V is an element in B(X). Each element in GZ(X), and in B(X), has a unique decomposition into sums of irreducible components. Each irreducible element has in turn a unique decomposition into components of various dimensions. We let B k (X) denote the elements in B(X) of pure dimension k. We also introduce a notion of effective generalized cycle µ in GZ(X), and class in B(X), generalizing the notion of effective cycle. Each µ in GZ(X), and in B(X), has a well-defined multiplicity, mult x µ, at each point x ∈ X, that is an integer and nonnegative if µ is effective. Moreover, for each µ in GZ k (X), or in B k (X), there is a unique decomposition (1.2) µ = µ f ix + µ mov , where µ f ix is an ordinary cycle of dimension k, whose irreducible components are called the fixed components of µ, and µ mov , whose irreducible components, the moving components, have Zariski support on varieties of dimension strictly larger than k.
We say that the image is the •-product of µ 1 , . . . , µ r . It is obtained, roughly speaking, in the following way: We first choose representatives for the B-classes µ 1 , . . . , µ r , then form a Stückrad-Vogel-type product of them. Even for cycles, this product depends on several choices. Taking a suitable mean value, we get a generalized cycle that turns out to define an element in B(P n ) that is independent of all choices. If µ j are cycles, then the fixed components in the Stückrad-Vogel product appear as fixed components of µ 1 • · · · • µ r . The formal definition, Definition 6.6, is expressed in terms of a certain Monge-Ampère type product, that can be obtained as a limit of quite explicit expressions, see Section 6.
Here is our main result.
One should keep in mind that the •-product of r factors is not a repeated •-product of two factors. In general, the •-product of two factors is not associative, see Example 8. 10. Notice that ρ equals n−(n−dim µ 1 +· · ·+n−dim µ r ), which is the "expected dimension" of the intersection. The Bézout formula (1.7) may hold even if ρ < 0: For instance, if µ j are different lines through the point a, then their •-product is a so that both sides of (1.8) are 1, see Example 8.8. Moreover, if we take a linear embedding P n ֒→ P n ′ , n ′ > n, and consider µ j as elements in B(P n ′ ), then the product is unchanged. In particular, the •-self-intersection of a k-plane is always the k-plane itself. The •-self-intersection of the cuspidal curve Z = {x 3 1 −x 0 x 2 2 = 0} in P 2 is in the classical sense represented by 9 points on Z obtained as the divisor of a generic meromorphic section of O P 2 (3) restricted to Z. The fixed part of the self-intersection in the Stückrad-Vogel sense is the curve itself plus 3 times the point a = [1, 0, 0], whereas the moving part consists of another three points on Z that are determined up to rational equivalence on Z. Our product Z •Z consists of the the fixed part Z + 3a of the Stückrad-Vogel(SV)product and a moving component µ of dimension zero and degree 3; we think of µ as three points "moving around" on Z, cf. Example 8.13. In this case the local intersection numbers are carried by the fixed components. In general also moving components can contribute, see, e.g., Example 8.6.
We also consider another intersection product that is a B-variant of the classical nonproper intersection product in [9]: For any regular embedding i, in [5] we introduced a B-analogue of the Gysin mapping i ! used in [9], see Section 2.7 below. Let i : P n → P n × · · · × P n = (P n ) r be the diagonal embedding in (P n ) r . In analogy with the classical intersection product in [9] we define, for pure-dimensional µ j , in B(P n ). We have the following relation to the •-product. Theorem 1.2. Assume that µ 1 , . . . , µ r ∈ B(P n ) have pure dimensions. Let V = ∩ j |µ j | and let ρ be as in (1.6). Then In particular, µ 1 · B(P n ) · · · · B(P n ) µ r = µ 1 · P n · · · · P n µ r if µ 1 , . . . , µ r are cycles that intersect properly, see (1.9).
In [5, Section 10] we introduced cohomology groups H * , * (V ) for a reduced subvariety V ֒→ P n of pure dimension d that coincide with usual de Rham cohomology H * , * (V ) when V is smooth. There are natural mappings Theorem 1.3. Assume that Z 1 , . . . , Z r are cycles in P N and let V = ∩ j |Z j |. The images in H * , * (V ) of the Chow class Z 1 · P n · · · · P n Z r and the B-class Z 1 · B(P n ) · · · · B(P n ) Z r coincide.
The plan of the paper is as follows. Sections 2 through 4 contain mainly material from [5] and well-known facts from [9], as well as the definition of local intersection numbers and of the notion of an effective generalized cycle. The product · B(Y ) is introduced in Section 5. In Section 6 we define the •-product and prove Theorem 1.1, whereas the connection to the · B(P n ) -product is worked out in Section 7. Finally we have collected several examples in Section 8.

Ackowledgement:
We are grateful to Jan Stevens for valuable discussions on the ideas in this paper.

Preliminaries
Throughout this section X is a reduced analytic space of pure dimension n. We will recall some basic notions from intersection theory that can be found in [9], and some notions and results from [5]; however the material in Section 2.4 and Lemma 2.1 is new. We formulate statements in terms of coherent sheaves, rather than schemes.

Currents and cycles.
We say that a current µ on X of bidegree (n − k, n − k) has (complex) dimension k. If f : X ′ → X is a proper mapping of analytic spaces, then f * is well-defined on currents and preserves dimension. If µ is a current on X ′ and η is a smooth form on X, then If µ has order zero then f * µ has order zero. If V ֒→ X is a subvariety, then If V ֒→ X has dimension k, then its associated Lelong current (current of integration) [V ] has dimension k. We will often identify V and [V ]. An analytic k-cycle µ on X is a formal locally finite linear combination a j V j , where a j ∈ Z and V j ⊂ X are irreducible analytic sets of dimension k. We denote the Z-module of analytic k-cycles on X by Z k (X). The support |µ| of µ ∈ Z k (X) coincides with the support of its associated Lelong current. Recall that mult x µ = ℓ x µ, where ℓ x µ denotes the Lelong number (of the Lelong current) of µ ∈ Z k (X) at x, and mult x µ is the multiplicity of µ at x, see [6,Chapter 2.11.1].
If f : X ′ → X is a proper mapping, then we have a mapping and the Lelong current of the direct image f * µ is the direct image of the Lelong current of µ. If i : V ֒→ X is a subvariety, then µ ∈ Z k (V ) can be identified with the cycle i * µ ∈ Z k (X). The cycle µ ∈ Z k (X) is rationally equivalent to 0 on X, µ ∼ 0, if there are finitely many subvarieties i j : V j ֒→ X of dimension k + 1 and non-trivial meromorphic functions g j on V j such that 1 We denote the Chow group of cycles Z k (X) modulo rational equivalence by A k (X). If f : X ′ → X is a proper morphism and µ ∼ 0 in A k (X ′ ), then f * µ ∼ 0 in A k (X) and there is an induced mapping f * : A k (X ′ ) → A k (X).

2.2.
Chern and Segre forms. Recall that to any Hermitian line bundle 2 L → X there is an associated (total) Chern form 3ĉ (L) = 1 +ĉ 1 (L) and that two Hermitian metrics give rise to Chern forms whose difference is dd c γ for a smooth form γ on X. We let c(L) denote the associated cohomology class.
Assume that E → X is a Hermitian vector bundle, and let π : P(E) → X be the projectivization of E, i.e., the projective bundle of lines in E. Let L = O(−1) be the tautological line bundle in the pullback π * E → P(E), and letĉ(L) be the induced Chern form on P(E). Since π is a submersion,ŝ(E) := π * (1/ĉ(L)) is a smooth form on X called the Segre form of E. If E is a line bundle, then P(E) ≃ X and hence For a general Hermitian E → X we take (2.4) as the definition of its associated Chern form. If f : X ′ → X is a proper mapping, then . Since π is a submersion two different metrics on E give rise to Segre forms and Chern forms that differ by dd c γ for a smooth form γ on X. The induced cohomology classes are denoted by s(E) and c(E), respectively. There are induced mappings 2.3. Generalized cycles. Generalized cycles on X were introduced in [5] and all statements in this subsection except Lemma 2.1 are proved in [5,Sections 3 and 4]. We say that a current µ is a generalized cycle if it is a locally finite linear combination over Z of currents of the form τ * α, where τ : W → X is a proper map, W is smooth and connected, and α is a product of components of Chern forms for various Hermitian vector bundles over W , i.e., where E j are Hermitian vector bundles over W . One can just as well use components of Segre forms, and one can in fact assume that all E j are line bundles. Notice that a generalized cycle is a real closed current of order zero with components of bidegree ( * , * ). We let GZ k (X) denote the Z-module of generalized cycles of (complex) dimension k (i.e., of bidegree (n − k, n − k)) and we let GZ(X) = GZ k (X). If µ ∈ GZ(X) and γ is a component of a Chern form on X, then γ∧µ ∈ GZ(X). If E → X is a Hermitian vector bundle we thus have mappings GZ k (X) → GZ k−ℓ (X) defined by µ →ĉ ℓ (E)∧µ.
If i : V ֒→ X is a subvariety and µ ∈ GZ(X), then 1 V µ ∈ GZ(X). More precisely, if where τ j : W j → X, then Each subvariety of X is a generalized cycle so we have an embedding Given µ ∈ GZ(X) there is a smallest variety |µ| ⊂ X, the Zariski support of µ, such that µ vanishes outside |µ|. If f : X ′ → X is proper, then we have a natural mapping that coincides with (2.3) on Z k . If i : V ֒→ X is a subvariety, then is an injective mapping whose image is precisely those µ ∈ GZ k (X) such that |µ| ⊂ V . Thus we can identify GZ(V ) with generalized cycles in X that have Zariski support on Z. We have the Dimension principle: Assume that µ ∈ GZ k (X) has Zariski support on a variety V . If dim V = k, then µ ∈ Z k (X). If dim V < k, then µ = 0. A nonzero generalized cycle µ ∈ GZ(X) is irreducible if |µ| is irreducible and 1 V µ = 0 for any proper analytic subvariety V ֒→ |µ|. If µ has Zariski support V ⊂ X it is irreducible if and only if V is irreducible and µ has a representation (2.7) where τ j (W j ) = V for each j. An irreducible µ ∈ GZ(X) has the decomposition µ = µ p + · · · + µ 1 + µ 0 , µ k ∈ GZ k (X), where p is the dimension of |µ|. Each µ ∈ GZ(X) has a unique decomposition where µ ℓ are irreducible with different Zariski supports. If 0 → S → E → Q → 0 is a short exact sequence of Hermitian vector bundles over X, then we say thatĉ(E) −ĉ(S)∧ĉ(Q) is a B-form. If β is a component of a B-form, then there is a smooth form γ on X such that dd c γ = β. We say that µ ∈ GZ k (X) is equivalent to 0 in X, µ ∼ 0, if µ is a locally finite sum of currents of the form where τ : W → X is proper, β is a component of a B-form, α is a product of components of Chern or Segre forms, and γ is a smooth form on W . If µ = µ 0 + · · · + µ n , where µ k ∈ GZ k (X) we say that µ ∼ 0 if µ k ∼ 0 for each k. Let B(X) denote the Z-module of generalized cycles on X modulo this equivalence. A class µ ∈ B(X) has pure dimension k, µ ∈ B k (X), if µ has a representative in GZ k (X). Thus B(X) = ⊕ k B k (X). The mapping Z(X) → B(X) is injective so we can consider Z(X) as a subgroup of B(X).
If µ ∈ B(X) andμ ∈ GZ(X) is a representative for µ, then the Zariski support |µ| ⊂ X of µ is the union of the Zariski supports of the irreducible components ofμ that are nonzero in B(X). Moreover, µ ∈ B(X) is irreducible if there is a representativê µ ∈ GZ(X) that is irreducible. The decomposition into irreducible components, as well as the decomposition into components of different dimensions, extend from GZ(X) to B(X).
In particular, if E and E ′ are the same vector bundle with two different Hermitian metrics, thenĉ ℓ (E)∧μ ∼ĉ ℓ (E ′ )∧μ so we have mappings If f : X ′ → X is a proper mapping, then we have a natural mapping If i : V ֒→ X is a subvariety, then is injective, and we can identify its image with the elements in B(X) that have Zariski support on V . Each µ ∈ B k (X) (and µ ∈ GZ k (X)) has a unique decomposition (1.2) where µ f ix is a cycle of pure dimension k and the irreducible components of µ mov have Zariski supports of dimension strictly larger than k. We say that the irreducible components of µ f ix are fixed and that the irreducible components of µ mov are moving.
We will need the following simple lemma.
2.4. Effective generalized cycles. We say that a generalized cycle µ is effective if it is a positive current, see, e.g., [7, Ch.III Definition 1.13]. Clearly effectivity is preserved under direct images. Lemma 2.2. Let µ = µ 1 + µ 2 · · · be the decomposition of µ ∈ GZ(X) into its irreducible components. Then µ is effective if and only if each µ j is effective.
Proof. The if-part is clear. For the converse, let V be an irreducible subvariety of X. We already know that 1 V µ is a generalized cycle. It is not hard to see that it is positive if µ is positive. It is also part of the Skoda-El Mir theorem, see, e.g., [7, Ch.III Theorem 2.3]. Now let V j be the Zariski supports of the various µ j and assume that V k has minimal dimension. Then V k ∩ V j has positive codimension in V j for each j = k. By the definition of irreducibility it follows that 1 V k µ = 1 V k µ k = µ k . We conclude that µ k is positive for each k such that V k has minimal dimension. Let V ′ be the union of these V k and let µ ′ be the sum of the remaining irreducible components. Clearly µ ′ is positive in X \ V ′ . Let A = ia 1 ∧ā 1 ∧ . . . ∧ia r ∧ā r for smooth (1, 0)-forms a j and some r. It follows that A∧µ ′ is positive outside V ′ by definition. However, 1 V ′ µ ′ = 0 and so A∧µ ′ = A∧1 X\V ′ µ ′ is positive. Since A is arbitrary, we conclude that µ ′ is positive. Now the lemma follows by induction.
We say that µ ∈ B(X) is effective if it has a representativeμ ∈ GZ(X) that is effective. It follows that µ is effective if and only each of its irreducible components is effective. Moreover, the multiplicities of an effective µ ∈ B(X) are nonnegative.
2.5. The Segre and B-Segre class. The material in this subsection is found in [5,Section 5] or in [9]. Let J → X be a coherent ideal sheaf over X with zero set Z. First assume that X is irreducible. If J = 0 on X, then we define the Segre class s(J , X) = s 0 (J , X) = 1 X ∈ A n (X). Otherwise, let π : X ′ → X be a modification such that π * J is principal 4 . For instance X ′ can be the blowup of X along J , or its normalization. Let D be the exceptional divisor, and let L D be the associated line bundle that has a section σ 0 that defines D and hence generates π * J . Then it is a well-defined element in A * (X). If X has irreducible components X 1 , X 2 , . . ., then s(J , X) = s(J , X 1 ) + s(J , X 2 ) + · · · . Notice that s(J , X) has support in Z so that it can be identified with an element s(J , X) in A * (Z). If J is the sheaf associated with the subscheme V of X, then s(J , X) coincides with the classical Segre class s(V, X), cf. [9, Corollary 4.2.2]. We can define the B-Segre class S(J , X) in an analogous way by just interpreting ∩ as the ordinary wedge product. However, we are interested in more explicit representations and also in a definition of a B-Segre class on µ ∈ B(X). To this end we assume that the ideal sheaf J → X is generated by a holomorphic section σ of a Hermitian vector bundle E → X. If X is projective one can always find such a σ for any coherent ideal sheaf J → X. We shall consider Monge-Ampère products on a generalized cycle µ. Theorem 2.3. Assume that σ is a holomorphic section of E → X and let J be the associated coherent ideal sheaf with zero set Z. For each µ ∈ GZ(X) the limits (dd c log |σ| 2 ) k ∧µ := lim ǫ→0 dd c log(|σ| 2 + ǫ) k ∧µ, k = 0, 1, 2, . . . , exist and are generalized cycles with Zariski support on |µ|. The generalized cycles M σ k ∧µ := 1 Z (dd c log |σ| 2 ) k ∧µ , k = 0, 1, 2, . . . , have Zariski support on Z ∩ |µ|. If µ ∼ 0, then M σ k ∧µ ∼ 0. If g is a holomorphic section of another vector bundle that also defines J , then M σ k ∧µ ∼ M g k ∧µ.
If J vanishes identically on |µ|, then it follows from the definition that S(J , µ) = µ.
One can define M σ k ∧µ by a limit procedure without applying 1 Z , see [5, Proposition 5.7 and Remark 5.9]: Proposition 2.6. Let σ be a holomorphic section of a Hermitian bundle E → X and let , and µ = f * µ ′ , then (2.1) and (2.12) imply that Let ξ be a section of a vector bundle in a neighborhood U ⊂ X of x such that ξ defines the maximal ideal at x. Notice that if µ ∈ GZ k (X), then by Theorem 2.3, M ξ ∧µ is a generalized cycle with Zariski support at x and its image in B(X) is independent of the choice of section ξ defining the maximal ideal. In view of the dimension principle, see Section 2.3, for some real number a. We say that a is the multiplicity, mult x µ, of µ at x, i.e., It is an integer that is independent of the choice of neighborhood U and only depends on the class of µ in B(X). If µ is effective (i.e., represented by a positive current), then mult x µ is the Lelong number of µ at x and hence nonnegative, see [5, Section 6].
Example 2.9. If µ ∈ GZ(X) is of the form µ = γ∧µ ′ in a neighborhood of x, where γ is a closed smooth form of positive degree and µ ′ ∈ GZ(X), then mult x µ = 0. In fact, by (2.13), M ξ ∧µ = γ∧M ξ ∧µ ′ which must vanish by the dimension principle, since M ξ ∧µ ′ has support at x and γ has positive degree.
2.6. Segre numbers. Let J → X be a coherent ideal sheaf over X of codimension p. In [13] and [10] Tworzewski, and Gaffney and Gassler, independently introduced, at each point x ∈ X, a list of numbers (e p (J , X, x), . . . , e n (J , X, x)), called Segre numbers in [10]. The Segre numbers generalize the Hilbert-Samuel multiplicity at x in the sense that if J has codimension n at x then e n (J , X, x) is the Hilbert-Samuel multiplicity at x. The definitions in [13] and [10], though slightly different, are both of geometric nature. There is also a purely algebraic definition, [1,2]. In [4] were introduced semiglobal currents whose Lelong numbers are precisely the Segre numbers. These currents are generalized cycles where they are defined. We can define Segre numbers for J over a generalized cycle µ ∈ GZ(X): In a neighborhood U of a given point x we can take a section σ of a trivial Hermitian bundle such that σ generates J and define the Segre numbers 2.7. Regular embeddings and Gysin mappings. Assume now that X is smooth and that J → X is locally a complete intersection of codimension κ. This means that ι : Z J ֒→ X is a regular embedding, where Z J is the non-reduced space of codimension κ defined by J . Then the normal cone N J X is a vector bundle over the reduced space i : Z ֒→ X and hence there is a well-defined cohomology class c(N J X) on Z. Therefore there is a well-defined mapping, the classical Gysin mapping 5 where the lower index k − κ denotes the component of dimension k − κ. We have the analogous B-Gysin mapping Our main interest is when J defines a submanifold; in this case Z = Z J and i = ι. By suitable choices we can represent (2.17) by a mapping on GZ(X): Assume that J is defined by a section σ of a Hermitian vector bundle E → X and let E ′ be the pull-back to Z. There is a canonical holomorphic embedding ϕ : N J X → E ′ , see [5,Section 7]. Let us equip N J X with the induced Hermitian metric and letĉ(N J X) be the associated Chern form, cf. Section 2.2. Then we have the concrete mapping which induces the mapping (2.17). We recall [5, Propositions 1.4 and 1.5]: 5 Since this is a map to A k−κ (Z), to be formally correct, we must insert i * in the formula defining ι ! , cf. Section 2.5. Proposition 2.10. If J → X defines a regular embedding, then Example 2.11. Let i : Z → X be the inclusion of a smooth submanifold of codimension κ and suppose that µ ∈ GZ k (X) is a smooth form. Then, in view of Proposition 2.10, Thus, i ! µ = i * µ is the usual pullback.
2.8. Intersection with divisors and the Poincaré-Lelong formula on a generalized cycle. See [5,Section 8] for proofs of the statements in this subsection. Let h be a meromorphic section of a line bundle L → X. We say that divh intersects the generalized cycle µ properly if h is generically holomorphic and nonvanishing on the Zariski support |µ j | of each irreducible component µ j of µ. If divh and µ intersect properly there is a generalized cycle divh · µ with Zariski support on |divh| ∩ |µ| that we call the proper intersection of divh and µ.
If τ : W → X such that µ = τ * α, where α is a product of components of Chern or Segre forms, then divh · µ = τ * ([divτ * h]∧α). Then divh · µ ∼ 0 if µ ∼ 0 so that the intersection has meaning for µ ∈ B(Y ). If h is holomorphic, i.e., divh is effective, then, in a local frame for L, where |h| • is the norm of the holomorphic function obtained from any fixed local frame for L so that dd c log |h| • is well-defined. It follows that divh · µ is effective if both divh and µ are effective. In light of (2.18) it is natural to write divh · µ as [divh]∧µ.
Proposition 2.12 (The Poincaré-Lelong formula on a generalized cycle). Let h be a nontrivial meromorphic section of a Hermitian line bundle L → X. Assume that divh intersects µ properly. Then where µ ′ j are the irreducible components of µ that divh intersects properly, see [5, Section 9].
2.9. Mappings into cohomology groups. In this subsection we assume that X is projective, in particular compact, cf. [5,Section 10]. Let H k,k (X) be the equivalence classes of d-closed (k, k)-currents µ on X of order zero such that µ ∼ 0 if there is a current γ of order zero such that µ = dγ. If X is smooth there is a natural isomorphism H n−k,n−k (X) → H n−k,n−k (X, C); the surjectivity is clear and the injectivity follows since a closed current of order zero locally has a potential of order zero. If i : X ֒→ M is an embedding into a smooth manifold M of dimension N , then there is a natural mapping i * : H n−k,n−k (X) → H N −k,N −k (M, C) induced by the push-forward of currents.
There are natural cycle class mappings in H(X), where the right hand side is represented by the wedge product of a smooth form and a current. There are natural mappings Example 2.14. Assume that h is a meromorphic section of a Hermitian line bundle L → X such that divh intersects µ ∈ GZ k (X) properly. It follows from Proposition 2.12 that [divh]∧µ andĉ 1 (L)∧µ coincide in H n−k+1,n−k+1 (X).
Let us recall, [5,Proposition 1.6], that the images of A k (X) and B k (X) in H n−k,n−k (X) coincide. We have the commutative diagram .
Example 2.15. It follows from the dimension principle that A n (X) = Z n (X) = B n (X). If X has the irreducible components X 1 , X 2 , . . ., then the image in H 0,0 (X) of the cycle a 1 X 1 + a 2 X 2 + · · · on X is the d-closed (0, 0)-current a 1 1 X 1 + a 2 1 X 2 + · · · . It follows that the mappings into H 0,0 (X) are injective.
More generally, we have [5, Proposition 1.7]: Proposition 2.16. Assume that J → X defines a regular embedding Z J ֒→ X of codimension κ and let µ be a cycle. The images in H * , * (Z) of the Gysin and the B-Gysin mappings of µ, (2.16) and (2.17), coincide.

Local intersection numbers
Let Y be a smooth manifold, let µ 1 , . . . , µ r be generalized cycles on Y of pure dimensions and let d = dim µ 1 + · · · + dim µ r . Following the ideas of Tworzewski [13] we define the local intersection numbers at x, cf. Lemma 2.1 and Section 2.6, is a Hermitian vector bundle and σ is a section of E that generates J ∆ , then M σ ∧(µ 1 × · · · × µ r ) is a global generalized cycle such that for ℓ ≤ d. More invariantly we have, cf. Definition 2.4, Given a point x, (3.1) holds as soon as σ defines J ∆ in a neighborhood of the point i(x) so we can assume that σ is a section of a trivial bundle. If the µ j are cycles, therefore these numbers coincide with the local intersection numbers (1.1) introduced by Tworzewski in [13], cf. Section 2.6 and [4, Section 10].
Remark 3.1. Tworzewski, [13], proved that there is a unique global cycle µ such that the sum of its multiplicities, of its components of various dimensions, at each point x ∈ V coincides with the sum of the local intersection numbers at x. Since this definition is local, it cannot carry global information. For instance, the self-intersection, in this sense, of any smooth curve Z in P 2 is just the curve itself, and therefore the Bézout formula, cf. (1.7), is not satisfied unless Z is a line.
. . , η m ) be a tuple of linear forms on C M +1 in general position. As usual we identify the η j with sections of the line bundle L = O(1) → P M and η with a section of E := ⊕ m 1 L. Similarly to Section 2.8 we let |η| • be the norm of the holomorphic tuple obtained from any fixed local frame for L so that dd c log |η| • is well-defined. Let Z be the plane of codimension m that η defines and let J → P M be the associated radical ideal sheaf.
Let µ be a fixed generalized cycle in P M of pure dimension d. For a generic choice of a = (a 1 , . . . , a d ) ∈ (∈ P m−1 ) d , the successive intersections 6 by divisors, cf. Section 2.8, in Proposition 4.1. If we take the mean value of (4.2) over (P m−1 ) d , with respect to normalized Haar measure, then we get the generalized cycle  Notice that M L,η ∧µ has support in Z ∩ |µ| so that we may identify V (J , L, µ) with an element in B(Z ∩ |µ|), cf. [5,Definition 9.6].
Let U ⊂ P M be an open set where we have a local frame e for L. For instance, each nontrivial section of L vanishes on a hyperplane H and thus gives rise to a local frame in the open set P M \ H. In U we have that with the metric on L| U such that |e| = 1, cf. [5,Remark 8.2]. It follows that local statements that hold for M η ∧ µ must hold for M L,η ∧ µ as well. In particular, if η defines the maximal ideal at x ∈ P M , then, in view of (2.15), By (2.12) and (4.4), in U we have the regularization  Assume that µ ∈ GZ d (X). We have the mass formula If m ≤ d, then the last term in (4.10) vanishes since (dd c log |η| 2 • ) m = 0 outside Z. For future reference we also point out the following invariance result. Assume that i : P M → P M ′ is a linear embedding of P M in P M ′ . Let p : P M ′ P M be a projective (generically defined) projection, i.e., induced by an affine projection C M ′ +1 → C M +1 , so that p • i is the identity on P M . Then p * η j are well-defined linear forms on P M ′ . Let η ′ be some additional linear forms on P M ′ that vanish on i(P M ). Proof. Since η ′ = 0 on the Zariski support of i * µ, M L,(p * η,η ′ ) ∧i * µ = M L,(p * η,0) ∧i * µ. Now the proposition follows from (2.1) and Proposition 4.3, or (4.6), since η = i * p * η.

B-intersection products on manifolds
Assume that µ 1 , . . . , µ r are cycles on a complex manifold Y of dimension n as in the introduction. It is well-known that if they intersect properly, then, see, e.g., [6,Chapter 12], one can define the wedge product [µ 1 ] ∧ · · · ∧ [µ r ] by means of appropriate regularizations, see, e.g., [7,Chapter III.3], and this current coincides with (the Lelong current of) the proper intersection cycle µ 1 · Y · · · · Y µ r , see, e.g., [6, page 212]. It is easy to see that the cycle µ = µ 1 × · · · × µ r and the diagonal ∆ in Y r = Y × · · · × Y intersect properly, and one can prove that if we identify ∆ and Y , then the proper intersection ∆ · Y r µ coincides with µ 1 · Y · · · · Y µ r . If the µ j do not intersect properly the basic idea is to define the intersection of ∆ and µ 1 × · · · × µ r , cf. Section 3. The advantage then is that one of the factors is a regular embedding.
Remark 5.2. If J is the radical ideal of a submanifold or a reduced locally complete intersection i : Z ֒→ Y of codimension κ and µ is a k-cycle in Y intersecting Z properly, then i * (Z ⋄ B(Y ) µ) is the proper intersection [Z] ∧ µ. In fact, in view of Definition 2.4 and Proposition 2.10, Thus, by (2.17), As above, notice that after identification of Y and ∆ we have Assume that µ 1 is a regular embedding. Contrary to the classical intersection product case it is not true in general that µ 1 ⋄ B(Y ) µ 2 and µ 1 · B(Y ) µ 2 coincide. Example 8.14 below shows that the B-self-intersection of the cusp µ = {x 3 1 This example also shows that the B-analogue of the classical self-intersection formula does not hold in general. However, it is true for smooth cycles.
Proposition 5.5 (Self-intersection formula). Let V ֒→ Y be a smooth subvariety of Y of codimension m. Then where the last equality follows from Proposition 2.10 and, since V is smooth, that

cf. (2.4). Thus we get (5.3).
Example 5.6. Let E be the exceptional divisor of the blow-up Y = Bl a P 2 → P 2 at a point a ∈ P 2 . Let L E → Y be the line bundle with a section that defines E. It follows which is expected in view of the classical self-intersection of E.
We have always coincidence of the various intersection products on cohomology level; recall the mappings (2.19) and (2.20).
Proposition 5.8. (i) If µ 1 , . . . , µ r are cycles in Y that intersect properly, then Proof. Assume that the µ j have dimensions d j , respectively. The assumption about proper intersection means that the set-theoretic intersection V = |µ 1 | ∩ · · · ∩ |µ r | has the expected dimension k := d 1 + · · · + d r − (r − 1)n and that µ 1 · Y · · · · Y µ r and µ 1 · B(Y ) · · · · B(Y ) µ r are elements in A k (V ) and B k (V ), respectively. Now (5.6) follows from (5.4) and Example 2.15. Let us now consider part (ii). We may assume that µ = τ * α, where τ : W → Y is proper holomorphic and α is a product of components of Chern or Segre forms, cf. (2.6). The assumption of proper intersection implies that h is not identically zero on |µ| = τ (W ) so that M h 0 ∧µ = 1 h=0 µ = τ * 1 τ * h=0 α = 0. Let ι be the regular embedding given by the ideal sheaf J h generated by h. We have N J h Y = L| h=0 , cf. Section 2.7. Thus We now consider the last equality in (5.7). Consider the commutative diagram where p is the projection on the first factor. By definition, cf.
(2.14) and (2.13), . Notice that g defines the graph G of τ in Y × W . Since divh and µ intersect properly, τ * h is generically non-vanishing on W and so h ⊗ 1 is generically non-vanishing on G. Thus, G and div(h ⊗ 1) intersect properly. The Zariski support of M g ℓ ∧[div(h ⊗ 1)] is G∩{h⊗1 = 0}, which thus has dimension dim W −1. Since M g ℓ ∧[div(h⊗1)] has dimension dim W + n − ℓ − 1 it follows from the dimension principle that M g ℓ ∧[div(h ⊗ 1)] = 0 for ℓ < n. Thus, S ℓ (J ∆ , divh × µ) = 0 for ℓ < n and from (5.9) we get , notice that g defines a regular embedding in Y × W of codimension n and that, since dim(G ∩ {h ⊗ 1 = 0}) = dim W − 1, the restriction of g to div(h ⊗ 1) defines a regular embedding in div(h ⊗ 1) of codimension n. Thus, by [5, Corollary 7.5], where J g is the ideal sheaf generated by g. Since (5.8) is commutative, (5.10) and (5.11) give • α) = dd c (log |h| 2 • µ) = divh · µ, finishing the proof. 6. The •-product on P n In this section we define the product (1.4) of generalized cycles on P n and prove Theorem 1.1. The first step is to define the join of two generalized cycles. For simplicity we first assume that r = 2. The mapping ). is well-defined outside the union of the two disjoint n-dimensional planes x = 0 and y = 0, and it has surjective differential. If µ 1 , µ 2 ∈ GZ(P n ), therefore p * (µ 1 × µ 2 ) is a well-defined current outside the indeterminacy set of p. We will see that p * (µ 1 × µ 2 ) extends in a natural way to a generalized cycle µ 1 × J µ 2 on P 2n+1 x,y .
Proof. Note that (ii) is a direct consequence of (i). Let X = P n ×P n and X ′ = Bl P 2n+1 x,y . We may assume that µ = τ * α, where τ : W → X is proper and α is a product of components of Chern forms. Consider the fibre square Since p is smooth it follows that the fibre product W ′ = W × X Y is smooth, cf. (6.5) below. The pullbackπ * α is a product of Chern forms on W ′ and thus ρ * π * α is a generalized cycle on X ′ . We claim that (6.4) ρ * π * γ = p * τ * γ for any smooth form γ. Taking (6.4) for granted we conclude that p * µ = p * τ * α is a generalized cycle, which proves (i). It is enough to prove (6.4) for all smooth forms γ with small support. Notice that locally in X, say in a small open set U , X ′ | U is biholomorphic to U × P 1 . Let us assume that τ * γ has support in an open set U ⊂ X, where X ′ = U × P 1 t . Letting W = τ −1 (U ), by the definition of fiber product, (6.5) W × U (U × P 1 t ) = {(w, x, t); τ (w) = p(x, t) = x} = {(w, τ (w), t); w ∈ W } ≃ W × P 1 and ρ(w, t) = (τ (w), t). Now (6.4) is obvious.
Proof. We may assume that the µ j have pure dimension. There are currents a j in P n such that dd c a j = µ j − (deg µ j )ω k j if dim µ j = n − k j , whereω is the Fubini-Study form on P n . It follows that there is a current A on P n x 1 × · · · × P n x r such that cf. Lemma 2.1. Applying π * p * , it is enough to show that deg (ω k 1 × J · · · × Jω kr ) = 1; but this is obvious if we just notice that π * p * of a hyperplane in P n x 1 × · · · × P n x r induced by a hyperplane in one of the factors P n x j is a hyperplane in P r(n+1)−1 x 1 ,...,x r and replace eacĥ ω k j by the intersection of k j generic hyperplanes.
For the last argument one can also observe that log (|x 1 | 2 + · · · + |x r | 2 )/|x j | 2 is a well-defined locally integrable function on P r(n+1)−1 x 1 ,··· ,x r and that be the parametrization of the join diagonal ∆ J in P r(n+1)−1 and let J J be the associated sheaf. Notice that J J is generated by the (r − 1)(n + 1) linear forms, i.e., sections of L = O(1), we see that η is a minimal generating set.
Proof. First notice that j * ω P r(n+1)−1 =ω P n , where j is defined in (6.8) andω P M denote the Fubini-Study form on P M . Therefore, for µ ∈ GZ k (P n ), , and thus (6.15) follows in view of Proposition 6.5.
However, as mentioned in the introduction, the condition (1.6) is not necessary for (1.7) to hold. For instance, by Proposition 6.7, the •-product is not affected if we perform the multiplication in a larger P n ′ . Thus, as mentioned already in the introduction, the self-intersection of a k-plane is the k-plane itself, in particular, the self-intersection of a point is the point itself. On the other hand, clearly the product of two distinct points vanishes. In this case the last term in (6.15) carries the "missing mass" in the Bézout formula.
We are now in position to prove Theorem 1.1.
Proof of Theorem 1.1. The first statements, about multlilinearity, commutativity and the support, are already discussed after Definition 6.6. Since local intersections numbers (multiplicities) are locally defined we can work in an affinization and use the results from [4, Sections 9 and 10] to prove (1.5). However, we omit the details since it is also a direct consequence of the global Proposition 7.1 below, cf. (3.2) and (7.3).
In the discussion after the proof of Proposition 6.8 is noticed that (1.7) holds if (1.6) is fulfilled. If µ j are effective, then so is µ 1 × J · · · × J µ r , and it follows that (6.11), and hence µ 1 • · · · • µ r , are effective, cf. (4.6). Moreover deg µ j are positive and the last term in (6.15) is non-positive so we get (1.8).
Remark 6.10. Consider (6.7) and the corresponding diagram (6.2). By abuse of notation, let ∆ J denote the preimage under π of the join diagonal, let J J denote the sheaf in Bl P r(n+1)−1 corresponding to ∆ J , and let j denote the embedding of P n in Bl P r(n+1)−1 as ∆ J induced by (6.8). Since (6.11) has support on ∆ J and Bl P r(n+1)−1 and P r(n+1)−1 coincide in a neighborhood of ∆ J we can alternatively think of (6.11) as a generalized cycle on Y .
7. Relation to the · B(P n ) product In this section we prove Theorem 1.2. For simplicity let us restrict from now on to the case r = 2; the general case is handled in a similar way.
Next, consider the mapping where we are using the notation from Section 6 and where j is given by (6.8).
Remark 7.2. There are classical mappings A(P n × P n ) → A(P n ) analogous to i ! and j ♭ . If µ 1 and µ 2 are cycles and µ = µ 1 × µ 2 , then, see [9,Example 8.4.5], the analogue of Proposition 7.1 holds for the component of dimension ρ, which is the component of main interest also for us. However, the argument given in [9] cannot be transferred to to the B-setting.
Proof of Proposition 7.1. Let Bl P 2n+1 x,y be as in Section 6. Since Bl P 2n+1 x,y coincides with P 2n+1 in a neighborhood of ∆ J , the restrictions of c(N J J Bl P 2n+1 x,y ) and c(N J J P 2n+1 ) to ∆ J coincide, and moreover, π * p * µ and p * µ coincide on ∆ J , cf. Remark 6.10. Therefore j ♭ coincides with the mapping x,y with P n . Hence it suffices to prove that i ! coincides with (7.4).
Let M = P n so that ∆ = i(M ) and ∆ J = j(M ) and let X = P n ×P n and Y = Bl P 2n+1 x,y . Then . Let E → Y and F → X be Hermitian vector bundles with holomorphic sections φ and σ that define J j(M ) and J i(M ) , respectively. Fix Hermitian metrics on N j(M ) Y and N i(M ) X and letĉ(N j(M ) Y ) andĉ(N i(M ) X) be the associated Chern forms. Moreover, let µ ∈ GZ(P n × P n ) denote also a fixed representative of µ ∈ B(P n × P n ).  and for any Hermitian metric on L, Taking this lemma for granted we can conclude the proof of Proposition 7.1. We have to prove that if µ 1 and µ 2 are the unique elements in GZ(M ) such that In view of (7.5) and (2.10) we havê . Therefore, cf. (2.5), From (7.6) and (7.7) we get which means that p * j * µ 2 ∼ i * µ 1 on X. Since p * j * = i * and (2.11) is injective, we conclude that µ 1 ∼ µ 2 on M . Thus Proposition 7.1 is proved.
Proof of Lemma 7.3. Let us use the notation N σ X for N i(M ) X etc. We first consider (7.5). Notice that, with the notation from [5,Section 7], for any columns of minimal sets of generators s, s ′ of J i(M ) = J σ at points on iM ⊂ X there is an invertible matrix g such that s ′ = gs. A section ξ of the normal bundle N σ X can be defined as a set of holomorphic tuples ξ(s) such that gξ ( Thus the restriction to j(M ) of the matrices To prove (7.6) we must return to the definition of p * , so let us assume that µ = τ * α and recall the fiber square (6.3). We may also assume that W is chosen so that τ * σ is principal and hence ρ * φ is a regular embedding of codimension 2 in W ′ . We claim that In fact, notice thatπ * τ * σ combined with the section ρ * η generate the same sheaf as ρ * φ. Arguing precisely as above for (7.5) we then get (7.8).
We now claim that where Z τ * σ is the fundamental cycle of the ideal sheaf generated by τ * σ etc. Since it is an equality of currents it is a local statement. By the dimension principle it is then enough to check it in an open set U ⊂ W where Z τ * σ is smooth andπ −1 U ≃ U × P 1 t in suitable coordinates (x, t) so thatπ is (x, t) → x, cf. the proof of Lemma 6.1. Thus, we may assume that the ideal generated by τ * σ is generated by x ℓ 1 in U . Then ρ * φ is generated by (x ℓ 1 , t) and (7.9) is reduced to the equality ℓ[x 1 = 0] =π * (ℓ[x 1 = 0] × [t = 0]). Next we claim that (7.10) M τ * σ ∼π * ĉ(ρ * L)∧M ρ * φ on W . In fact, from [5, Proposition 1.5] we have By (7.8), noting that (2.10) holds for Segre forms as well in view of (2.4), we have that By (2.1) and (2.5) for Segre forms, thus In view of (7.9) and [5, Proposition 1.5], now (7.10) follows.

Examples
We shall now present some further results on our products and various examples. We first consider an embedding i : P M → P M +1 as a linear hyperplane defined by the linear form ξ. Let a ∈ P M +1 be a point outside this hyperplane and let p : P M +1 P M be the induced projection. If Y is the blowup of P M +1 at a we have the diagram As in Section 6 we see that given µ ∈ GZ(P M ) the current p * µ has a well-defined extension to an element π * p * µ in GZ(P M +1 ), cf. Lemma 6.1. Proof. Since the support of i * µ is contained in the hyperplane i(P M ) and Y and P M +1 coincide in a neighborhood of i(P M ), the right-hand side of (8.1) is well-defined. Now (8.1) follows from (2.14) and (4.4) since p • i = id so that i * p * η = η.
For the second equality first notice that both sides of (8.2) have support on i(P M ) and that Y and P M +1 coincide in a neighborhood of i(P M ). For the rest of this proof let i denote also the inclusion of P M in Y . Since η defines a regular embedding, it follows from [5,Example 7.8] that i * M η ∧µ =ĉ(π * L)∧M (p * η,π * ξ) ∧p * µ if µ is a smooth form; here we use the standard metric on L. It follows in general, by assuming that µ = τ * α, τ : W → P M , and pulling back to W and W ′ according to the fibre square cf. the proofs of Lemmas 6.1 and 7.3 above. Sinceĉ(π * L) = 1 + π * ω we get Thus, in view of (4.8), where we for the last equality have used that M p * η,π * ξ 0 ∧p * µ = 0 so that we may let the sum start from j = 0; indeed, M p * η,π * ξ 0 ∧p * µ = 0 since ξ is generically non-vanishing on the Zariski support of p * µ. Thus, (8.2) follows by applying π * .
We will now deduce a formula for A•µ when A is a linear subspace. Proposition 8.2. Assume that A is a linear subspace of P n of dimension m, defined by n − m linear forms σ 1 , . . . , σ n−m . If µ ∈ GZ d (P n ), then Proof. Let us use the notation from Section 6. By (6.13) the •-product is not affected by a linear change of coordinates on C n+1 x and therefore we can assume that x = (x ′ , x ′′ ) and σ = x ′′ . Then we need to prove that in B(P n ). Recall that η = x − y. By definition we have, cf. (2.14) and (4.4), Proposition 8.3. Assume that µ ∈ B(P n ). Then If a is a point, then Proof. From Proposition 8.2 we have that 1 P n •µ = M L,0 ∧µ = µ and so (8.10) follows. To see (8.11) let ξ be linear forms that define a. By (8.3) and (4.5) we have a • µ = M L,ξ ∧µ = mult a µ · [a].
Example 8.6. Let a = [1, 0, . . . , 0] ∈ P n and let θ = dd c log(|x 1 | 2 + · · · + |x n | 2 ) in P n x 0 ,...,xn . For each k, θ k is a well-defined positive closed current, see, e.g., [7,Chapter III]. It is an irreducible generalized cycle of dimension n − k and degree 1, with mult a θ k = 1 and mult x θ k = 0 for x = a; for k < n, θ k has Zariski-support equal to P n whereas θ n = [a], see [5, Example 6.3] and cf. Example 2.9. One can think of θ k as an (n−k)-plane through a moving around a. We claim that In fact, notice that both sides coincide outside a in virtue of Proposition 8.4. Thus they can only differ on a generalized cycle with Zariski support at a, that is, m[a] for some integer m. Since the degree of θ is 1, also the degree of θ k• must be 1 by the Bézout formula (1.7); indeed note that ρ in (1.6) in this case equals n − k ≥ 0. Since the degree of the right hand side is 1 it follows that m = 0 and hence (8.18) holds.
Example 8.7. Let n = 2, let a and θ be as in the previous example, and let ℓ be a line through a. Then In fact, in view of (8.14), outside a, θ • [ℓ] = θ∧[ℓ], which vanishes since the pullback of θ to ℓ vanishes. By the same argument as in Example 8.6, using Bézout's formula (1.7), we get (8.19).
Example 8.8. Let µ 1 , . . . , µ r , r ≥ 2, be different lines through a ∈ P n . We claim that µ 1 • · · · •µ r = [a]. In fact, since the set-theoretic intersection is a, the product must be m[a] for some integer m. Since the µ j are effective it follows from (1.8) that m is 1 or 0. By (1.5) it is enough to determine the local intersection number ǫ 0 (µ 1 , . . . , µ r , a), and thus we can assume that the µ j are lines through a = 0 in C n . In view of (3.1) and (4.4) this equals the multiplicity of M L,η r ∧ (µ 1 × · · · × µ r ), where η is a tuple of linear forms defining the diagonal in (C n ) r = C n × · · · × C n . This, in turn, can be computed by intersecting µ 1 × · · · × µ r by r generic hyperplanes div(α · η), see Section 4. Doing this, we get [0] with multiplicity 1, which proves the claim. Example 8.9. Let G be the graph in C 6 x,y = C 6 x 1 ,x 2 ,x 3 ,y 1 ,y 2 ,y 3 of the function , and let Z be the closure in P 6 x 0 ,x,y . Clearly Z is irreducible of dimension 3. We want to compute A•Z, where A = {y = 0}. By (8.3), In view of Section 4 we can compute the right hand side by successively intersecting [Z] by hyperplanes divh j , where h 1 = α · y, h 2 = β · y, and h 3 = γ · y for generic α, β, γ ∈ P 2 , and then taking averages. The map P 3 ], lifts to an injective holomorphic map from the blow-up Y = Bl t 0 =t 3 =0 P 3 to P 6 with image Z. Then Z can be parametrized by two copies of P 2 × C, identified by s = uv, sσ = u. Let Z 1 and Z 2 be the image of the first and second map, respectively. Since Z 2 ∩ A = ∅, the SV-cycle we are to compute is contained in Z 1 .
We now give an example that shows that the •-product is not associative. It follows that neither · B(P N ) is associative in B(P n ). In fact, it follows from (8.21), (8.22), and Theorem 1.2, that

whereas
(H 3 · B(P n ) H 2 ) · B(P n ) Z = ω∧A + m[a] and these right-hand sides are not equal in B(P n ).
Example 8.11. Let γ be a smooth curve in P 2 of degree d. It is well-known, see, e.g., [4], that local intersection numbers are biholomorphic invariants. Therefore, since the •-self-intersection of a line is the line itself, cf. the discussion after Proposition 6.8, it follows from (1.5) that at each x ∈ γ, mult x (γ • γ) 1 = 1 and mult x (γ • γ) 0 = 0. Thus, since |γ • γ| ⊂ γ, in view of the dimension principle, γ • γ = γ + µ where µ has dimension 0 and Zariski support equal to γ. By the Bézout formula (1.7) the degree of µ must be d 2 − d. We can think of µ as d 2 − d points moving around on γ.
Example 8.14. Let Z ⊂ P 2 be the cusp as in the previous example. Since Z is a regular embedding in P 2 we can also form the product Z⋄ B(P 2 ) Z. Let J → P 2 be the sheaf defining Z. If i : Z ֒→ P 2 , then i * J Z = 0 so that S(J Z , Z) = S(0, Z) = [Z], cf. Section 2.5. Moreover, N Z P 2 = O(3)| Z , so that c 1 (N Z P 2 ) = 3ω. Thus It is somewhat less obvious that µ is cohomologous with ω on Z.
Example 8.14 also shows that the self-intersection formula, Proposition 5.5, does not generalize to non-smooth Z.