On intersection product of analytic cycles

Sławomir Rams

IMUJ PREPRINT 1998/01

Abstract

We prove that generalized index of intersection of an analytic set with a closed submanifold (Thm 4.3) and intersection product of analytic cycles (Thm 5.4) , which are defined in [T], are intrinsic. We define intersection product of analytic cycles on a reduced analytic space (Def. 5.8) and prove a relation of its degree with the exponent of proper separation (Thm 6.3).

1 Introduction

In the paper [T] an intersection product of two analytic cycles on a (complex) manifold is constructed. This construction is based on a pointwise defined so-called extended index of intersection g̃(Z, S)(c) of an analytic subset Z of a manifold N with a submanifold S in a point c (see [T] p. 185). The notion of intersection product of arbitrary analytic cycles defined in [T] generalizes the intersection product constructed in [Dr], [BH], and the isolated improper intersection multiplicity studied in [ATW].

This paper is a continuation of [T] and its main aim is to prove that both the extended index of intersection and the intersection product (in the sense of [T]) are intrinsic. In the case of isolated improper intersection multiplicity this fact is proved in [ATW]. The idea of the proof from [ATW] does not work in the general case. Thank to the results from the first five sections of our paper the construction from [T] can be applied to analytic cycles on reduced analytic spaces (Def. 5.8), which leads to a generalization (Thm 6.3) of some estimates of the exponent of proper separation (see [Tw], [CT], [Cg]).

The reader is expected to be familiar with the notion of the multiplicity of proper intersection of analytic sets (see [Dr]), intersection product of analytic cycles (see [T]) and some properties of positive holomorphic cycles (a.k.a. chains) (see [Ch, §12.2], [T, §3]). For the convenience of reader we present the construction of intersection product of analytic cycles on a manifold (in the sense of [T]) in the third section of this paper. The definition of the topology of chains, which plays crucial role in some proofs, can be found in [R1]. We apply it only on the space of positive chains, when it coincides with the classical weak topology of currents (see [Ch]). Throughout the text we work over the field ℂ of complex numbers. All the manifolds are assumed to be second-countable.

2 Preliminaries

Let N be an n-dimensional complex manifold, M be an m-dimensional closed submanifold of N, and let S be an s-dimensional closed submanifold of M. We fix a purely k-dimensional analytic set Z ⊂ M, and a point c ∈ Z ∩ S. Let U be a neighbourhood of c in N.

We define the following family of hypersurfaces:

B(U) := {H ⊂ U : H is a smooth hypersurface in U, H ⊃ S ∩ U}.

We endow B(U) with the topology of (n − 1)-chains (see [R1]) and obtain a metrizable topological space (see [R1] Thm 3.6), which allows us to use the notion of sequence to prove that a subset of B(U) is open. Since the elements of B(U) are positive cycles we can apply the results from [T].

We will often restrict our attention to polydiscs. Throughout the text E denotes the unit disc in ℂ.

Let us recall some basic notions associated to analytic cycles. We will use them frequently in the next two sections. Let A = ∑_{j∈J} α_{j}C_{j} be an analytic cycle on N. The degree of such a cycle at the point c is defined to be the sum ∑_{j∈J} α_{j}ν(C_{j}, c), where ν(C_{j}, c) stands for the degree of the component C_{j} at the point c. By Prop 2.1 from [T] the function N ∋ c ↦ ν(A, c) ∈ ℤ is analytically constructible, and for each analytically constructible function f : N ↦ ℤ there exists a unique cycle such that its degree equals the value of f at every point of N. This cycle will be called the cycle defined by the function f. The analytic cycle A has unique decomposition into the sum of j − cycles (formal combinations of analytic sets of pure dimension j) A = ∑_{j=0}^{n} T_{(j)}. The extended degree of A at c is defined by the formula:

ν̃(A, c) := (ν(T_{(n)}, c), …, ν(T_{(0)}, c) ∈ ℤ.

For the cycle A = ∑_{j∈J} α_{j}C_{j} the part of A supported by S is defined to be

A^{S} := ∑_{j∈J,C_{j}⊂S} α_{j}C_{j}.

Let V be an open subset of N. The restriction of the cycle A to this open set is defined to be ∑_{j∈J} α_{j}(C_{j} ∩ V) and denoted by A ∩ V.

We end this section with the following simple lemma.

Lemma 2.1 Let N = E^{n}, S = E^{s}×{0}^{n−s}, and let {F_{ν}}_{ν=1}^{∞} be a sequence of orthogonal mappings such that each F_{ν} = idℂ^{s} ⊕f_{ν}, where f_{ν} : ℂ^{n−s} ↦ ℂ^{n−s}. If F_{ν} → idℂ^{n} and a hypersurface H belongs to B(E^{n}), then

F_{ν}(H) → H.

Proof: Let us suppose that there exists a compact set K ⊂ (E^{n} \ H) and a sequence {x_{ν}}_{ν=1}^{ν=∞} ⊂ H such that all F_{ν}(x_{ν}) belong to the set K. We can assume that x_{ν} converges to a point x_{0} which belongs to H , and the sequence F_{ν}(x_{ν}) converges to an y_{0}. Since F_{ν} → idℂ^{n}, we have

x_{0} = y_{0} ∈ H ∩ K,

which contradicts our choice of K and {x_{ν}}_{ν=1}^{ν=∞}.

It suffices now to check the condition (3) of Lemma 3.2 from [T]. Fix x ∈ Reg(H). We can assume that H = V (f) and f′(x) ≠ 0. Choose v ∈ ℂ^{n} such that f′(x).v ≠ 0 and positive ε which satisfies the following conditions:

(a) (x + (2ε)\overline{E}v) ∩ H = \{x\},

(b) each function f ∘ F_{ν}^{−1} is analytic in a neighbourhood of x + (2ε)\overline{E}v.

Since (f ∘ F_{ν}^{−1})|(x + (2ε)\overline{E}v) converges uniformly to f|(x + (2ε)\overline{E}v), we can apply Hurwitz theorem and obtain the equality

deg(F_{ν}(H) · (x + εEv)) = 1

for sufficiently large ν. □

3 Intersections on manifolds

In this section we maintain the setup of the previous one. We recall here some basic facts from [T] but our main aim is to study the behaviour of systems of hypersurfaces which appear in Algorithm 4.1 from [T].

Let us recall the definition of a family of systems of hypersurfaces, which apears in [T] (see p. 184). By H(U, Z) we denote the family of all H := (H_{1}, …, H_{n−s}) satisfying the following conditions:

(a) H_{j} is a smooth hypersurface of U containing U ∩ S for j = 1, …, n − s,

(b) ⋂_{j=1}^{j=n−s} T_{x}(H_{j}) = T_{x}(S) for each x ∈ U ∩ S,

(c) ((U \ S) ∩ Z) ∩ H_{1} ∩ … ∩ H_{j} is analytic subset of U \ S of pure dimension k − j (or empty) for j = 0, 1, …, k.

Obviously the family H(U, Z) depends on the chosen submanifold S and we will write H_{S}(U, Z) to point out the submanifold S when necessary.

Any system H := (H_{1}, …, H_{n−s}) belonging to H(U, Z) enables us to construct an analytic cycle Z · H in S ∩ U by the following procedure:

Algorithm (4.1) from [T] Step 0: Let Z_{0} = Z ∩ U. Then Z_{0} = Z_{0}^{S} + (Z_{0} − Z_{0}^{S}), where Z_{0}^{S} is the

part of Z_{0} supported by S ∩ U (see [T] p. 182).

Step 1: Let Z_{1} = (Z_{0} − Z_{0}^{S}) · H_{1}. Then Z_{1} = Z_{1}^{S} + (Z_{1} − Z_{1}^{S}), where Z_{1}^{S}

is the part of Z_{1} supported by S ∩ U. Step 2: Let Z_{2} = (Z_{1} − Z_{1}^{S}) · H_{2}. Then Z_{2} = Z_{2}^{S} + (Z_{2} − Z_{2}^{S}), where Z_{2}^{S}

is the part of Z_{2} supported by S ∩ U.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Step n − s: Let Z_{n−s} = (Z_{n−s−1} − Z_{n−s−1}^{S}) · H_{n−s}. Now we have a decomposition Z_{n−s} = Z_{n−s}^{S}+(Z_{n−s}−Z_{n−s}^{S}), where |Z_{n−s}−Z_{n−s}^{S}|∩S = ∅.

The cycle Z · H := Z_{0}^{S} + … + Z_{n−s}^{S} will be called the result of Algorithm 4.1 applied to the system H, the set Z and the submanifold S.

We define the extended index of intersection g̃(Z, S)(c) of Z and S at c by the formula (see [T] Def 4.2):

g̃(Z, S)(c) := min_{lex}{ν̃(Z · H, c) : H ∈ H(V, Z), V is a neighbourhood of c},

where lex denotes the lexicographical ordering. The sum of all elements of g̃(Z, S)(c) is called index of intersection of Z and S at c and denoted by g(c) (we will write sometimes g(Z, S)(c) to specify the set and the manifold in question). Apparently both g̃ and g depend on the ambient manifold. Therefore we will write g_{N} (resp. g̃_{N}) instead of g (resp. g̃) when necessary.

Let X, Y be irreducible analytic subsets of the manifold N, a ∈ N, and let Δ_{N} denote the diagonal in N^{2}. By Thm. 6.2 from [T] the function

N ∋ x ↦ g_{N^{2}}(X × Y, Δ_{N}, (x, x)) ∈ ℕ

is analytically constructible. The cycle defined by this function is called the intersection product of the sets X, Y and denoted by X • Y. We extend this definition to the case of arbitrary analytic cycles in the usual way i.e. by ℤ-linearity (see [T] Def 6.4).

Having presented the construction which plays fundamental role in this paper, we can introduce some useful notation. Given a system of hypersurfaces (H_{1}, …, H_{p}) and an open set V, we define a system of locally analytic sets by the equality:

(H_{1}, …, H_{p})|_{V} := (H_{1} ∩ V, …, H_{p} ∩ V).

Let us now fix an integer 1 ≤ i ≤ n − s and a system of elements of B(U) H = (H_{1}, …, H̆_{i}, …, H_{n−s}). For an H ∈ B(U) we define the system

H_{i}(H) := (H_{1}, …, H_{i−1}, H, …, H_{n−s}).

Let us obseve that for any system H = (H_{1}, …, H_{n−s}) of n−s elements of B(U) which satisfies the condition (c) of the definition of the family H(U, Z) we can construct the cycle H·Z. In this case the equality |Z_{n−s}−Z_{n−s}^{S}|∩S = ∅ may no longer hold.

• By G(U, Z) we will denote the family of systems H = (H_{1}, …, H_{n−s}), where each H_{j} belongs to B(U), which satisfy the condition:

((U \ S) ∩ Z) ∩ H_{1} ∩ … ∩ H_{j} is an analytic subset of U \ S of pure dimension k − j (or empty) for j = 1, …, k.

If V is a neighbourhood of the point c and a system of hypersurfaces H satisfies the condition H|_{V} ∈ G(V, Z), then we will write ν̃(H ·Z, c) to denote ν̃((H|_{V} · Z), c).

In the sequel we will need the following families of hypersurfaces:

• By I_{i}(H, U, Z) we denote the set of H ∈ B(U) for which there exists a neighbourhood V of c such that H_{i}(H)|_{V} belongs to H(V, Z).

• J_{i}(H, U, Z) := {H ∈ I_{i}(H, U, Z) : ν̃(H_{i}(H) · Z, c) = g̃(Z, S)(c)}. The following lemma holds:

Lemma 3.1 Let H = (H_{1}, …, H_{n−s}) ∈ H(U, Z) and let a sequence {H^{j}}_{j=1}^{∞} of elements of B(U) converge to H_{1}. If H_{1} ∈ J_{1}(H, U, Z) then there exists a neighbourhood V of c such that for almost all j we have

H_{1}(H^{j})|_{V} ∈ G(V, Z).

Proof: We introduce the following notation:

• If Ũ is a neighbourhood of c such that we can apply first l steps of Algorithm (4.1) from [T] to the system H_{1}(H^{j})|Ũ, the set Z and S, then Z_{l}(j, Ũ), Z_{l}^{S}(j, Ũ) stand for the results of this procedure.

We can assume that no component of Z is included in the submanifold S. Consequently there exists a neighbourhood U_{0} of c and an integer j_{0} such that H_{1}^{j} ∩ U_{0} meet Z ∩ U_{0} properly for j > j_{0}, which means that the cycles Z_{1}(j, U_{0}) are well-defined for j > j_{0}.

We can apply Thm 3.6 from [T] and obtain Z_{1}(j, U_{0}) → Z_{1}(0, U_{0}).

By Thm 3.4 from [T] there exists a neighbourhood Ũ_{1} ⊂ U_{0} of c such that

(Z_{1}(j, U_{0}) − Z_{1}^{S}(j, U_{0})) ∩ Ũ_{1} → (Z_{1}(0, U_{0}) − Z_{1}^{S}(0, U_{0})) ∩ Ũ_{1}.

Since ν((Z_{1}(0, U_{0}) − Z_{1}^{S}(0, U_{0}))^{H_{2}}, 0) = 0, we can apply Thm 3.4 from [T] to find a neighbourhood Û_{1} ⊂ Ũ_{1} such that

((Z_{1}(j, U_{0}) − Z_{1}^{S}(j, U_{0}))^{H_{2}}) ∩ Û_{1} → 0,

where 0 is the zero cycle in Û_{1}. Consequently there exists a neighbourhood U_{1} ⊂ Û_{1} of c and an integer j_{1} such that Z_{2}(j, U_{1}) are well-defined for j > j_{1}.

Since Z_{2}(j, U_{1}) = (H_{2} ∩ U_{1}) · (Z_{1}(j, U_{1}) − Z_{1}^{S}(j, U_{1})) we apply Thm 3.6 from [T] again and obtain the convergence Z_{2}(j, U_{1}) → Z_{2}(0, U_{1}). Consequently this procedure can be repeated until we find a neighbourhood U_{n−s−1} of c and an integer j_{n−s−1} such that the cycles Z_{n−s}(j, U_{n−s−1}) are well-defined for j > j_{n−s−1}. Put V := U_{n−s−1}. □

We can now prove the following property of the family J_{i}(H, U, Z) .

Proposition 3.2 Under the main assumptions of this section the following inclusion

J_{i}(H, U, Z) ⊂ Int( I_{i}(H, U, Z))

holds, where I_{i}(H, U, Z) is considered as a subset of the space B(U).

Proof: Let H̆_{i} belong to J_{i}(H, U, Z). It suffices to prove that there exists a neighbourhood V of H̆_{i} in B(U) such that every H_{i} ∈ V satisfies the following conditions:

(a) ⋂_{j=1}^{j=n−s} T_{c}H_{j} = T_{c}S,

(b) there exists a neighbourhood Ũ of c such that Z ∩(Ũ \S)∩H_{1} ∩…∩H_{j} is analytic subset of Ũ \ S of pure dimension k − j (or empty) for j ≤ k.

To find a neighbourhood consisting only of hypersurfaces satisfying (a) we can assume that U = (1 + ε)E^{n}, where ε > 0, S = (1 + ε)E^{s} × {0}^{n−s}, c = 0. Let us observe that ⋂_{j≠i} T_{0}H_{j} = T_{0}S + l, where l is a line. Choose v ∈ (T_{0}S + l) \ T_{0}S. By Def. 2.4 from [R1] the condition v ∉ T_{0}H defines a neighbourhood of T_{0}H̆_{i}. Therefore, it suffices to prove continuity of the mapping:

B(U) ∋ H ↦ T_{0}H ∈ B(U).

Let {H^{ν}}_{ν=0}^{ν=∞} ⊂ B(U) and let H^{ν} converge to H^{0}. Without loss of generality H^{0} = (1 + ε)E^{n−1} × {0}. For sufficiently large ν we have the equalities

(\overline{E^{n−1}} × ∂E) ∩ H^{ν} = ∅ and μ(π |_{H^{ν}∩E^{n}}) = 1,

where π(x_{1}, …, x_{n}) := (x_{1}, …, x_{n−1}) and μ denotes the multiplicity of holomorphic covering . Consequently almost all hypersurfaces H^{ν} ∩ E^{n} are graphs of holomorphic mappings G_{ν} : E^{n−1} ↦ E, and G_{ν} converge uniformly to G_{0}. The latter fact yields convergence G′_{ν}(0) → G′_{0}(0) and we get

T_{0}H^{ν} → T_{0}H^{0}.

To prove the existence of a neighbourhood satisfying (b) we can assume that i = 1. Suppose that there exists a sequence {H_{1}^{j}}_{j=0}^{∞} ⊂ B(U) such that H_{1}^{j} → H_{1}^{0}, H_{1}^{0} ∈ J_{1}(H, U, Z), and H_{1}^{j} do not satisfy the condition (b) for nonzero j. Existence of such a sequence contradicts Lemma 3.1. □

Now we can prove the main theorem of this section.

Theorem 3.3 The family J_{i}(H, U, Z) is open subset of B(U).

Proof: By the previous proposition it suffices to prove that J_{i}(H, U, Z) is open subset of I_{i}(H, U, Z) . Let us suppose that there exists a sequence {H^{j}}_{j=1}^{∞} ⊂ I_{i}(H, U, Z) \ J_{i}(H, U, Z) which converges to a hypersurface H^{0}, where H^{0} ∈ J_{i}(H, U, Z).

Let us introduce the following notation:

• Z_{l}(j), Z_{l}^{S}(j) stand for the results of the l^{th} step of Algorithm (4.1) from [T] applied to the systems H_{i}(H^{j}), (observe that Lemma 3.1 allows us to assume that all those cycles are well-defined),

• (ν_{s}^{j}, …, ν_{0}^{j}) := ν̃(H_{i}(H^{j}) · Z, c),

• α := max{l : lim sup ν_{l}^{j} > ν_{l}^{0}}.

Obviously Z_{0}^{S}(j) = Z_{0}^{S}(0) and ν_{l}^{j} = ν_{l}^{0} for l ≥ k − i + 1. We apply Thm 3.6 from [T] and get Z_{1}(j) → Z_{1}(0). By Thm 3.4 from [T] there exists a neighbourhood U_{i} of c such that

(a_{i}) Z_{i}^{S}(j) ∩ U_{i} → Z_{i}^{S}(0) ∩ U_{i}
(b_{i}) (Z_{i}(j) − Z_{i}^{S}(j)) ∩ U_{i} → (Z_{i}(0) − Z_{i}^{S}(0)) ∩ U_{i} .

The convergence (a_{i}) yields that lim sup ν_{i}^{j} = ν_{i}^{0}, which gives α < i.

Since H^{0} ∈ J_{i}(H, U, Z) and (b_{i}) holds, we can apply Thm 3.4 from [T] again to find U_{i+1} satisfying the conditions (a_{i+1}), (b_{i+1}). Finally we get the inequality α < 0, which contradicts our choice of the sequence {H^{j}}_{j=1}^{∞}. □

4 Extended index of intersection

We keep the setup of the previous section. In this section our main aim is to prove that the extended index of intersection of the set Z and the submanifold S at the point c does not depend on the ambient manifold N.

In the sequel we will need the following two simple lemmas about systems of hypersurfaces in polydiscs.

Lemma 4.1 Let N = E^{n}, M = E^{m} × {0}^{n−m}, S = E^{s} × {0}^{n−s} and let Z be a purely k-dimensional analytic subset of M. If V is a neighbourhood of 0 in M, and H = (H_{1}, …, H_{m−s}) ∈ H(V, Z), then there exist hypersurfaces H_{m−s+1}, …, H_{n−s} such that the system

H̃ := (H_{1} × E^{n−m}, …, H_{m−s} × E^{n−m}, H_{m−s+1}, …, H_{n−s})

belongs to H(V × E^{n−m}, Z), and the equality ν̃(H̃ · Z, 0) = ν̃(H · Z, 0) holds.

Proof: We introduce the following notation:

• Z_{j}, Z_{j}^{S} (resp. Z̃_{j}, Z̃_{j}^{S}) denote the result of the j^{th} step of Algorithm (4.1) from [T] applied to the system H (resp. H̃), the set Z and S.

Choose a system of hyperplanes H_{m−s+1}, …, H_{n−s} which satisfies the condition ℂ^{s} ×{0}^{m−s} ×ℂ^{n−m} ∩⋂_{j=m−s+1}^{j=n−s} H_{j} = ℂ^{s} ×{0}^{n−s}. The assumption Z ⊂ M gives the inclusion

⋂_{j=1}^{m−s}(H_{j} × E^{n−m}) ∩ Z ⊂ ( ⋂_{j=1}^{m−s} H_{j}) × {0}^{n−m} = S,

which yields that H̃ := (H_{1} × E^{n−m}, …, H_{m−s} × E^{n−m}, H_{m−s+1}, …, H_{n−s}) belongs to H(V × E^{n−m}, Z) and the cycles Z̃_{j}^{S} vanish for j > m − s.

It sufices to prove the equality Z_{j} = Z̃_{j} for j ≤ m − s. We can assume that V = M and proceed by induction on j.

Obviously the equality Z_{0} = Z̃_{0} holds. Suppose that Z_{j−1} = Z̃_{j−1}. Thm 2.2 from [TW] yields the equality

(H_{j} × E^{n−m}) ·_{N} (Z̃_{j−1} − Z̃_{j−1}^{S}) = ((H_{j} × E^{n−m}) ·_{N} M) ·_{M} (Z̃_{j−1} − Z̃_{j−1}^{S}).

Since (H_{j} × E^{n−m}) ·_{N} M = H_{j} and Z̃_{j−1} − Z̃_{j−1}^{S} = Z_{j−1} − Z_{j−1}^{S} we get the equality Z_{j} = Z̃_{j}. □

Lemma 4.2 Let N = E^{n}, M = E^{m} × {0}^{n−m}, S = E^{s} × {0}^{n−s} and let Z be a purely k-dimensional analytic subset of M. If U is a neighbourhood of 0 in N, and the system H̃ = (H̃_{1}, …, H̃_{n−s}) ∈ H(U, Z) satisfies the condition:

⋂_{1}^{m−s} T_{0}(H̃_{i}) ∩ M = S,  (1)

then there exists a neighbourhood V of 0 in M such that the system H := (H̃_{1} · M, …, H̃_{m−s} · M)|_{V} belongs to H(V, Z) and ν̃(H̃ · Z, 0) = ν̃(H · Z, 0).

Proof: Since H̃ ∈ H(U, Z), the system (H̃_{1} ∩ M, …, H̃_{m−s} ∩ M) satisfies the condition (c) of the definition of the family H(U ∩ M, Z). By (1) the hypersurfaces H̃_{1}, …, H̃_{m−s} meet M transversally in a neighbourhood V of 0 in M, which gives that the cycles (H̃_{j} · M) ∩ V are smooth hypersurfaces for 1 ≤ j ≤ m − s. The same assumption yields that the system H := (H̃_{1} · M, …, H̃_{m−s} · M) satisfies the condition (b) of the definition of the family H(V, Z).

Let Z_{j}, Z_{j}^{S} (resp. Z̃_{j}, Z̃_{j}^{S}) denote the result of the j^{th} step of Algorithm (4.1) from [T] applied to the system H (resp. H̃), the set Z and S. As in the proof of Lemma 4.1 we have the inclusion ⋂_{j=1}^{m−s} H_{j} ∩ Z ⊂ S , which yields that the cycles Z̃_{j}^{S} vanish for j > m − s. Therefore it sufices to prove that Z_{j} = Z̃_{j} ∩ (V ∩ S) for j ≤ m − s. We proceed by induction on j. Obviously the equality Z_{0} = Z̃_{0} holds. Suppose that Z_{j−1} = Z̃_{j−1} ∩ (V ∩ S). As in the proof of the previous lemma, Thm 2.2 from [TW] yields the equality Z_{j} = Z̃_{j} ∩ (V ∩ S). □

Theorem 4.3 Let S be an s-dimensional closed submanifold of a manifold M, and let Z be a purely k-dimensional analytic subset of M, c ∈ S. If M is submanifold of a manifold N, then the following equality holds:

g̃_{N}(Z, S)(c) = g̃_{M}(Z, S)(c) .

Proof: Without loss of generality N = E^{n}, M = E^{m} × {0}^{n−m}, S = E^{s} × {0}^{n−s}, c ∈ Z ∩ S and c = 0. We can also assume that there exists a system H = (H_{1}, …, H_{n−s}) ∈ H(N, Z), which satisfies the condition ν̃(H · Z, c) = g̃(Z, S)(c).

We will use H to construct H^{m−s} := (H̃_{1}, …, H̃_{m−s}, H_{m−s+1}, …, H_{n−s}) and find a positive ε_{q} > 0 such that H^{m−s} satisfies the assumption (1) of Lemma 4.2, the system H^{m−s}|_{ε_{m−s}E^{n}} belongs to the family H(ε_{m−s}E^{n}, Z) and the equality ν̃(H^{m−s} · Z, c) = g̃(Z, S)(c) holds. Construction of H̃_{1} : If H_{1} meets M at 0 transversally, we put H̃_{1} := H_{1}. Otherwise we choose a sequence {F_{ν}}_{ν=1}^{∞} consisting of mappings satisfying the following conditions:

(a) F_{ν} = idℂ^{s} ⊕ f_{ν}, where f_{ν} : ℂ^{n−s} ↦ ℂ^{n−s} are orthogonal mappings ,

(b) F_{ν} → idℂ^{n},

(c) F_{ν}(H_{1}) is transversal to the submanifold M at the point 0.

Lemma 2.1 gives the convergence F_{ν}(H_{1}) → H_{1}. By Thm 3.3 we can find an integer ν_{1} such that F_{ν}(H_{1}) ∈ J_{1}(H, E^{n}, Z) for ν ≥ ν_{1}. In this case we put H̃_{1} := F_{ν_{1}}(H_{1}). Define H^{1} := H_{1}(H̃_{1}). Obviously there exists positive ε_{1} such that the system H^{1}|_{ε_{1}E^{n}} belongs to H(ε_{1}E^{n}, Z). Let q be an integer, 2 ≤ q ≤ m−s. Suppose that we have found hypersurfaces H̃_{1}, …, H̃_{q−1} which satisfy the condition:

dim( ⋂_{j=1}^{j=q−1} T_{0}(H̃_{j}) ∩ M) = m − q + 1,  (2)

for which there exists a positive ε_{q−1} such that the system H^{q−1}|_{ε_{q−1}E^{n}} belongs to H(ε_{q−1}E^{n}, Z), and the equality ν̃(H^{q−1} · Z, 0) = g̃(Z, S)(0) holds, where H^{q−1} := (H̃_{1}, …, H̃_{q−1}, H_{q}, …, H_{n−s}). Then we can find the q^{th} element of H^{m−s} in the following way: Construction of H̃^{q} : Observe that condition (2) implies that 0 belongs to Reg(⋂_{j=1}^{j=q−1} H̃_{j} ∩ M). If H_{q} is transversal to ⋂_{j=1}^{j=q−1} H̃_{j} ∩ M at the point 0, then we put H̃_{q} := H_{q}. Otherwise we choose a sequence {F_{ν}^{q}}_{ν=1}^{∞} consisting of mappings satisfying the conditions (a), (b) (from the construction of H̃_{1}) and such that all the hypersurfaces F_{ν}(H_{q}) are transversal to the set ⋂_{j=1}^{j=q−1} H̃_{j} ∩ M at 0. Lemma 2.1 gives the convergence F_{ν}(H_{q}) → H_{q}. By Thm 3.3 we can find an integer ν_{q} such that F_{ν}(H_{q}) ∈ J_{q}(H^{q−1}, E^{n}, Z) for ν ≥ ν_{q}. In this case we put H̃_{q} := F_{ν_{q}}(H_{q}). Now we define H^{q} := H_{q−1}(H̃_{q}). One can see that there exists positive ε_{q} such that the system H^{q}|_{ε_{q}E^{n}} belongs to H(ε_{q}E^{n}, Z).

Since H^{m−s} satisfies the condition (2) for q = m − s + 1, the system H^{m−s}|_{ε_{m−s}E^{n}} satisfies the assumption (1) of Lemma 4.2. Therefore we can apply this lemma and get the inequality:

g̃_{M}(Z, S)(c) ≤lex ν̃(H^{m−s} · Z, c) = g̃_{N}(Z, S)(c) .

Now let us assume that H = (H_{1}, …, H_{m−s}) ∈ H(M, Z) satisfies the equality ν̃(H · Z, c) = g̃_{M}(Z, S)(c). Lemma 4.1 yields

g̃_{M}(Z, S)(c) ≥lex g̃_{N}(Z, S)(c) . □

5 Intersections on analytic spaces

Let A, B be analytic cycles on a manifold M. If M is a submanifold of a manifold N, we denote by A •_{M} B (resp. A •_{N} B) their intersection product in M (resp. N). In the first part of this section we prove that those cycles are equal. In the second part we construct the intersection product of analytic cycles on a reduced analytic space.

We will need the following lemmas about systems of hypersurfaces in polydiscs.

Lemma 5.1 Let N = E^{2n}, M = E^{2m} × {0}^{2n−2m}, S = E^{m} × {0}^{2n−m}, S′ = E^{m} × {0}^{n} × E^{n−m} and let Z be a purely k-dimensional analytic subset of M. If H = (H_{1}, …, H_{m}) ∈ H_{S}(M, Z), then there exists a system of hypersurfaces H_{m+1}, …, H_{n} such that

Ĥ := (H_{1} × E^{2n−2m}, …, H_{m} × E^{2n−2m}, H_{m+1}, …, H_{n})  (3)

belongs to H_{S′}(N, Z) and the following equality holds

(\underbrace{0, …, 0}_{n−m}, ν̃(H · Z, 0)) = ν̃(Ĥ · Z, 0).

Proof: Put H_{j} := V (x_{j}) ∩ E^{n} for i = m + 1, …, n and define Ĥ by the equality (3). The inclusion ⋂_{j=1}^{m} H_{j} ∩ Z ⊂ S′ yields that Ĥ ∈ H_{S′}(N, Z), The same inclusion yields that the cycles Ẑ_{j}^{S} vanish for j > m, where Ẑ_{j}, Ẑ_{j}^{S′} (resp. Z_{j}, Z_{j}^{S}) denote the result of the j^{th} step of Algorithm (4.1) from [T] applied to the system Ĥ (resp. H), the set Z and S′ (resp. S). The equality S′ ∩ M = S yields the equality Ẑ_{j}^{S} = Ẑ_{j}^{S′} for j ≤ m. Therefore it suffices to prove that Z_{j} = Ẑ_{j} for j ≤ m. It is obvious that the cycles Z_{0}, Ẑ_{0} are equal. Suppose that Z_{j−1} = Ẑ_{j−1}. Thm 2.2 from [TW] gives Z_{j} = Ẑ_{j} (see the proof of Lemma 4.1). □

Lemma 5.2 Under the assumptions of Lemma 5.1, if Ĥ = (H_{1}, …, H_{n}) ∈ H_{S′}(M, Z), then there exist hypersurfaces H_{n+1}, …, H_{2n−m} such that H := (H_{1}, …, H_{2n−m}) belongs to H_{S}(N, Z) and the following equality holds

(\underbrace{0, …, 0}_{n−m}, ν̃(H · Z, 0)) = ν̃(Ĥ · Z, 0).

Proof: Put H_{j} := V (x_{m+j}) ∩ E^{2n} for j = n + 1, …, 2n − m and H := (H_{1}, …, H_{2n−m}). The inclusion ⋂_{j=1}^{n} H_{j} ∩ Z ⊂ S yields that H ∈ H_{S}(N, Z). Let Z_{j}, Z_{j}^{S} (resp. Ẑ_{j}, Ẑ_{j}^{S}) denote the result of the j^{th} step of Algorithm (4.1) from [T] applied to the system H (resp. Ĥ), the set Z and S (resp. S′). By the same inclusion the cycles Ẑ_{j}^{S} vanish for j > m. It is obvious that Z_{j}^{S} = Ẑ_{j}^{S′} for j ≤ m. □

Given a pair of analytic sets X, Y ⊂ M of pure dimension and a point a ∈ M, we define the integer d_{M}(X, Y)(a) := g_{M^{2}}(X ×Y, Δ_{M})(a), where Δ_{M} denotes the diagonal in M^{2}.

Proposition 5.3 Let M be an m-dimensional complex manifold, a ∈ M, and let X, Y be analytic subsets of M of pure dimension. If M is submanifold of an n-dimensional manifold N, then the following equalities hold

1. (\underbrace{0, …, 0}_{n−m}, g̃_{N^{2}}(X × Y, Δ_{M})(a)) = g̃_{N^{2}}(X × Y, Δ_{N})(a),

2. d_{M}(X, Y)(a) = d_{N}(X, Y)(a).

Proof: It suffices to prove the first equality. Lemma 5.2 yields the inequality (0, …, 0, g̃_{N^{2}}(X × Y, Δ_{M})(a)) ≤lex g̃_{N^{2}}(X × Y, Δ_{N})(a) . By Lemma 5.1 the following inequality holds

(\underbrace{0, …, 0}_{n−m}, g̃_{M^{2}}(X × Y, Δ_{M})(a)) ≥lex g̃_{N^{2}}(X × Y, Δ_{N})(a) .

Thm 4.3 concludes the proof. □

As an immediate consequence we obtain the following theorem.

Theorem 5.4 Let M be an m-dimensional complex manifold, and let A, B be analytic cycles on M. If M is submanifold of a manifold N, then the following equality holds

A •_{N} B = A •_{M} B .

Let W be a reduced analytic space. We introduce the following notation • K(W) := {f : W ↦ ℂ : f(W) ⊂ ℤ, f is analytically constructible },

• G(W) denotes the family of analytic cycles on W. We have the following generalization of Prop. 2.1.3 from [T].

Lemma 5.5 The mapping ν : G(W) ∋ A ↦ ν(A) ∈ K(W) is a bijection.

Proof: Obviously the mapping in question is an additive injection. Fix f ∈ K(W). Our aim is to construct a cycle T satisfying ν(T) = f.

Put W_{0} := W, f_{0} := f. Let {W_{0}^{j}}_{j∈J} be irreducible components of the space W. Since the proof of Prop. 2.1.3 from [T] is valid for irreducible analytic spaces, we can find a family of analytic cycles {T_{0}^{j}}_{j∈J} such that T_{0}^{j} ∈ G(W_{0}^{j}) and ν(T_{0}^{j}) = f |_{W_{0}^{j}}. Observe that the sum T_{0} := ∑_{j∈J} T_{0}^{j} is an analytic cycle and define W_{1} := Sing W_{0}, f_{1} := f_{0} − ν(T_{0}). Since the inclusion supp(f_{1}) ⊂ W_{1} holds, we can decompose W_{1} and construct the cycle T_{1}.

Repeating this procedure we obtain a sequence {W_{j}, T_{j}, f_{j}}_{j=0}^{j=∞}, where T_{j} ∈ G(W_{j}), f_{j} ∈ K(W_{j}), W_{j} are analytic subsets of W, which satisfies the following conditions:

(a_{i}) f_{i} = f_{i−1} − ν(T_{i−1}),
(b_{i}) supp(f_{i}) ⊂ W_{i},
(c_{i}) W_{i} = Sing W_{i−1}.

The condition (a_{i}) yields the equality

(d_{i}) ∑_{j=0}^{j=i} f_{j} − ∑_{j=0}^{j=i} ν(T_{j}) = ∑_{j=1}^{j=i+1} f_{j}.

By (b_{i}), (c_{i}) the formal sum T := ∑_{j=0}^{j=∞} T_{j} is a cycle. Fix a point x ∈ W. By the same conditions there exists a j_{0} such that the functions f_{j}, ν(T_{j}) vanish at x for j ≥ j_{0}. The equality (d_{i}) yields f(x) − ν(T, x) = 0. □

As an immediate consequence of Prop. 5.3 we get the following lemma.

Lemma 5.6 Let W be a reduced analytic space, X, Y ⊂ W be analytic sets of pure dimension, a ∈ W. Let ψ_{i} : U_{i} ↦ Z_{i} be charts for i = 1, …, 2 , where U_{i} are neighbourhoods of a and Z_{i} are analytic subsets of complex manifolds M_{i}. Then the following equality holds

d_{M_{1}}(ψ_{1}(X), ψ_{1}(Y))(ψ_{1}(a)) = d_{M_{2}}(ψ_{2}(X), ψ_{2}(Y))(ψ_{2}(a)).

Now we can state the definition of the multiplicity of intersection of sets X and Y at a point a.

Definition 5.7 Let W be a reduced analytic space, X and Y be its irreducible analytic subsets, a ∈ W. Let ψ : U ↦ Z be a chart , where U is a neighbourhood of a and Z is analytic subset of a complex manifold M. The multiplicity of intersection of X and Y at the point a is defined to be

d(X, Y)(a) := d_{M}(ψ(X), ψ(Y))(ψ(a)).

By Thm 5.5 from [T] the function W ∋ x ↦ d(X, Y)(x) is analytically constructible. Consequently Lemma 5.5 enables us to formulate the main definition of this section.

Definition 5.8 Let X, Y be irreducible analytic subsets of a reduced analytic space W. The intersection product X • Y of the sets X and Y is defined to be the unique cycle such that ν(X • Y) = d(X, Y).

As in the smooth case (see [T] Def. 6.4) we extend this definition to the case of arbitrary analytic cycles on a reduced analytic space W by ℤ-linearity.

6 Regular separation on analytic spaces

We will now study the relation between the degree of the intersection product and the exponent of proper separation. We begin by recalling some basic facts about the notion of proper separation.

Let X, Y be locally analytic subsets of a normed complex vector space M. Let | · | denote the norm in M and ϱ(·, Z) stand for the distance function to the set Z ⊂ M. For p ∈ [1, ∞) we say that X, Y are p-separated at a point a ∈ X ∩ Y if there exist C, r > 0 such that

ϱ(z, X) + ϱ(z, Y) ≥ Cϱ(z, X ∩ Y)^{p}

provided |z − a| < r.

Obviously p-separation is biholomorphic invariant. We extend the above definition to the case of analytic subsets X, Y of a reduced analytic space W in the natural way.

Definition 6.1 Let W be a reduced analytic space and let X, Y be analytic subsets of W. We say that X and Y are p-separated at a ∈ X ∩Y if for every chart φ : V ↦ G, where V is a neighbourhood of a, the sets φ(X), φ(Y) are p-separated at φ(a).

By Corollary 3.10 from [Cg] if W is smooth, then X and Y are p-separated at a ∈ X ∩ Y for p = ν(X •_{W} Y, a). We will prove that the same statement holds in the case of analytic subsets of a reduced analytic space.

Lemma 6.2 Let W be a reduced analytic space, X and Y be its analytic subsets, a ∈ X ∩ Y. If ψ : U ↦ G, where U is a neighbourhood of a, is a chart, p ∈ [1, ∞), then the following conditions are equivalent:

(1) X and Y are p-separated at a,

(2) the sets ψ(X), ψ(Y) are p-separated at ψ(a).

Proof: Let k < n be positive integers. It suffices to prove that if X and Y are analytic subsets of E^{k}, then the following conditions are equivalent:

(a) X and Y are p-separated at 0 (in E^{k}),

(b) X × {0}^{n−k} and Y × {0}^{n−k} are p-separated at 0 (as subsets of E^{n}). Obviously the condition (b) implies (a). To prove the other implication consider ℂ^{n} with the norm |(z_{1}, …, z_{n})| = max{|z_{1}|, …, |z_{n}|} and denote π(z_{1}, …, z_{n}) := (z_{1}, …, z_{k}). One can see that the following inequality holds max{ϱ_{n}(z, E^{k} × {0}^{n−k}), ϱ_{k}(π(z), X)} ≥

min{C, 1} · (max{ϱ_{n}(z, E^{k} × {0}^{n−k}), ϱ_{k}(π(z), X ∩ Y)})^{p},

where ϱ_{n} (resp. ϱ_{k}) is the distance function from a subset of ℂ^{n} (resp. ℂ^{k}), C, p are the constants from the proper separation inequality for X and Y in E^{k}, and z belongs to a neighbourhood of 0 in E^{n}. □

We can now prove the main theorem of this section.

Theorem 6.3 Let X, Y be analytic subsets of a reduced analytic space W and let a belong to X ∩ Y. Then the sets X and Y are p-separated at the point a for p = ν(X • Y, a).

Proof: Choose a chart ψ : U ↦ G, where U is a neighbourhood of a in W. By Lemma 6.2 it sufices to prove that the sets ψ(X), ψ(Y) are p-separated at the point ψ(a) for p = ν(ψ(X) • ψ(Y), ψ(a)). Corollary 3.10 from [Cg] concludes the proof. □ Acknowledgements. I would like to express my sincere thanks to Piotr Tworzewski for suggesting the subject and many helpful conversations.

References

[ATW] R. Achilles, P. Tworzewski, T. Winiarski, On improper isolated intersection in complex analytic geometry , Ann. Polon. Math. 51 (1990), 21-36.

[Ch] E. M. Chirka, Complex analytic sets, Kluwer Acad. Publishers, 1989.

[Cg] E. Cygan, Theory of intersection and regular separation, (in Polish) Thesis, Jagiellonian University, Poland (1997).

[CT] E. Cygan, P. Tworzewski, Proper intersection multiplicity and regular separation of analytic sets, Ann. Polon. Math. 59 (1994), 293-298.

[Dr] R. N. Draper, Intersection theory in analytic geometry, Math.Ann. 180 (1969), 175-204.

[BH] A.Borel, A.Haeflinger La classe d’homologie fondamentale d’un espace analytique, Bull. Soc. Math. France 89 (1961),461-513.

[R1] S. Rams, Convergence of holomorphic chains, Ann. Polon. Math. 65.3 (1997), 227-234.

[R2] S. Rams, Bézout-type theorems for certain analytic sets, preprint.

[S] E. Selder, Eine algebraische Definition lokaler analytischer Schnittmultiplizitäten, Rev. Roumaine Math. Pures Appl. 5, 29 (1984), 417-432.

[Tw] P. Tworzewski, Isolated intersection multiplicity and regular separation of analytic sets, Ann. Polon. Math. 58 (1993), 213-219.

[T] P. Tworzewski, Intersection theory in complex analytic geometry, Ann. Polon. Math. 62.2 (1995), 177-191.

[TW] P. Tworzewski, T. Winiarski, Cycles of zeros of holomorphic mappings, Bull. Polish. Acad. Sci. Math 37 (1989), 95-101.

[W] T. Winiarski, Continuity of total number of intersection, Ann. Polon. Math. 47 (1986), 155-187.