On intersections with hypersurfaces and normal pseudo-flatness

Sławomir Rams

IMUJ PREPRINT 1999/27 KRAKÓW

Abstract We study the behaviour of the families of hypersurfaces that are applied in [4] to estimate the Łojasiewicz exponent and the exponent of separation at infinity.

1 Introduction

In the recent paper [4] the so-called index of contact is applied to estimate the exponent of separation at infinity and the Łojasiewicz exponent. This number is obtained in a procedure based on intersecting an analytic set Z with a hypersurface H that contains a fixed submanifold S, extracting the part of the cycle H · Z supported by S and counting its multiplicity ν((H · Z)^{S}, c) (see [3], [4] for details). Similar procedures are applied in [10], [13], [6], [7] and [9]. The index of contact is also closely related to the extended index of intersection (see [13], [12], [1]).

The main aim of this note is to characterize the set of hypersurfaces for which ν((Z · H)^{S}, c) is minimal (i.e. generic). According to [10], p. 91 no such characterization is known. We will prove that the hypersurfaces in question are to be smooth along a subset of Z ∩S and to meet some algebraic cones along the tangent space to S.

1991 Mathematics Subject Classification : 32C25, 14B05, 32C99

Key words : index of contact, multiplicity of intersection, normal pseudo-flatness.

Partially supported by KBN Grant 2 P03A 001 15.

The reader is expected to be familiar with the notion of proper intersection multiplicity of analytic sets (see [5]), its generalization given in [2], relative tangent cone (see [2]), basic properties of holomorphic cycles (also known as chains) (see [13], [11]) and the notion of normal pseudo-flatness (see [8]). Throughout the text we work over the field C of complex numbers.

2 Preliminaries

Let N be an n-dimensional manifold and let S be its s-dimensional closed submanifold. For an analytic cycle A = ∑_{j∈J} α_{j}C_{j} in U, where U is an open subset of N, we define the part of A supported by S as A^{S} := ∑_{j∈J,C_{j}⊂S} α_{j}C_{j}. The multiplicity of the cycle A at the point a ∈ U is defined to be ν(A, a) := ∑_{j∈J} α_{j}ν(C_{j}, a), where ν(C_{j}, a) denotes the multiplicity of the analytic set C_{j} at the point a. |A| denotes the suport of A (i.e. |A| := ⋃_{j∈J} C_{j}). The tangent cone to C_{j} at a is denoted by C_{a}(C_{j}). If all the irreducible sets C_{j} are of dimension p, we say that A is a p-cycle. A sequence {A_{j}}_{j=1}^{∞} of p-cycles in U converges to A iff it converges to a p-cycle A in the weak topology of currents. If X, Y are analytic subsets of U that meet properly, C is an irreducible component of X ∩ Y, then i(X, Y; C) stands for the multiplicity of their intersection along C. The intersection product ∑ i(X, Y; C) · C, where the summation runs over all components of X ∩ Y, is denoted by X · Y. Let X, Y be locally analytic subsets of N and let a be an isolated point of X ∩ Y. By i(X, Y; a) we denote the isolated intersection multiplicity defined in [2], and C_{a}(X, Y) stands for the relative tangent cone (see [2]). The unit ball in C is denoted by E.

Let Z be a purely k-dimensional analytic subset of N such that S contains no component of Z. For an open U ⊂ N, and a ∈ U we define the following families:

• F^{1}(U, Z) := {H : H is a closed hypersurface in U, dim(H ∩ Z) = k − 1, U ∩ S ⊂ H}, • F_{a}^{2}(U, Z) := {H ∈ F^{1}(U, Z) : H_{a} ∩ Z_{a} = (Z ∩ S)_{a}}.

Let H ∈ F^{1}(U, Z). If {X_{i}^{j}}_{j∈J_{i}} is the family of all (k − i)-dimensional irreducible components of Z∩S, then |(Z·H)^{S}| = ⋃_{j∈J_{1}} X_{1}^{j}, and the following equality holds

ν((Z · H)^{S}, a) = ∑ ν(X_{1}^{j}, a) · i(H, Z; X_{1}^{j}).  (1)

Definition 2.1 Let a ∈ reg(Z ∩S), dim_{a}(Z ∩S) = k−1. We define T_{a}(Z, S) to be the family of all (n − k + 1)-dimensional submanifolds P ⊂ N that are transversal to Z ∩ S at a, and meet Z properly.

One can see that T_{a}(Z, S) is nonempty (by Rückert”s lemma) and every P ∈ T_{a}(Z, S) determines the 1-cycle P · Z = ∑ β_{j}C_{j}. Both i((Z · P), S; a) := ∑ β_{j}i(C_{j}, S; a) and the cone C_{a}(Z, S) depend only on the germs Z_{a}, S_{a}, P_{a}. If H ∈ F_{a}^{2}(U, Z) and a ∈ reg(Z ∩ S) ∩ X_{1}^{1} then the following equality holds

i((P · Z), H; a) = i(Z, H; X_{1}^{1}).  (2)

In the sequel we will need the following lemmas.

Lemma 2.2 Let U be a neighbourhood of a, H ∈ F^{2}(U, Z), a ∈ reg(Z ∩ H), P ∈ T_{a}(Z, S). Then the following conditions are equivalent:

1. i((P · Z), H; a) = i((P · Z), S; a),

2. a ∈ reg(H) and C_{a}(P ∩ Z, S) ∩ T_{a}H = T_{a}S.

Proof: Let V be a neighbourhood of the point a such that P ∩ V is a closed submanifold of V and (S ∩ Z) ∩ V = X_{1}^{1} ∩ V, where X_{1}^{1} is the irreducible component of Z ∩ S to which a belongs. Let (P ∩V)·(Z ∩V) = ∑_{j=1}^{j=m} β_{j}C_{j}. By [2], Th. 4.4 the equality i(C_{j}, H; a) = i(C_{j}, S; a) holds iff a ∈ reg(H) and ⋃_{j=1}^{j=m} C_{a}(C_{j}, S) ∩ T_{a}H = T_{a}S. [2], Property 2.6 yields the equality ⋃_{j=1}^{j=m} C_{a}(C_{j}, S) = C_{a}(P ∩ Z, S). Hence, the inequality i((P · Z), H; a) ≥ i((P · Z), S; a) becomes an equality iff the condition 2 is satisfied. □

Lemma 2.3 Let N = E^{n}, S = E^{s} × {0}^{n−s}, {a_{j}}_{j=1}^{∞} ⊂ S. Let C_{j} be locally analytic curves such that a_{j} ∈ C_{j} and (C_{j})_{a_{j}} ∩ S_{a_{j}} = {a_{j}}. Then the set of hyperplanes H ⊂ C^{n−s} that satisfy the condition

C_{a_{j}}(C_{j}, S) ∩ C^{s} × H = C^{s} × {0}^{n−s}  (3)

for all j, is a dense subset of Grassman manifold G_{n−s−1}(C^{n−s}).

Proof:By [2], Lemma 2.4 the equality C_{a_{j}}(C_{j}, S) = C^{s} × C_{0}(π(C_{j} ∩ U_{j})), holds, where π : C^{s} ⊕ C^{n−s} ↦ C^{n−s} is the projection and U_{j} is a sufficiently small neighbourhood of a_{j}. Therefore, (3) is equivalent to the condition that H does not contain a finite family of lines. □

3 Intersections with hypersurfaces

Let N, S, Z be as in the previous section. We fix a point c ∈ Z ∩ S.

• g_{1}(Z, S)(c) := min{ν((Z · H)^{S}, c) : H ∈ F^{1}(V, Z), c ∈ V },

• F(S, Z) := {a ∈ Z | h(I_{S,a}) = s(I_{S,a})}, where h(I_{S,a}) (resp. s(I_{S,a}) denotes the height (resp. the analytic spread) of I_{S,a} O_{Z,a} (for more details about normal pseudo-flatness locus see [8]) In this section the following lemma will play crucial role.

Lemma 3.1 Let a ∈ reg(Z ∩ S) ∩ F(S, Z), dim_{a}(Z ∩ S) = k − 1 and let P ∈ T_{a}(Z, S). Then the equality i((P · Z), S; a) = g_{1}(Z, S)(a) holds.

Proof: Lemma is an immediate consequence of [12], Th. 3.7. □

Let U be a neighbourhood of c ∈ Z ∩ S. Our aim is to characterize the following family:

P_{c}(U, Z) = {H ∈ F^{1}(U, Z) : ν((Z · H)^{S}, c) = g_{1}(Z, S)(c)}.

We can assume that dim_{c}(Z ∩ S) = k − 1 (otherwise P_{c}(U, Z) = F^{1}(U, Z)). For a given H ∈ F^{1}(U, Z) we define the following sets:

• reg_{H}(Z, S) := {a ∈ reg(Z ∩S) : dim_{a}(Z ∩S) = k −1, a ∉ |Z ·H −(Z ·H)^{S}|, the component of Z ∩ S to which a belongs passes through c, },

• sreg_{H}(Z, S) := reg_{H}(Z, S) ∩ F(S, Z).

Now we are in position to prove the main result of this note:

Theorem 3.2 Under the assumptions of this section, the following conditions are equivalent:

1. H ∈ P_{c}(U, Z),

2. for every point a ∈ reg_{H}(Z, S) and for every P ∈ T_{a}(Z, S) we have a ∈ reg(H) and the equality T_{a}H ∩ C_{a}(Z ∩ P, S) = T_{a}S holds, 3. for each (k − 1)-dimensional component X_{j} of Z ∩ S such that c ∈ X_{j}

there exists an a_{j} ∈ sreg_{H}(Z, S) ∩ X_{j} and a submanifold P_{j} ∈ T_{a_{j}}(Z, S) such that a_{j} ∈ reg(H) and T_{a_{j}}H ∩ C_{a_{j}}(Z ∩ P_{j}, S) = T_{a_{j}}S.

Proof: If k = 1, then c is an isolated point of Z ∩ S and the theorem results immediately from [2], Th. 4.4. Therefore we can assume k ≥ 2. Let {X_{j}}_{j=1}^{j=q} denote the family of all (k − 1)-dimensional components of the set Z ∩ S that pass through the point c. Fix H ∈ F^{1}(U, Z) and consider the functions:

• Φ : sreg_{H}(Z, S) ∋ a → i((Z · R), S; a),

• Ψ : sreg_{H}(Z, S) ∋ a → ν(Z · H, a), where R ∈ T_{a}(Z, S). Observe, that by Lemma 3.1, Φ(a) does not depend on the choice of R ∈ T_{a}(Z, S). According to Lemma 2.2 the equality Φ(a) = Ψ(a) holds iff a ∈ reg(H) and T_{a}(H) ∩ C_{a}((P ∩ Z), S) = T_{a}S. By Lemma 3.1, [12], Th. 4.1(e) and [12], Cor. 4.4 the integer Φ(a) equals the index of intersection of Z and S at a. Hence, [13], Th. 5.4 yields that Φ is locally constant and sreg_{H}(Z, S) is a Zariski open and dense subset of ⋃ X_{j}. (1.⇒2.) Fix a ∈ reg_{H}(Z, S) ∩ X_{1}, P ∈ T_{a}(Z, S). Assume that a ∈ sreg(Z, S). In this case we are to prove that Φ(a) = Ψ(a). Therefore, it suffices to find b ∈ sreg(Z, S) ∩ X_{1} such that Φ(b) = Ψ(b). Without loss of generality N = E^{n}, S = E^{s} × {0}^{n−s}, c = 0. Choose a dense subset {b_{j}}_{j=1}^{∞} of sreg_{H}(Z, S) and a family {P_{j}}_{j=1}^{∞} such that P_{j} ∈ T_{a_{j}}(Z, S). By Lemma 2.3, there exists a linear space L ∈ G_{n−s−1}(C^{n−s}) such that H̃ := E^{s} × L meets Z properly and satisfies C_{b_{j}}(C_{j}, S) ∩ H̃ = T_{b_{j}}S for all j.

We can assume that b_{j} ∈ X_{j} ∩ regH̃(Z, S) for j = 1, …, q. By Lemma 2.2 the equality i(Z, H̃; b_{j}) = i((Z · P_{j}), S; b_{j}) holds for j = 1, …, q. Consequently, ν(Z · H̃, b_{j}) ≤ ν(Z · H, b_{j}), and (see the equality (1)) we have

ν((Z · H̃)^{S}, b_{j}) ≤ ν((Z · H)^{S}, b_{j}).

By our assumption, the opposite inequality holds, which yields (by (2)) that i((Z ·P_{j}), H; b_{j}) equals i((Z ·P_{j}), S; b_{j}) for j = 1, …, q. Hence Φ(b_{1}) = Ψ(b_{1}).

To prove the implication (1.⇒2.) for an arbitrary a ∈ reg_{H}(Z, S) we can assume that a = 0, P = {0}^{k−1} × E^{n−k+1}, Z ∩ S = E^{k−1} × {0}^{n−k+1}, and the projection π : Z → E^{k} × {0}^{n−k} is proper. Observe, that i((P · Z), H; 0) = i(P, (Z · H); 0) = i(Z · H; X_{1}). Hence, by Lemma 2.2 it suffices to prove the inequality

i((P · Z), S; 0) ≥ i(Z, H; X_{1}).

We choose a sequence {a_{j}}_{j=1}^{∞} ⊂ sreg_{H}(Z, S) ∩ X_{1} such that a_{j} → 0, and define P_{j} := a_{j} +{0}^{k−1} ×E^{n−k+1}. Observe that P_{j} ∈ T_{a_{j}}(Z, S) and P_{j} → P, which yields P_{j} · Z → P · Z (by [13], Th. 3.6). Choose a hypersurface H̃ such that i(P · Z, H̃; 0) = i(P · Z, S; 0). We can assume that H̃ is an analytic subset of E^{n} which meets all the sets |P_{j} · Z| properly (by [14], Th. 2). Consequently, (P_{j} ·Z)·H̃ → (P ·Z)·H̃, which yields i((P · Z), H̃; 0) ≥ i((P_{j} · Z), H̃; a_{j}) for sufficiently large j (by [11], Def 2.4). Therefore

i((P · Z), S; 0) ≥ i((P_{j} · Z), S; a_{j}).

Since i((P_{j} · Z), S; a_{j}) = i(Z, H; X_{1}) we obtain the inequality

i((P · Z), S; 0) ≥ i(Z, H; X_{1}).

(2.⇒3.) Obvious.

(3. ⇒1.) By our assumption the equality Φ ≡ Ψ holds. We can assume that there exists an H̃ ∈ P_{c}(U, Z). Let b_{j} ∈ X_{j} ∩ regH̃(Z, S) ∩ sreg_{H}(Z, S). Then

i((P · Z), H̃; b_{j}) ≥ i((P · Z), S; b_{j}) = i((P · Z), H; b_{j}) = i(Z · H; X_{j}),

which yields (see (1)) ν((H̃ · Z)^{S}, c) ≥ ν((H · Z)^{S}, c). □

Acknowledgements. I would like to express my sincere thanks to Piotr Tworzewski for inspiring remarks.

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Sławomir Rams Institute of Mathematics Jagiellonian University Reymonta 4 30-059 Kraków (Poland) Email: rams@im.uj.edu.pl