Abstract. We endow the module of analytic p-chains with the structure of a second- countable metrizable topological space.

POLONICI MATHEMATICI LXV.3 (1997)

Convergence of holomorphic chains

by S lawomir Rams (Krak´ow)

Abstract. We endow the module of analytic p-chains with the structure of a second- countable metrizable topological space.

1. Introduction. A holomorphic p-chain in an open subset Ω of C

is a formal locally finite sum Z = P

j∈J

k

Z

where Z

are pairwise distinct irreducible analytic subsets of Ω of pure dimension p and k

∈ Z \ {0} for j ∈ J . The set S

j∈J

Z

is called the support of the chain Z and denoted by

|Z|. Each Z

is called a component of Z and the number k

is the multiplicity of Z

.

A holomorphic p-chain Z is positive if the multiplicities of all its com- ponents are positive. G

(Ω) denotes the set of positive p-chains in Ω. The set of holomorphic p-chains in Ω is endowed with the structure of a free Z-module. We denote it by G

(Ω).

Given a 0-chain and an open relatively compact subset U of Ω the total multiplicity of Z in U is defined as the sum of multiplicities of all its com- ponents contained in U . We denote the total multiplicity by deg

Z. When J is finite we extend this definition putting deg Z = P

j∈J

k

.

One can define convergence of chains as the classical weak convergence of the associated currents (for details see e.g. [Ch, §14.1-2]). An attempt to explain the geometrical meaning of this convergence is made in [Ch].

In [Ch, § 12.2] the author proves that proper intersection is sequentially continuous and also states that this operation is continuous [Ch, §12.4]. How- ever, he neither defines a topology nor proves the equivalence of sequential continuity and continuity.

The main aim of this note is to define a topology on G

(Ω) and to study some properties of this topological space. We shall prove that the result of this construction is second-countable, metrizable, and convergence in it coincides with the one defined in [Ch, §12.2].

1991 Mathematics Subject Classification: 32B15, 32C25, 32C30, 32C99.

Key words and phrases: holomorphic chains, currents, convergence of chains.

[227]

The topology constructed here is useful in studying the intersections of analytic sets (see [Tw], [R]).

2. Topology of p-chains. Let 0 ≤ p < n be integers, Ω be an open subset of C

. We shall use the following notation:

• E = {z ∈ C : |z| < 1},

• Λ(n, p) = {λ : {1, . . . , p} → {1, . . . , n} : λ(1) < . . . < λ(p)},

• e

, . . . , e

—the canonical basis of C

,

• π

: (z

, . . . , z

) → (z

, . . . , z

), π = π

| E

,

• A(Ω) = {f : C

→ C

: f an affine isomorphism, f (E

) ⊂ Ω},

• µ(h)—order of a finite branched holomorphic covering h,

• for Z = P k

Z

, z ∈ Z

\ S

j6=s

Z

, m(z, Z) = k

.

Suppose that Ω

, Ω

are open subsets of C

and Ω ⊂ Ω

. Given a biholomorphic mapping f : Ω

→ Ω

and Z = P

j∈J

k

Z

belonging to G

(Ω), a new p-cycle in f

(Ω) can be defined by f

(Z) = P

j∈J

k

f

(Z

).

Definition 2.1. Let V be an open subset of C

containing E

, and Z ∈ G

(V ), Z = P

j∈J

k

Z

, such that |Z| ∩ (E

× ∂E

) = ∅. Define µ(Z) = X

j∈J

k

µ(π | Z

∩ E

).

Definition 2.2. Let f

∈ A(Ω), c

∈ Z for j = 1, . . . , m and let K be a compact subset of Ω. Define U ({(f

, c

), . . . , (f

, c

)}, K) to be the set of all p-chains Z in Ω such that |Z| ∩ K = ∅ and

|Z| ∩ f

(E

× ∂E

) = ∅, µ(f

(Z)) = c

for j = 1, . . . , m.

It is easy to verify the following

Proposition 2.3. If Ω is an open subset of C

, then in G

(Ω) the family U (Ω) = {U (A, K) : A is a finite subset of A(Ω) × Z, K is compact in Ω}, is a base of a topology.

Definition 2.4. The topology of p-chains in Ω is defined to be the topology generated by U (Ω).

The next proposition is an immediate consequence of the last definition.

Proposition 2.5. Let Z, Z

, e Z

, e Z ∈ G

(Ω).

1. If Z

→ Z, e Z

→ e Z, and |Z + e Z| = |Z| ∪ | e Z|, then Z

+ e Z

→ e Z + Z.

2. If Z

→ Z, a ∈ Z, then a · Z

→ a · Z.

3. P

ν=0

Z

is convergent iff Z

→ 0.

4. If f is an affine isomorphism, then G

(Ω) 3 Z 7→ f

(Z) ∈ G

(f

(Ω))

is a homeomorphism.

Example 2.6. Ω = C

, Z

= {1/ν} × C, Z = {0} × C, e Z

= (−1) · ({−1/ν} × C), e Z = (−1) · ({0} × C). Then Z

→ Z and e Z

→ e Z but Z

+ e Z

does not converge to e Z + Z. Hence addition is not continuous on G

(Ω). Proposition 2.5.1 and Theorem 2.9 give its continuity on G

(Ω).

Given Z

, . . . , Z

belonging to G

(Ω), . . . , G

(Ω), respectively, and sat- isfying the conditions

1. the sum of the codimensions of |Z

| is equal to n, 2. T

j=1

|Z

| is zero-dimensional, a 0-chain is defined by

Z

∧ . . . ∧ Z

= X

a∈|Z¹|∩...∩|Z^k|

i(Z

∧ . . . ∧ Z

, a) · {a}

where i(Z

∧ . . . ∧ Z

, a) denotes the intersection multiplicity defined in [Dr]

(see also [Ch]). It is easy to prove that in Definition 2.1, (1) µ(Z) = deg

(({w} × E

) ∧ Z) for w ∈ E

.

If f : Ω

→ Ω

⊃ Ω is biholomorphic, then by [Ch, §12.3], (2) i(Z

∧ . . . ∧ Z

, f (a)) = i(f

(Z

) ∧ . . . ∧ f

(Z

), a).

Proposition 2.7. Let Z

, Z ∈ G

(Ω). If for each compact K ⊂ Ω \ |Z|

we have |Z

| ∩ K = ∅ for almost all ν, then the following conditions are equivalent :

1. For each point a ∈ Reg |Z|, each (n−p)-dimensional plane transversal to |Z| at a and each open set U relatively compact in L such that U ∩ |Z| = {a} there is an index ν

such that dim(|Z

| ∩ U ) = 0, deg

(Z

∧ L) = deg

(Z ∧ L) for all ν > ν

.

2. For each point a from a given dense subset D ⊂ Reg |Z|, each (n − p)- dimensional plane transversal to |Z| at a and each open set U relatively compact in L such that U ∩ |Z| = {a} there is an index ν

such that dim(|Z

| ∩ U ) = 0, deg

(Z

∧ L) = deg

(Z ∧ L) for all ν > ν

.

3. Z

→ Z in the topology of p-chains.

P r o o f. The proposition is obvious for p = 0 or Z = 0. Let p > 0, Z 6= 0.

1⇒2. Obvious.

2⇒3. Let Z ∈ U (A, K), A = {(f

, c

), . . . , (f

, c

)}. We check that Z

∈ U (A, K) for sufficiently large ν. Since U (A, K) = T

j=1

U ({(f

, c

)}, K) we

can assume m = 1. By Proposition 2.5.4 it suffices to consider f

= id

.

Fix w ∈ E

such that {w} × E

is transversal to |Z| at each point of the

set ({w} × E

) ∩ |Z| = {z

, . . . , z

}. There exist ε > 0 and open pairwise

disjoint relatively compact subsets U

, . . . , U

of E

such that:

• w + εE

⊂ E

,

• |Z| ∩ ({w} × U

) = {z

} for j = 1, . . . , s,

• |Z| ∩ K

= ∅ where K

= (w + εE

) × (E

\ (U

∪ . . . ∪ U

)).

Choose z e

∈ D ∩ ((w + εE

) × U

) for j = 1, . . . , s. Then µ(Z) =

s

X

j=1

deg({w} × U

) ∧ Z =

s

X

j=1

deg({π( z e

)} × U

) ∧ Z.

For sufficiently large ν we have |Z

| ⊂ Ω \ (K ∪ K

), and so

s

X

j=1

deg({π( z e

)} × U

) ∧ Z =

s

X

j=1

deg({π( z e

)} × U

) ∧ Z

= µ(Z

).

Then Z

∈ U (A, K) for sufficiently large ν and condition 3 follows.

3⇒1. Fix a = (a

, . . . , a

), L, U as in 1. By Proposition 2.5.4 and (2) we can assume that a = 0, L = C{e

, . . . , e

} and E

⊂ U .

There is ε > 0 such that |Z| ∩ (εE

× ∂E

) = ∅ and εE

× E

⊂ Ω.

Moreover,

|Z

| ∩ ((εE

× ∂E

) ∪ ({0}

× (U \ E

))) = ∅ and

µ(f

(Z

)) = µ(f

(Z)), where f = (ε id

, id

) and ν is large enough.

The set |Z

| ∩ U is compact and non-empty, hence dim(|Z

| ∩ U ) = 0.

By (1),

deg

(Z

∧ L) = deg

(Z ∧ L).

R e m a r k. Condition 2 resembles the one given in [Ch, §12.2]. The fol- lowing example shows the slight difference between them.

Example 2.8. Ω = C

, Z

= ({1/ν} × C) + ({1 − 1/ν} × C), Z = ({0} × C) + ({1} × C), e Z = ({0}×C) + 2({1}×C). One can see that Z

→ Z and Z

→ e Z in the sense of [Ch, §12.2]. The definition in [Ch, §12.2] seems to be erroneous, for [Z

] does not converge to [ e Z] as a sequence of currents.

Neither does it converge to e Z in the topology of p-chains.

Let us define:

• A

(Ω) = {f ∈ A(Ω) : f (0), f (e

), . . . , f (e

) ∈ (Q + iQ)

},

• e K = {[q

, q

] × . . . × [q

, q

] : q

, . . . , q

∈ Q},

• K = { S B : B ⊂ K, B is finite}, e

• U

(Ω) = {U (A, K) : A ⊂ A

(Ω) × Z, A is finite, K ∈ e K, K ⊂ Ω},

• E(r

, r

) = r

E

× r

E

for r

, r

> 0.

Theorem 2.9. U

(Ω) is a base for the topology of p-chains in Ω.

P r o o f. The assertion is obvious for p = 0. Suppose that p > 0 and let Z = P

j∈J

k

Z

∈ U (A, K). We can assume A = {(f

, c

)} (see the proof of Proposition 2.7). Then there are e K ∈ K and ε > 0 satisfying

(3) K ∩ |Z| = ∅, e K ⊂ e K ⊂ Ω, f

(E(1 + ε, 1 + ε)) ⊂ Ω, (4) f

(E(1 + ε, 1 + ε) \ E(1 + ε, 1 − ε)) ⊂ e K.

Fix 0 < r < 1. By a simple computation there is a neighborhood U ⊂ A(Ω) of f

in the Banach space of affine mappings C

→ C

such that each f ∈ U satisfies the following conditions:

(5) f (E(1 + ε/2, 1 + ε/2) \ E(1 + ε/2, 1 − ε/2)

⊂ f

(E(1 + ε, 1 + ε) \ E(1 + ε, 1 − ε)), (6) f

({0}

× E

) ⊂ f (E(r/2, 1 + ε/2)) ⊂ f

(E(r, 1 + ε)),

(7) (f

◦ f )({0}

× E

) ∩ (E

\ f

( e K))

= (f

◦ f )({0}

× C

) ∩ (E

\ f

( e K)), (8) (f

◦ f )({0}

× C

) ∩ E

⊂ E(r, 1),

(9) π

|(f

◦ f )({0}

× C

) is a bijection.

Let f ∈ U and W = P l

W

∈ U ({(f, c

)}, e K). Inclusions (4) and (5) give

(f

(|W |) ∪ f

(|W |)) ∩ (E

× ∂E

) = ∅.

If f

(W

) ∩ E

= ∅ then by (4), f

(W

) ∩ E(1, 1 + ε) = ∅. So, according to (6), f

(W

) ∩ E(r/2, 1 + ε/2) = ∅. Thus, by Remmert’s theorem we have f

(W

) ∩ E

= ∅. Similarly f

(W

) ∩ E

= ∅ ⇒ f

(W

) ∩ E

= ∅, which gives {j : f

(W

) ∩ E

6= ∅} = {j : f

(W

) ∩ E

6= ∅}.

By (1),

µ(f

(W

)) = deg(f

(W

) ∧ ({0}

× E

)).

By [Wi, Theorem 9.1] and (8), (9),

deg

(f

(W

)∧({0}

×E

)) = deg

(f

(W

)∧(f

◦f )({0}

×C

)).

From (7),

deg

(f

(W

) ∧ (f

◦ f )({0}

× C

))

= deg

(f

(W

) ∧ (f

◦ f )({0}

× E

)).

By (4) and (6),

deg

(f

(W

) ∧ (f

◦ f )({0}

× E

))

= deg(f

(W

) ∧ (f

◦ f )({0}

× E

)), deg(f

(W

) ∧ (f

◦ f )({0}

× E

)) = deg(W

∧ f ({0}

× E

))

= deg(f

(W

) ∧ ({0}

× E

)) = µ(f

(W

)).

We have obtained Z ∈ U ({(f, c

)}, e K) ⊂ U ({(f

, c

)}, K). Density of A

(Ω) in A(Ω) ends the proof.

3. Metric on G

(Ω). Let Z ∈ G

(Ω). For each compact subset K of Ω we fix 0 < d

< min{1, dist(K, ∂Ω)} and define H(K) = S

x∈K

B(x, d

).

Definition 3.1.

d(Z, K) =  dist(|Z| ∩ H(K), K) if |Z| ∩ H(K) 6= ∅,

d

if |Z| ∩ H(K) = ∅.

Lemma 3.2. d(·, K) is continuous.

P r o o f. Let Z

→ Z and d(Z, K) > 0. Fix e d < d(Z, K). Then we have

|Z

| ∩ S

x∈K

B(x, e d) = ∅ for almost all ν. We obtain lim inf

d(Z

, K) ≥ d(Z, K). If |Z| ∩ H(K) = ∅ then d(Z, K) = d

and the lemma follows.

If |Z| ∩ H(K) 6= ∅ then dist(|Z| ∩ H(K), K) = |z − y| where y ∈ K, z ∈ |Z| ∩ H(K). By R¨ uckert’s lemma there is a sequence {z

}, z

∈ |Z

|, z

→ z, which gives

lim sup

ν→∞

d(Z

, K) ≤ d(Z, K).

By the same argument the previous inequality holds when d(Z, K) = 0.

Let l ∈ Z and let E

⊂ Ω.

Definition 3.3. If |Z| ∩ (E

× ∂E

) = ∅, |Z| ∩ E

6= ∅, µ(Z) = l we define m

(Z) = d(Z, E

× ∂E

). We put m

(Z) = 0 otherwise.

Lemma 3.4. m

is continuous.

P r o o f. Let Z

→ Z. If m

(Z) 6= 0 then m

(Z

) = d(Z

, E

× ∂E

) for sufficiently large ν and we can use Lemma 3.2 . If m

(Z) = 0 and

|Z| ∩ (E

× ∂E

) 6= ∅ then |m

(Z

)| ≤ |d(Z

, E

× ∂E

)| → 0. Suppose that |Z| ∩ (E

× ∂E

) = ∅ and |Z| ∩ E

= ∅. By Remmert’s theorem

|Z

| ∩ E

= ∅ for almost all ν. If m

(Z) = 0, |Z| ∩ (E

× ∂E

) = ∅ and

|Z| ∩ E

6= ∅, then

µ(Z

) = µ(Z) 6= l for sufficiently large ν.

Set P(Ω) = {m

◦ f

: f ∈ A

(Ω), l ∈ Z}, and observe that we have n Y

h∈J

h · d(·, K) : J ⊂ P(Ω), J is finite, K ∈ K o ,

a countable family of continuous functions. Let {G

} denote a sequence of

all its elements.

Definition 3.5. Let X, Z ∈ G

(Ω). We define

%(X, Z) =

∞

X

j=0

1

2

|G

(X) − G

(Z)|.

Theorem 3.6. % is a metric on G

(Ω). The topology induced by % coin- cides with the topology of p-chains.

P r o o f. It is sufficient to prove that the sequence {G

} gives an embed- ding of G

(Ω) in the Hilbert cube. According to [En, 2.3, Theorem 10] we need to prove that:

1. {G

}

separates elements of G

(Ω),

2. {G

}

separates elements of G

(Ω) from closed subsets of G

(Ω).

1) We can assume that |Z| 6= ∅. If |X| 6= |Z| then there is K ∈ e K satisfying |X| ∩ K = ∅, |Z| ∩ K 6= ∅. We obtain

0 = d(Z, K) 6= d(X, K).

Suppose |X| = |Z|. There is z ∈ Reg |X| satisfying m(z, X) 6= m(z, Z).

Fix g ∈ A

(Ω) such that µ(π|g

(|Z|)) = 1. Consequently, (m

◦ g

)(Z) 6= (m

◦ g

)(X) = 0.

2) Let X ∈ U ({(f

, c

), . . . , (f

, c

)}, K) ⊂ G

(Ω)\C, where C is a closed subset of G

(Ω). Without loss of generality U ({(f

, c

), . . . , (f

, c

)}, K)

∈ U

(Ω).

If |X| 6= ∅ set G

= Q

j=1

(m

◦ f

) · d( , K). If |X| = ∅ choose G

= d( , e K) where e K ∈ K and

U ( e K) ⊂ U ({(f

, c

), . . . , (f

, c

)}, K).

In both cases we obtain G

|

= 0, G

(X) 6= 0.

Acknowledgements. I would like to express my sincere thanks to Piotr Tworzewski for many helpful conversations.

References

[Ch] E. M. C h i r k a, Complex Analytic Sets, Kluwer Acad. Publishers, 1989.

[Dr] R. N. D r a p e r, Intersection theory in analytic geometry , Math. Ann. 180 (1969), 175–204.

[En] R. E n g e l k i n g, General Topology , Heldermann, 1989.

[Gu] R. C. G u n n i n g, Introduction to Holomorphic Functions of Several Complex Vari- ables. Local Theory , Wadsworth & Brooks/Cole, 1990.

[Ha] R. H a r v e y, Holomorphic chains and their boundaries, in: Proc. Sympos. Pure

Math. 30, Part 1, Amer. Math. Soc., 1977, 309–382.

[ Lo] S. L o j a s i e w i c z, Introduction to Complex Analytic Geometry , Birkh¨ auser, Basel, 1991.

[R] S. R a m s, Bezout-type theorem for certain analytic sets, in preparation.

[Tw] P. T w o r z e w s k i, Intersection theory in complex analytic geometry , Ann. Polon.

Math. 62 (1995), 177–191.

[TW] P. T w o r z e w s k i and T. W i n i a r s k i, Continuity of intersection of analytic sets, ibid. 42 (1983), 387–393.

[Wi] T. W i n i a r s k i, Continuity of total number of intersection, ibid. 47 (1986), 155–

178.

Institute of Mathematics Jagiellonian University Reymonta 4

30-059 Krak´ ow, Poland E-mail: rams@im.uj.edu.pl

Re¸ cu par la R´ edaction le 3.4.1996

R´ evis´ e le 26.7.1996