• Nie Znaleziono Wyników

HOLOMORPHIC MAPS OF UNIFORM TYPE

N/A
N/A
Protected

Academic year: 2021

Share "HOLOMORPHIC MAPS OF UNIFORM TYPE"

Copied!
6
0
0

Pełen tekst

(1)

VOL. LXIX 1995 FASC. 1

HOLOMORPHIC MAPS OF UNIFORM TYPE

BY

L E M A U H A I

AND

T H A I T H U A N Q U A N G (HANOI)

Let E be a locally convex space and X complex manifold modelled on a locally convex space. A holomorphic map f from E to X is called a map of uniform type if f can be factorized holomorphically through the canonical map ω % from E to E % for some continuous seminorm % on E. Here for each continuous seminorm % on E we denote by E % the canonical Banach space associated with %, and by ω % the canonical map from E to E % . Now let H(E, X) and H u (E, X) denote the sets of holomorphic maps and of holomorphic maps of uniform type from E to X respectively. The aim of the present note is to find some necessary and sufficient conditions for the equality

(UN) H(E, X) = H u (E, X)

to hold. This problem for vector-valued holomorphic maps, i.e. for the case where X is a locally convex space, was investigated by some authors. The first result on this problem belongs to Colombeau and Mujica. In [2] they have shown that the equality (UN) holds when E is a dual Fr´ echet–Montel space and X a Fr´ echet space. Next, a necessary and sufficient condition for (UN) to hold in the class of scalar holomorphic functions on a nuclear Fr´ echet space was established by Meise and Vogt [7]. An important sufficient condition for (UN) for scalar holomorphic functions on such a space was also found recently by those two authors [8]. However, until now, when X does not have a linear structure, the problem has not been investigated.

Here we consider this problem for holomorphic maps with values in a complex manifold of infinite dimension, in particular, in the projective space associated with a Fr´ echet space (see the definition in §2). In the first section, by the method of [4], we give a characterization of the uniformity of holo- morphic maps with values in complex Banach manifolds. The scalar case has been proved by Meise and Vogt [7] by a different method. Section 2 is devoted to proving the main result (Theorem 2.1) of this note: every holomorphic map from a dual space of a nuclear Fr´ echet space (i.e., from

1991 Mathematics Subject Classification: 32A20, 32C10, 46A04.

[81]

(2)

a (DFN)-space) to the projective space CP(F ) associated with a Fr´echet space F is of uniform type. This is a variant of a result of Colombeau and Mujica [2].

The main tools for the proof of Theorem 2.1 are the solvability of

∂-equations for C closed differential (0, 1)-forms together with the uni- formity of C functions on a (DFN)-space which have been shown in [1]

and [2] respectively. However, the factoriality of the ring of germs of holo- morphic functions in infinitely many complex variables is also used here (see [6]).

Finally, we shall use standard notations from the sheaf theory of germs of holomorphic functions as presented in [3] for the finite-dimensional case and in [6] for the infinite-dimensional case, and from the theory of nuclear locally convex spaces in [10].

1. An extension characterization of uniformity. In this section we shall prove the following.

1.1. Theorem. Let E be a nuclear locally convex space and X a complex Banach manifold. Then the following two conditions are equivalent :

(i) Every holomorphic map from E to X is of uniform type.

(ii) If E is a subspace of a locally convex space F then every holomorphic map on E with values in X can be holomorphically extended to F .

P r o o f. (i)⇒(ii). Let f : E → X be a holomorphic map. By hypothesis there exists a continuous seminorm % on E and a holomorphic map g from E % to X such that f = gω % . Take a continuous seminorm % 1 ≥ % on E such that the canonical map ω %

1

% from E %

1

to E % is nuclear. We can write ω %

1

% in the form ω %

1

% = α ◦ β, where β : E %

1

→ ` and α : ` → E % are continuous linear maps. By the Hahn–Banach theorem, β can be extended to a continuous linear map b β : F % ˆ

1

→ ` , where % b 1 is a continuous seminorm on F such that % b 1 | E = % 1 . Then gα b βω % ˆ

1

is a holomorphic extension of f to F .

(ii)⇒(i). Let cs(E) denote the set of all continuous seminorms on E.

Consider the locally convex space F = Y

{E % : % ∈ cs(E)}

containing E as a subspace. By the hypothesis for every holomorphic map f from E to X there exists a holomorphic map g : F → X such that g| E = f . Let V be a coordinate neighbourhood in X and U = g −1 (V ). Since V is isomorphic to an open set in a Banach space we can find a finite set A in cs(E) and a non-empty open subset W of U such that

g(z) = g({z % } %∈A )

(3)

for every z ∈ W . Put

G = {z ∈ F : there exists a neighbourhood Z of z in F such that g(y) = g({y % } %∈A ) for every y ∈ Z}.

Since G is a non-empty open subset of F , to complete the proof it suffices to show that G is closed in F .

Let z 0 ∈ ∂G. Take a connected neighbourhood W 0 of z 0 in F such that g(W 0 ) is contained in a coordinate neighbourhood of X. Consider a holomorphic map h : W 0 → X given by

h(z) = g({z % } %∈A ) for z ∈ W 0 .

Since h and g are holomorphic on W 0 with h = g on G ∩ W 6= ∅, we have h = g| W

0

. Hence z 0 ∈ G and G is closed.

2. Uniformity of holomorphic maps with values in the projec- tive space associated with a Fr´ echet space. Before formulating The- orem 2.1 we describe the projective space CP(F ) associated with a locally convex space F . As in the case where dim F < ∞, CP(F ) is the space of all complex lines in F passing through 0 ∈ F . This space is equipped with the quotient topology under the canonical map F \ {0} → CP(F ) : x 7→ [x], the complex line passing through x and 0 ∈ F . For each α ∈ F \ {0} we consider the open subset V α of CP(F ) and the map θ α : V α → ker α given by

V α = {[x] ∈ CP(F ) : α(x) 6= 0} and θ α ([x]) = α(x)e α − x α(x) , where e α ∈ F is chosen such that α(e α ) = 1. It is easy to see that θ α is a homeomorphism between V α and ker α with

θ −1 α (z) = [z − e α ] for z ∈ ker α.

Moreover,

θ β θ α −1 (z) = β(z − e α )e β − (z − e α ) β(z − e α )

is holomorphic on θ α (V α ∩ V β ). Thus CP(F ) is a complex manifold with the local coordinate system {(V α , θ α ) : α ∈ F \ {0}}. From the above relation it follows that θ α ([x]) is meromorphic on V β for every β ∈ F \ {0}, β 6= α.

Thus θ α can be considered as a meromorphic function on CP(F ) with values in ker α ⊂ F .

2.1. Theorem. Let E be a (DFN )-space and F a Fr´echet space. Then every holomorphic map from E to CP(F ) is of uniform type.

For proving the theorem we need the following two lemmas.

(4)

2.2. Lemma. Let f : D → CP(F ) be a holomorphic map, where D is an open set in a locally convex space E and F is a Fr´ echet space. Then for each z ∈ D there exists a neighbourhood U of z in D and two holomorphic functions h and σ on U with values in F and C respectively such that

Z(h, σ) = ∅ and f | U = [h : σ], where Z(h, σ) denotes the common zero-set of h and σ.

P r o o f. For each z ∈ D we can find a neighbourhood U of z in D such that if we consider f as a meromorphic function on U with values in F then f can be written in the form

f | U = h σ

with z 6∈ Z(h, σ), where h and σ are holomorphic functions on U . Then Z(h, σ) = ∅ in a neighbourhood of z in U .

2.3. Lemma. Let β and σ be scalar holomorphic functions on an open set D in a locally convex space E and let g be a holomorphic function on D with values in a locally convex space. Assume that βg/σ is holomorphic on D and Z(g, σ) = ∅. Then β/σ is holomorphic on D.

P r o o f. Let z 0 ∈ D. Since the local ring O E,z

0

of germs of holomorphic functions at z 0 is factorial [6, Proposition 5.15] we can write

σ = σ 1 p

1

. . . σ p n

n

in a neighbourhood U of z 0 with the germs (σ 1 ) z

0

, . . . , (σ n ) z

0

being irre- ducible. By hypothesis and from the equality βg/σ 1 = (βg/σ)σ 1 p

1

−1 . . . σ n p

n

it follows that βg/σ 1 is holomorphic at z 0 . On the other hand, since by hypothesis Z(g, σ) = ∅ and Z(σ) = S n

i=1 Z(σ i ) it follows that Z(g, σ i ) = ∅ for i = 1, . . . , n. Hence, from the irreducibility of σ 1 we infer that Z(σ 1 ) z

0

⊆ Z(β) z

0

. Since (σ 1 ) z

0

is irreducible, it follows that β = β 1 σ 1 in a neighbour- hood U 1 of z 0 in U , where β 1 and σ 1 are holomorphic on U 1 . Hence β/σ 1

is holomorphic on U 1 . Applying the above argument to β 1 , σ 1 and g we get the holomorphy of β/σ 1 2 at z 0 . Continuing this process we infer that β/σ is holomorphic at z 0 .

Now we can prove Theorem 2.1 as follows.

Let f : E → CP(E) be a holomorphic map, where E is a (DFN)-space.

We denote by O E (resp. M E ) the sheaf of germs of holomorphic (resp.

meromorphic) functions on E. Let

O E = {σ ∈ O E : σ is invertible},

M E = M E \ {0} and D E = M E /O E .

(5)

Here as in the finite-dimensional case, D E is called the sheaf of germs of divisors on E. We denote by Z the sheaf of integers on E. Then we have two exact sheaf sequences on E:

0 → Z → O E

−→ O exp E → 0, 0 → O E → M E −→ D η E → 0,

where exp(σ) = e 2πiσ and η is the canonical projection. By [6, p. 266, Propo- sition 3.6] we have H 1 (E, O E ) = 0. On the other hand, since H 2 (E, Z) = 0, considering the exact cohomology sequences associated with the above exact sheaf sequences it follows that for every divisor d ∈ H 0 (E, D E ) there exists a meromorphic function σ ∈ H 0 (E, M E ) such that η(σ) = d, where b b η is the map from H 0 (E, M E ) to H 0 (E, D E ) induced by η. By applying Lemma 2.2 to f we can find an open cover {U j } of E and holomorphic functions h j and σ j on U j such that

Z(h j , σ j ) = ∅ and f | U

j

= [h j : σ j ]

for every j. Since h i /σ i = h j /σ j on U i ∩ U j , Lemma 2.3 implies that the formula

z 7→ (σ j ) z O E,z

for z ∈ U j defines a divisor d on E. Thus there exists a meromorphic function β on E with β 6= 0 such that β z /d z ∈ O E,z for z ∈ E.

These relations imply that β is holomorphic on E and h = βf is holo- morphic on E with Z(h, β) = ∅. Let {x j } be a sequence of continuous linear functionals on F which separates the points of h(E). Since Z({x j h}, β) = ∅, it follows from [6, p. 247, Proposition 3.2] that there exist C functions ϕ j , j ≥ 0, such that

X

j≥1

ϕ j |x j h| 2 + ϕ 0 |β| 2 = 1.

By applying a result of Colombeau and Mujica [2] we find a continuous seminorm % on E and C functions ϕ b j , j ≥ 0, together with holomorphic functions b h j , j ≥ 1, b β and b h on E % such that ϕ j = ϕ b j ω % , x j h = b h j ω % for j ≥ 1 and ϕ 0 = ϕ b 0 ω % , β = b βω % , h = b hω % . Then P

j≥1 ϕ b j |b h j | 2 + ϕ b 0 | b β| 2 = 1 on E % . Thus Z({b h j } j≥1 , b β) = ∅ and, hence, Z(b h, b β) = ∅. Consequently, the formula

f (z) = [b b h(z) : b β(z)] for z ∈ E %

defines a holomorphic map b f from E % to CP(F ) such that f = b f ω % . Theo- rem 2.1 is proved.

Acknowledgements. The authors would like to thank Prof. N.V. Khue

for helpful advice, and also the referees for their suggestions.

(6)

REFERENCES

[1] J. F. C o l o m b e a u, Differential Calculus and Holomorphy , North-Holland Math.

Stud. 65, North-Holland, Amsterdam, 1982.

[2] J. F. C o l o m b e a u and J. M u j i c a, Holomorphic and differentiable mappings of uniform bounded type, in: Functional Analysis, Holomorphy and Approximation Theory, J. A. Barroso (ed.), North-Holland Math. Stud. 71, North-Holland, Ams- terdam, 1982, 179–200.

[3] G. F i s c h e r, Complex Analytic Geometry , Lecture Notes in Math. 538, Springer, Berlin, 1976.

[4] N. V. K h u e, On the extension of holomorphic maps on locally convex spaces with values in Fr´ echet spaces, Ann. Polon. Math. 44 (1984), 163–175.

[5] —, On meromorphic functions with values in locally convex spaces, Studia Math.

73 (1982), 201–211.

[6] P. M a z e t, Analytic Sets in Locally Convex Space, North-Holland, Amsterdam, 1984.

[7] R. M e i s e and D. V o g t, Extension of entire functions on locally convex spaces, Proc. Amer. Math. Soc. 92 (1984), 495–500.

[8] —, —, Holomorphic functions of uniformly bounded type on nuclear Fr´ echet spaces, Studia Math. 83 (1986), 147–166.

[9] J. M u j i c a, Domains of holomorphy in (DFC )-spaces, in: Functional Analysis, Holo- morphy and Approximation Theory, Lecture Notes in Math. 843, Springer, 1981, 500–533.

[10] A. P i e t s c h, Nuclear Locally Convex Spaces, Ergeb. Math. Grenzgeb. 66, Springer, 1972.

DEPARTMENT OF MATHEMATICS PEDAGOGICAL INSTITUTE 1 HANOI TULIEM, HANOI, VIETNAM

Re¸ cu par la R´ edaction le 26.1.1993;

en version modifi´ ee le 11.7.1994

Cytaty

Powiązane dokumenty

Let F be locally integrable then we can define local leaves, F -topology and global leaves in a similar way as in the regular case (comp. Therefore we get:.. Then X/F is a

For supersingular elliptic curves over finite fields, it turns out that there are some interesting isogeny invariants... In order to identify another isogeny invariant, we will

In §2 we study F -quotients of uniform type and introduce the concept of envelope of uF -holomorphy of a connected uniformly open subset U of E.. We have been unable to decide if an

F. We prove that if A is a basin of immediate attraction to a periodic at- tracting or parabolic point for a rational map f on the Riemann sphere, then the periodic points in

In this paper a relationship between subordination and inclusion the maps of some concentric discs is investigated in a case when f ranges over the class Nn, (n&gt;2) and P

Pointwise Bounded Families of Holomorphic Functions Punktowo ograniczone rodziny funkcji holomorficznych Точечно ограниченные семейства голоморфных функций.. Let 7

, On the domain of local univalence and starlikeness in a certain class of holomorphic functions, Demonstr. , Geometric Theory of Functions of a Complex

Note that from the well-known estimates of the functionals H(.f) a |a2| and H(,f) = |a^ - ot a22j in the class S it follows that, for «6S 10; 1) , the extremal functions