On the extension of analytic sets into a neighborhood of the edge of a wedge in nongeneral position Текст научной статьи по специальности «Математика»

УДК 517.55

On the Extension of Analytic Sets into a Neighborhood of the Edge of a Wedge in Nongeneral Position

Evgeniya V. Yurieva*

Institute of Mathematics and Computer Science, Siberian Federal University, Svobodny, 79, Krasnoyarsk, 660041

Russia

Received 10.05.2013, received in revised form 10.06.2013, accepted 20.06.2013 Let K = D+ U Tn U D- be an n-circled two-sided wedge in Cn where the unit polycircle (torus) Tn plays a role of the edge, and domains D± adjoined to Tn may not contain any full-dimensional cone near Tn. In this case we say that the wedge K is in nongeneral position. Consider a question when the closures of pure n-dimensional analytic sets A± С D± x Cm compose a single analytic set in a neighborhood of the wedge K x Cm. If K is in general position then the answer to the question is given by the theorem of S.I.Pinchuk. In the present article we expand this theorem to the case when the two-circled wedge K is in nongeneral position, and m = 1.

Keywords: the edge of the wedge theorem, analytic sets, currents.

Introduction

Besides the Hartogs' theorem, one of the important phenomenon of an analytic continuation of holomorphic functions in several variables is the edge of the wedge theorem obtained by N. N. Bogolubov in 1956 [1,2]. This theorem asserts that if f (z) is a function holomorphic in a tubular domain T = Rn + ir, whose base r is the two-sided light cone y2 > + • • • + vl., and continuous in its closure T, then f (z) admits an analytic continuation in Cn. In [3] S. I. Pinchuk generalized the Bogolubov's result taking instead of the light cone an arbitrary wedge with the edge on a generating manifold bounded by smooth hypersurfaces in general position. The condition on the general position means that both sides of the wedge contain full-dimensional cones near the points of the edge. In article [4] we studied the problem of holomorphic extension of functions into a neighborhood of the edge of a two-sided n-circled wedge in nongeneral position. It dealt with wedges formed by two n-circled domains D± adjoined to the unit torus Tn = ||zi| = • • • = |zn| = 1} while the union K = D+ U Tn U D_ may not contain any full-dimensional cone near the edge Tn; in this case we say that the wedge K is it nongeneral position.

The problem of description the removable singularities of analytic sets was considered in works of H.Alexander, J. Bekker, B. Shiffman, K.Funahashi (one can find the statements, proofs and references in the book of E. M.Chirka [5]). Following E. M.Chirka [5] we can pose this problem in a such way: let E be a closed subset of complex manifold X and A is a pure dimensional analytic subset in X \ E; then a question arises: under what conditions on E and A the closure A will be an analytic set in X. The most general result was obtained by E.M.Chirka ([5], Theorem 18.5). He remarked that if E has a rich complex structure then an analytic subset in X \ E may approach E with uncontrolled "wobbles". Thus to remove such singularities we need some additional conditions. On the other hand, if the complex structure of E is poor, for example, if E is a smooth manifold that does not contain any maximal complex

* evg_yurieva@fromru.com © Siberian Federal University. All rights reserved

submanifold of the appropriate dimension (the candidate on the boundary of the analytic set) then one can hope that an analytic subset A in X \ E "won't notice" E, i. e. A will be analytic in X. This was confirmed by the Shiffman's result [6] about the extension of analytic sets of dimension p ^ 2 through the skeleton of the polycircle. K. Funahashi showed [7] that in Cn the singularities of the type Rn-m x Cm are also removable. The Chirka's theorem [5] states that if E is a C^submanifold in X such that the dimension of the complex tangent plane T^E in almost every point Z is smaller than p — 1, and A is a pure p-dimensional analytic subset in X \ E, then A is an analytic subset in X.

S.I.Pinchuk interpreted in [8] the result [3] (on the extension of holomorphic functions) as an phenomenon of continuation of analytic sets defined as graphs of holomorphic functions in a wedge in Cn+1. He received in [8] the generalization of the Bogolubov's theorem for analytic sets defined in a wedge of general position in Cn+m. The goal of this article is to expand the Pinchuk's result to the case of an n-circled wedge in nongeneral position when n = 2, m = 1.

1. Admissible analytic subsets in a wedge and their values on the edge

Let us introduce necessary notations. Let Q be a domain in C3 = C2 x Cw of the type

Q = U(K) x w,

where U(K) is a neighborhood of the two-circled wedge of type K = D+ U T2 U D_ c C and w is a bounded domain in Cw. We assume that K contains the "diagonal" A = {|z1| = |z2|}. Let A± be pure two-dimensional analytic subsets in D± x w. We will call such subsets admissible if the following properties hold:

1. The closures of A± do not cross the piece K x dw of the boundary d(K x w).

2. A± n (A x w) has a finite Hausdorff's 3-measure.

3. For any form f G D2'0(Q) there exist limits

i .

lim

£—>0

(T12±exw)nA±

defining currents d0A± of bidimension (2,0) on Q. Here T12±£ = {|zi| = |z2| = 1 ± e} is a family of tori which lie on the "diagonal" A and exhaust it. These currents we will call the values of A± on

t2

x w. We say that values of A+ and A_

coincide on

t2

x w, if d0A+ = d°A_ (see Fig. 1).

T2 x C

Fig. 1

a

Remark that the given notion of admissible subsets is inspired by the definition of S. I. Pinchuk [8], but they are different. The essential point is that we are able to introduce the currents d°A± without any restriction condition on the wedge K, like a restriction "of a general position".

Theorem. Assume that the wedge K = D+ U T2 U D_ contains the "diagonal" A and A± are admissible pure two-dimensional analytic sets in D± x w with the same values on T2 x w:

d°A+ = d°A_.

Then the closures A+ and A_ compose a single analytic set in a neighborhood of K x w.

2. The proof of the theorem

In view of the property 1 for admissible analytic sets, the projection n : A± ^ D± is a proper mapping and therefore it is a branched covering. Hence the sets A± are defined by Weierstrass' polynomials

- Wv(z)) = wk + a+(z)wfc-1 + • • • + a+(z) = 0,

v

n<w - wM(z)) = w1 + a-(z)w1-1 +-----+ a-(z) = 0

with a±(z) being holomorphic functions in D±.

Denote p1(z) := |z1| — |z2| and let p2(z) = p2(|z11, |z2|) be a smooth function in U(K) such that the image of {p2 = 0} on the Rheinhard diagram lies in D+ bellow the diagonal A and in D_ above A (see Fig. 2). Using functions p1 and p2 we introduce the following domains:

Q±i = {(z, w) G Q : ±Pi(z) > 0}, i = 1, 2, = Q±1 n Q±2.

In domains we define currents T± of bidimension (2,2):

a

I i i

fc^-=0

I

2'

M

Fig. 2

(T±2,^) := ±/ (p GD2'2 (Q±2)), T±1 := 0,

in±nA±

and currents Aa± of bidimension (2,1):

<Aa± := ± f

(a± xw)nA±

(1)

(2)

here A+ = {|z1| = |z21 > 1} and A_ = {|z1| = |z21 < 1}. By property 2 the last integrals are well defined.

1

Since A± are branched coverings over the points (z, w) e A± are parametrized by z e therefore we can consider Aa± as currents on A defined on forms in V2,1(U(K)). Similarly we consider T± as currents on defined on forms in D2'2(Q±i), where are the images of i. e.

= {z e U(K) : ±pi(z) > 0}, i = 1, 2, = n n±2. Taking into account this remark we rewrite the sum of currents in (2) as

(aa, ¥>) := lini ( J V + Jv) ,

a-

where A- = {|z11 = |z2| < 1 — e}, A+ = {|z1| = |z2| > 1+ e}. By property 3 and by assumption of Theorem we have

(aa,v) := lim i^J V ^y v),

ae

i. e. AA is a d-closed current.

For q =1, 2,... we define the current <rq = wqAa. Of course, <rq is d-closed. In a small Stein neighborhood U(A) it is d-exact:

<rq = drq, where tq e V'(U(K)).

Here we used the distribution space notation V' since currents of bidimension (2,2) in U(K) are generalized functions.

Let T±i := wqT±i — tq. We will show that these currents are d-closed in For v e

V2,1(n±2) one has

(dT±2, v) = (dwqT±2 — dTq, v) = J wqdv — (aq, v) = J d(wqv) — (aq, v).

n± n±

Using the Stokes' formula we get

(dT±2,V) = / wqV — / wqV — (^q,V), a P2 = 0

where the first integral gives (aq, v) and the second one vanishes, because having a compact support in n±2 the differential form v = 0 in a neighborhood of {p2 = 0}. Hence, dr±2 =0 in 0±2. Similarly, dTq 1 =0 in So, Tqi are holomorphic functions in

Currents Tq = Tq2 — Tq 1 are equal to wqT±2, i. e. they act in by formulas

(T±,V) = /[£(w„(z))q] v, v e V2'2(n±)

where wv are the roots of the corresponding Weierstrass' polynomial, so t± are polynomials in coefficients a±(z).

Note that t+ 2 and t- 1, t+ 1 and t-2 coincide and are equal to —tq in the intersections n Q-1, Q+1 n n-2 correspondingly. So they form holomorphic functions in U U

correspondingly. By Bochner's theorem [1] these functions admit an analytic continuation into the convex hull of the wedge K. So we get that coefficients of Weierstrass' polynomials are the same: a+(z) = a-(z). Thereby we constructed a Weierstrass' polynomial that defines a single analytic set in a neighborhood of K x w.

a+

a+

n±

References

[1] V.S.Vladimirov, Methods of the Theory of Functions of Complex Variables, Nauka, Moscow, 1964 (in Russian).

[2] N.N.Bogolubov, B.V.Medvedev, M.K.Polivanov, Problems of the Theory of Dispersion Relations, Fizmatgiz, Moscow, 1958 (in Russian).

[3] S.I.Pinchuk, Bogolubov's theorem on the "edge of the wedge" for generic manifolds, Mat. Sb, 94(136)(1974), no. 3(7), 468-482 (in Russian).

[4] E.V.Yurieva, On the holomorphic extention into a neighborhood of the edge of a wedge in nongeneral position, Siberian Math. J., 52(2011), no. 3, 563-568.

[5] E.M.Chirka, Complex Analytic Sets, Nauka, Moscow, 1985 (in Russian).

[6] B.Shiffman, On the continuation of analytic sets, Math. Ann., 185(1970), 1-12.

[7] K.Funahashi, On the extension of analytic sets, Proc. Japan Acad., 54(1978), Ser. A., 24-26.

[8] S.I.Pinchuk, The edge of the wedge theorem for analytic sets, Dokl. Akad. Nauk SSSR, 285(1985), 563-566 (in Russian).

Продолжение аналитических множеств в окрестность острия клина необщего положения

Евгения В. Юрьева

Пусть K = D+ U Tn U D- n-круговой двусторонний клин в Cn с острием на остове Tn единичного поликруга. При этом примыкающие к остову области D± могут не содержать вблизи Tn никакого полномерного конуса. В этом случаем мы говорим, что K — клин необщего положения. Рассматривается вопрос о том, когда чисто n-мерные аналитические множества A± С D± х Cm продолжаются до единого аналитического множества в окрестности клина K х Cm. Если K — клин общего положения, то ответ на поставленный вопрос дает теорема С. И. Пинчука. В статье для случая n = 2, m =1 эта теорема 'распространяется на клин необщего положения.

Ключевые слова: теорема об острие клина, аналитические множества, потоки.