On holomorphic one-forms transverse to closed hypersurfaces

On holomorphic one-forms transverse to closed hypersurfaces

Toshikazu Ito^I; Bruno Scárdua^II

^IDepartment of Natural Science, Ryukoku University, Fushimi-ku, Kyoto 612, Japan

^IIInstituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, 21945-970 Rio de Janeiro, RJ, Brasil

^{Correspondence}

ABSTRACT

In this note we announce some achievements in the study of holomorphic distributions admitting transverse closed real hypersurfaces. We consider a domain with smooth boundary in the complex affine space of dimension two or greater. Assume that the domain satisfies some cohomology triviality hypothesis (for instance, if the domain is a ball). We prove that if a holomorphic one form in a neighborhood of the domain is such that the corresponding holomorphic distribution is transverse to the boundary of the domain then the Euler-Poincaré-Hopf characteristic of the domain is equal to the sum of indexes of the one-form at its singular points inside the domain. This result has several consequences and applies, for instance, to the study of codimension one holomorphic foliations transverse to spheres.

Key words: Holomorphic one-form, vector field, Euler-Poincaré characteristic, foliation, distribution.

RESUMO

Nesta nota anunciamos alguns resultados obtidos no estudo de distribuições holomorfas admitindo hipersuperfícies reais fechadas transversais. Consideramos um domínio com bordo suave no espaço afim complexo de dimensão dois ou maior. Suponha que o domínio satisfaz uma certa hipótese de trivialidade cohomológica (por exemplo, se o domínio é uma bola). Provamos que se uma um-forma holomorfa em uma vizinhança do domínio é tal que a distribuição holomorfa correspondente é transversal ao bordo do domínio então a característica de Euler-Poincaré-Hopf do domínio é igual à soma dos índices da um-forma nos seus pontos singulares dentro do domínio. Este resultado tem várias conseqüências e se aplica, por exemplo, ao estudo de folheações holomorfas de codimensão um transversais a esferas.

Palavras-chave: Um-forma holomorfa, campo de vetores, característica de Euler-Poincaré, folheação, distribuição.

1 INTRODUCTION

The classical theorem of Poincaré-Hopf (Milnor 1965) implies that for a smooth (real) vector field X defined in a neighborhood of the closed ball and transverse to the boundary ¶ B²ⁿ (0, R) = S^2n–1(0; R) there is at least one singular point p Î sing(X) Ç B²ⁿ(0; R). Moreover, if the singularities of X in B²ⁿ (0; R) are isolated then å Ind(X; p) = 1 where p runs through all the singular points p Î sing(X) Ç B²ⁿ(0; R) and Ind(X; p) is the index of X at the singular point p. In (Ito 1994) one can find a version of this theorem for holomorphic vector fields on . This motivated the study of codimension one holomorphic foliations on open subsets of with the transversality property with real submanifolds and, particularly, the case of foliations transverse to spheres S^2n–1(0; R) Ì (see [Ito and Scardua 2002a] for more information). Let us recall the notion of transversality we shall use:

Given a holomorphic one form W in U Ì for each p Î U with W(p) ¹ 0 we define a (n – 1)-dimensional linear subspace P_W(p) : = {v Î T_p(); W(p)·v = 0}. If W(p) = 0 we set P_W(p) : = {O_p} < T_p() and we shall say that the distribution P_Wdefined by W is singular at p. As usual we assume that cod sing(W) > 2 so that if W is integrable i.e., W Ù d W º 0 in U (equivalently if P_W is integrable) then P_W = for a unique singular holomorphic foliation of codimension one in U having as singular set sing() = sing(P_W) = sing(W). Including the non-integrable case we have the following definition of transversality.

DEFINITION 1. (Ito and Scárdua 2002a). Given a smooth (real) submanifold M Ì U we shall say that P_Wis transverse to M if for every p Î M we have T_pM + P_W(p) = T_p() as real linear spaces.

In particular, since P_W(p) = {0} for any singular point p, we conclude that sing(P_W) Ç M = Æ if M < U. We also point out that if W is a holomorphic integrable one-form in U then given a submanifold M Ì U the distribution P_W is transverse to M if, and only if, for each p Î M we have p Ï sing() and also T_p(L_p) + T_p(M) = T_p() (as real linear spaces); where L_p is the leaf of that contains the point p Î M. Thus P_W is transverse to M if, and only if, the foliation defined by W is transverse to M in the sense of (Ito and Scárdua 2002a) which is the ordinary sense.

Let now W =

f_j(z)dz_j in holomorphic coordinates in a neighborhood of the closed domain in , then sing(W) = {p; f_j(p) = 0 , "j} and we define the gradient of W as the complex C^¥ vector field

Given any isolated singularity p Î sing(W) we define the index of W at p by

Our main result is the following:

THEOREM 1. (Ito and Scárdua 2002a,b). Let D Ì Ì be a relatively compact domain with smooth boundary ¶D Ì . Assume that the (canonical) exact sequence H¹(D, ) ® H¹(¶D, ) ® 0 is exact. Then given any holomorphic one-form W in a neighborhood U of in such that the corresponding holomorphic distribution P_Wis transverse to the boundary ¶D we have

where c(D) is the Euler-Poincaré-Hopf characteristic of D.

Write W =

f_j(z)dz_j in holomorphic coordinates in a neighborhood of o in . We shall say that o is a simple singularity of W if:

As an immediate consequence of Theorem 1 we obtain:

THEOREM 2. (Ito and Scárdua 2002a). Let W be a holomorphic one-form in a neighborhood U of the closed ball in , n > 2. Assume that P_W is transverse to the sphere S^2n–1(0; R) = ¶B²ⁿ(0; R). Then n is even and W has exactly one singular point o Î . Moreover this singular point is simple.

In (Ito and Scárdua 2002b) one finds a natural extension of the above result for holomorphically embedded closed balls in Stein spaces. In case D Ì Ì is Stein and n > 3 we also obtain:

COROLLARY 1. (Ito and Scárdua 2002b). Let D Ì be a relatively compact Stein domain with smooth boundary ¶D Ì and suppose n > 3. Given any holomorphic one-form W in a neighborhood of with P_W transverse to ¶D we have

Since Ind(W; p) > 1 for all (isolated) singular point we obtain:

COROLLARY 2. Let D Ì Ì and W be as in Theorem 1. If c(D) = 0 then sing(W) Ç = Æ. If c(D) > 1 then sing(W) Ç D ¹ Æ and necessarily n is even.

We also refer to (Ito and Scárdua 2002c) for further results.

2 SKETCH OF THE PROOF OF THEOREM 1

We have the canonical exact sequence H¹(D) ® H¹(¶D) ® H²(D, ¶D) and by hypothesis H¹(D) ® H¹(¶D) ® 0 is exact. Take a holomorphic vector field in a neighborhood of such that for each q Î ¶D the vector (q) is non-zero and ortogonal to the complex tangent space (¶D) < T_q (). Given W as in Theorem 1 we introduce the analytic set

Then for each q Î ¶D we have q Î S_W if and only if grad(W)(q) Î (¶D). Since the vector field grad(W) is orthogonal to P_W we conclude that there exists a smooth bump-function such that

is transverse to ¶D.

Using the hypothesis that H¹(D) ® H¹(¶D) ® 0 is exact we obtain a real smooth section (ie. a C^¥ real vector field) Î TZ over a neighborhood of which is transverse to ¶D; indeed is obtained as extension of a suitable vector field x(z) = a(z)X(z) + b(z)Y(z) defined in a neighborhood of ¶D and which is transverse to ¶D, where X and Y are given by

Theorem 1 now follows from Poincaré-Hopf Index theorem (Milnor 1965) applied to the vector field once we have the following lemma:

LEMMA 1. (Ito and Scárdua 2002a). In the above situation we have:

(i) sing(W) Ç D is finite, sing(W) Ç D = sing();

(ii) Given any p Î sing(W) Ç D we have Ind(W; p) = (–1)ⁿInd(; p).

Theorem 2 is a straightforward consequence of Theorem 1. Corollary 1 is proved recalling that by Poincaré-Lefschetz duality (Griffiphs and Harris 1978) we have that H²(D, ¶D) @ H_2n–2(D) = 0 in the case of a Stein domain and n > 3.

3 ACKNOWLEDGMENT

We are grateful to Professor N. Kawazumi for his interest and valuable suggestions in improving our original results.

Manuscript received on May 26, 2003; accepted for publication on June 3, 2003; presented by MARCIO SOARES

Mathematics Subject Classification: 32S65; 57R30.

• GRIFFITHS P AND HARRIS J. 1978. Principles of Algebraic Geometry; John Wiley & Sons, N.Y.
• ITO T. 1994. A Poincaré-Bendixson Type Theorem for Holomorphic Vector Fields; RIMS publication, Kyoto, nº 878.
• ITO T AND SCÁRDUA B. 2002a. A Poincaré-Hopf type theorem for holomorphic 1-forms; Pre-publication Ryukoku University, Kyoto.
• ITO T AND SCÁRDUA B. 2002b. On complex distributions transverse to spheres; submitted.
• ITO T AND SCÁRDUA B. 2002c. On real transverse sections of holomorphic foliations; submitted.
• MILNOR J. 1965. Topology from the differential view point, The University Press of Virginia, Charlottesville.

Publication Dates

• Publication in this collection
  25 Aug 2003
• Date of issue
  Sept 2003

History

• Accepted
  03 June 2003
• Received
  26 Mar 2003