On holomorphic one-forms transverse to closed hypersurfaces

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 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.


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 B 2n (0; R) ⊂ R 2n and transverse to the boundary ∂B 2n (0, R) = S 2n−1 (0; R) there is at least one singular point p ∈ sing(X) ∩ B 2n (0; R). Moreover, if the singularities of X in B 2n (0; R) are isolated then Ind(X; p) = 1 where p runs through all the singular points p ∈ sing(X) ∩ B 2n (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 C n . This motivated the study of codimension one holomorphic foliations on open subsets of C n with the 266 TOSHIKAZU ITO and BRUNO SCÁRDUA transversality property with real submanifolds and, particularly, the case of foliations transverse to spheres S 2n−1 (0; R) ⊂ C n (see [Ito and Scardua 2002a] for more information). Let us recall the notion of transversality we shall use: Given a holomorphic one form in U ⊂ C n for each p ∈ U with (p) = 0 we define a (n − 1)-dimensional linear subspace P (p) := {v ∈ T p (C n ); (p) · v = 0}. If (p) = 0 we set P (p) := {O p } < T p (C n ) and we shall say that the distribution P defined by is singular at p. As usual we assume that cod sing( ) ≥ 2 so that if is integrable i.e., ∧ d ≡ 0 in U (equivalently if P is integrable) then P = T F for a unique singular holomorphic foliation F of codimension one in U having as singular set sing(F ) = sing(P ) = sing( ). Including the non-integrable case we have the following definition of transversality.
Definition 1. (Ito and Scárdua 2002a). Given a smooth (real) submanifold M ⊂ U we shall say that P is transverse to M if for every p ∈ M we have T p M + P (p) = T p (R 2n ) as real linear spaces.
In particular, since P (p) = {0} for any singular point p, we conclude that sing(P Thus P is transverse to M if, and only if, the foliation F defined by is transverse to M in the sense of (Ito and Scárdua 2002a) which is the ordinary sense.
f j (z)dz j in holomorphic coordinates in a neighborhood of the closed domain D in C n , then sing( ) = {p; f j (p) = 0, ∀j } and we define the gradient of as the complex C ∞ vector field Given any isolated singularity p ∈ sing( ) we define the index of at p by Our main result is the following: Theorem 1. (Ito and Scárdua 2002a,b). Let D ⊂⊂ C n be a relatively compact domain with smooth boundary ∂D ⊂ C n . Assume that the (canonical) exact sequence H 1 (D, Z) → H 1 (∂D, Z) → 0 is exact. Then given any holomorphic one-form in a neighborhood U of D in C n such that the corresponding holomorphic distribution P is transverse to the boundary ∂D we have As an immediate consequence of Theorem 1 we obtain: Theorem 2. (Ito and Scárdua 2002a). Let be a holomorphic one-form in a neighborhood U of the closed ball B 2n (0; R) in C n , n ≥ 2. Assume that P is transverse to the sphere S 2n−1 (0; R) = ∂B 2n (0; R). Then n is even and has exactly one singular point o ∈ B 2n (0; R). 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 ⊂⊂ C n is Stein and n ≥ 3 we also obtain: We also refer to (Ito and Scárdua 2002c) for further results.

SKETCH OF THE PROOF OF THEOREM 1
We have the canonical exact sequence H 1 (D) → H 1 (∂D) → H 2 (D, ∂D) and by hypothesis H 1 (D) → H 1 (∂D) → 0 is exact. Take a holomorphic vector field n in a neighborhood of D such that for each q ∈ ∂D the vector n(q) is non-zero and ortogonal to the complex tangent space T C q (∂D) < T q (C n ). Given as in Theorem 1 we introduce the analytic set := q; (q) · n(q) = 0 .
Then for each q ∈ ∂D we have q ∈ if and only if grad( )(q) ∈ T C q (∂D). Since the vector field grad( ) is orthogonal to P we conclude that there exists a smooth bump-function ϕ : C n C ∞ −→ R such that is transverse to ∂D. Using the hypothesis that H 1 (D) → H 1 (∂D) → 0 is exact we obtain a real smooth section (ie. a C ∞ real vector field) ξ ∈ T Z over a neighborhood of D which is transverse to ∂D; indeed ξ is obtained as extension of a suitable vector field ξ(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 ξ once we have the following lemma: Lemma 1. (Ito and Scárdua 2002a). In the above situation we have: 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 H 2 (D, ∂D) H 2n−2 (D) = 0 in the case of a Stein domain and n ≥ 3.

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