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.
Highlights
The classical theorem of Poincaré-Hopf (Milnor 1965) implies that for a smooth vector field X defined in a neighborhood of the closed ball B2n(0; R) ⊂ R2n and transverse to the boundary ∂B2n(0, R) = S2n−1(0; R) there is at least one singular point p ∈ sing(X) ∩ B2n(0; R)
If the singularities of X in B2n(0; R) are isolated Ind(X; p) = 1 where p runs through all the singular points p ∈ sing(X) ∩ B2n(0; R) and Ind(X; p) is the index of X at the singular point p
Let us recall the notion of transversality we shall use: Given a holomorphic one form in U ⊂ Cn for each p ∈ U with (p) = 0 we define a (n − 1)-dimensional linear subspace P (p) := {v ∈ Tp(Cn); (p) · v = 0}
Summary
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 B2n(0; R) ⊂ R2n and transverse to the boundary ∂B2n(0, R) = S2n−1(0; R) there is at least one singular point p ∈ sing(X) ∩ B2n(0; R). Let us recall the notion of transversality we shall use: Given a holomorphic one form in U ⊂ Cn for each p ∈ U with (p) = 0 we define a (n − 1)-dimensional linear subspace P (p) := {v ∈ Tp(Cn); (p) · v = 0}.
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