Transversality of complex linear distributions with spheres, contact forms and Morse type foliations

Let AA be a nonsingular n×nn×n complex matrix. We prove that the corresponding codimension one linear distribution K(A)K(A) is transverse to the unit sphere centered at the origin if and only if ĀA has no positive eigenvalue. If K(A)K(A) is integrable then the corresponding foliation is of Morse type if and only if the eigenvalues of ĀA are pairwise distinct. In this case the variety of contacts with the spheres centered at the origin is a union of nn complex lines. We also give a canonical normal form similar to Takagi’s unitary normal form. We shall show that the tangency property is robust for small perturbations of the distribution. Finally, we will give a detailed study of an example of a non-Morse type non-linear holomorphic foliation given by the Pham polynomial.