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Asian Journal of Mathematics
Transversality of complex linear distributions with spheres, contact forms and Morse type foliations, II
DOI: https://dx.doi.org/10.4310/AJM.2015.v19.n2.a8
Toshikazu Ito (Department of Natural Science, Ryukoku University, Fushimi-ku, Kyoto, Japan)
Bruno Scárdua (Institute of Mathematics, Federal University of Rio de Janeiro, Brazil)
Yoshikazu Yamagishi (Department of Natural Science, Ryukoku University, Fushimi-ku, Kyoto, Japan)
The study of holomorphic foliations transverse to real submanifolds has its own interest, as for instance its connections with the construction or existence of complex structures. The comprehension of the transverse dynamics of such foliations is also granted by that study. As for the non-integrable case, the study of contact forms in the holomorphic framework is related to the study of (non-integrable) codimension one distributions which are transverse to spheres in the affine space.
The starting point for our work is the following question: Is there any codimension one holomorphic foliation $\mathcal{F}$ in a neighborhood of the closed unit disk $\overline{D^{2n}(1)} \subset \mathbb{C}^n$ such that $\mathcal{F}$ is transverse to the boundary sphere $S^{2n-1}(1)$ for $n \geq 3$? In this paper we study transversality of (integrable or not) holomorphic perturbations of codimension one linear distributions, with spheres in the complex affine space. So far, the examples of such distributions are related to contact forms and are as a kind of counterpart of the integrable case. Based on an extension of Takagi's factorization theorem for nonsingular matrices in terms of Jordan canonical forms of its generalized coneigenvectors, we prove that given a generic nonsingular $n \times n$ complex matrix $A$ and any holomorphic one-form ω having its linear part at the origin given by $A$, the corresponding distribution $\mathcal{K}(\omega) : \lbrace \omega = 0 \rbrace$ is not transverse to the spheres $S^{2n-1}(r)$ for small $r \gt 0$. Here, by generic we mean that $\overline{A}A$ has a simple positive eigenvalue $\lambda \gt 0$, and any other eigenvalue has absolute value different from $\lambda$. Using this, we are able to conclude that, in $\mathbb{C}^n$, a distribution which is a perturbation of a linear Morse foliation, is not transverse to small spheres and its variety of contacts has at least n branches in a neighborhood of the origin.
Takagi's factorization theorem, holomorphic distribution, holonomy group, linear Morse foliation
2010 Mathematics Subject Classification
Primary 15A23, 15A63, 37F75. Secondary 15A06, 37J30.