A Poincaré-Hopf-type theorem for holomorphic one-forms
The authors study the singularities of codimension one (not necessarily integrable) holomorphic distributions which are transverse to the \(2n-1\) sphere in \(\mathbb C^{n}\), \(n>1\). The main result is an interesting generalization of the classical Poincaré-Hopf index theorem. More in detail, let \(\Omega\) be a holomorphic one form in a neighborhood of a closed ball \(B\subset\mathbb C^n\). The associated (possibly singular) codimension one distribution given by \(\operatorname {Ker}\Omega\subset T\mathbb C^n\) is transverse to the boundary sphere \(S^{2n-1}=\partial B\) provided that \(S^{2n-1}\) contains no singularities of \(\Omega\) and \(\operatorname {Ker}\Omega_p\not\subset T_pS^{2n-1}\) for all \(p\in S^{2n-1}\). The authors prove that if \(\operatorname {Ker}\Omega\) is transverse to \(S^{2n-1}\) then \(n\) is even and \(\Omega\) has exactly one simple singularity in \(B\). In particular, the theorem applies to codimension one (singular) foliations, providing a negative answer for odd dimensions \(n\) to the following question: is there any codimension one holomorphic foliation in a neighborhood of a ball \(B\subset\mathbb C^n\) transverse to \(\partial B\) for \(n>2\)? Among other consequences, the previous result is used to prove that for \(n\geq 3\) any holomorphic one-form which annihilates the radial vector field and has no singularities on the boundary of a ball is not integrable
scientific article; zbMATH DE number 2137340