On Real Analytic Levi-Flat Hypersurfaces Associated with Milnor Fibrations

Let G and X be germs of holomorphic vector fields at \(0\in {\mathbb {C}}^n\) $0\in {\mathbb{C}}^n$ . Consider the real analytic map \(\psi _{G,X}:{\mathbb {C}}^n\rightarrow {\mathbb {C}}\) ${\psi }_{G,X}:{\mathbb{C}}^n\to \mathbb{C}$ defined by \(\psi _{G,X}(z)=\langle G(z),X(z)\rangle \) ${\psi }_{G,X}(z)=⟨G(z),X(z)⟩$ , where \(\langle \cdot ,\cdot \rangle \) $⟨·,·⟩$ represents the usual Hermitian product. In this paper, we investigate the following question: under which conditions on the germs of holomorphic vector fields G and X is the real analytic hypersurface \(M=\{F(z)=2{\textrm{Re}}(\psi _{G,X}(z))=0\}\) $M=\{F(z)=2\text{Re}({\psi }_{G,X}(z))=0\}$ Levi-flat? This problem was posed by Maria A. Soares Ruas.