Some variants of the Poincaré theorem on tubes in the complement of surfaces in complex Euclidean and projective spaces (Q2716189)

A classical Poincaré's theorem states that any two-dimensional cycle in the complement of an algebraic variety in \(\mathbb{C}^2\) is homologous to a tube over a one-dimensional cycle in the set of regular points of the variety. It was shown in [\textit{N. A. Buruchenko} and \textit{A. K. Tsikh}, Sb. Math. 186, No. 10, 1417-1427 (1995; Zbl 0865.32004)] that the result fails in \(\mathbb{C}^n\) for \(n>2\), and a class of analytic hypersurfaces was presented for which the theorem remains true (these are, roughly speaking, hypersurfaces with `thin' set of singularities).NEWLINENEWLINENEWLINEIn the present paper, an analog of the latter result is obtained for analytic varieties of higher codimensions. In addition, it is shown that to verify Poincaré's theorem for a homogeneous hypersurface in \(\mathbb{C}^{n+1}\), it suffices to do that for its projection to \(\mathbb{P}^n\). As a consequence, it gives Poicaré's theorem for homogeneous hypersurfaces in \(\mathbb{C}^3\).NEWLINENEWLINEFor the entire collection see [Zbl 0959.00006].