On the homology of subvarieties of toric varieties
In this paper the author considers a quasiaffine variety \(Y\backslash V\) represented as the difference of two algebraic varieties in \(\mathbb C^n\). The structure of the compact homology groups \(H_p(Y\backslash V)\) is described by Froissart's theorem [\textit{T. Fotiadi, M. Froissart, J. Lascoux} and \textit{F. Pham}, Topology 4, 159--191 (1965; Zbl 0173.09301)], which asserts that if \(V_i=S_i\cap Y\), the family of hypersurfaces \(S\) in \(\mathbb P^n\) is generic and \(V=\bigcup V_i\), then \[ H_p(Y\backslash V)=H_p(Y) \oplus [\bigoplus_i \delta H_{p-1}(V_i)] \oplus [\bigoplus_{i,j} \delta^2 H_{p-2}(V_i \cap V_j)] \oplus \dots, \] where the direct sum on the right is taken over all possible intersections of the family \({V_i}_i\) and \(\delta, \delta^2,\dots\) are iterated coboundary (tube) Leray operators [\textit{J. Leray}, Bull. Soc. Math. Fr. 87, 81--180 (1959; Zbl 0199.41203)]. The first theorem in the paper explains the reason of the decomposition of \(H_p(Y\backslash V)\) above. Indeed, they show a generalization of Froissart's theorem for compact toric \(n\)-dimensional spaces. Quasitoric varieties are defined as the difference \(Y\backslash V\) of two analytic subvarieties in some compact complex toric space \(X\). It turns out that the quasitoric variety \(Y\backslash V\) being generic is insufficient for the validity of a decomposition of \(H_p(Y\backslash V)\) as the one before. They show that for the validity of the decomposition it is necessary to impose additional constraints on the rational equivalence of cycles. Precisely, in Theorem 2 they give conditions on the class of cycles of \(X\) for which a decomposition \(H_p(Y\backslash V)=H_p(Y) \oplus [\bigoplus_i \delta H_{p-1}(V_i)] \oplus\dots\) holds. As a corollary, they conclude that for a generic algebraic hypersurface \(F\) in the complex torus \(\mathbb T^n\), \(H_p(F,\mathbb Z)\cong \mathbb Z^{n\choose p}\) for all \(p=0,\dots,n-2\).
scientific article; zbMATH DE number 5816874