On~the homological structure of a~fibering of an algebraic variety with isolated sungularities

A. B. Zhizhchenko

Sbornik. Mathematics, Tome 26 (1975) no. 2, pp. 280-293

Résumé

We study a fibering of a complex projective variety $W$ with isolated singularities of Pham type in the fibers. We investigate the connection between the homology groups $H_*(W)$ and the homology groups $H_*(F_0)$ of a general fiber $F_0$ and local data. We shall explain how the relation between the structure of the homology groups $H_{n+2}(W),\dots,H_{2n}(W)$, where $n=\dim_{\mathbf C}W$, and the homology groups of a nonsingular fiber is trivial, while the mean groups $H_{n+1}(W)$, $H_n(W)$ and $H_{n-1}(W)$ can be expressed in terms of the homology groups of the general fiber $F_0$ together with the kernels and cokernels of the morphisms $r_*^i\colon H_*(F_0)\to H_*(F_i)$, where $F_1,\dots,F_s$ are the singular fibers of the fibering, and the $r_*^i$ correspond to the mappings (retractions) of the nonsingular fiber onto the singular ones. Bibliography: 10 titles.