On the homology theory of analytic sheaves

V. D. Golovin

Izvestiya. Mathematics , Tome 16 (1981) no. 2, pp. 239-260

Résumé

The homology groups of analytic sheaves constitute a natural tool in the theory of duality on complex spaces. In algebraic geometry, homology theory was worked out back in the fifties by Grothendieck. But the corresponding theory remained undeveloped in complex analytic geometry, although important work has been done in this direction by Ramis and Ruget, and by Andreotti and Kas. Homology sheaves and homology groups of analytic sheaves have been defined by the author in an earlier paper (Dokl. Akad. Nauk SSSR 225, № 1 (1975), 41–43). In the present paper, their basic properties are investigated. In particular, it is shown that there exist spectral sequences relating the homology groups to the $\operatorname{Ext}$ functors and the cohomology groups. For complex manifolds, a Poincare duality is obtained. It is also shown that there are spectral sequences relating the homology groups of analytic sheaves to the Aleksandrov-Chekh homology and the canonical homology defined by Sklyarenko. Bibliography: 26 titles.