Aleksei Rosly

We construct the so-called polar complex for an arbitrary locally free sheaf on a smooth variety over a field of characteristic zero. This complex is built from logarithmic forms on all irreducible subvarieties with values in a locally free sheaf. We prove that cohomology groups of the polar complex are canonically isomorphic to the cohomology groups of the locally free sheaf. Relations of the polar complex with Rost's cycle modules, algebraic cycles, Cousin complex, and adelic complex are discussed. In particular, the polar complex is a subcomplex in the Cousin complex. One can say that the polar complex is a first order pole part of the Cousin complex, providing a much smaller, but, in fact, quasiisomorphic subcomplex.

We construct the so-called polar complex for an arbitrary locally free sheaf on a smooth variety over a field of characteristic zero. This complex is built from logarithmic forms on all irreducible subvarieties with values in a locally free sheaf. We prove that cohomology groups of the polar complex are canonically isomorphic to the cohomology groups of the locally free sheaf. Relations of the polar complex with Rost's cycle modules, algebraic cycles, Cousin complex, and adelic complex are discussed. In particular, the polar complex is a subcomplex in the Cousin complex. One can say that the polar complex is a first-order pole part of the Cousin complex, providing a much smaller, but, in fact, quasiisomorphic subcomplex.

Polar homology and linkings arise as natural holomorphic analogues in algebraic geometry of the homology groups and links in topology. For complex projective manifolds, the polar k -chains are subvarieties of complex dimension k with meromorphic forms on them, while the boundary operator is defined by taking the polar divisor and the Poincaré residue on it. We also define the corresponding analogues for the intersection and linking numbers of complex submanifolds, and show that they have properties similar to those of the corresponding topological objects. Finally, we establish the relation between the holomorphic linking and the Weil pairing of functions on a complex curve and its higher-dimensional counterparts.