  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.
