A polar complex for locally free sheaves

TL;DR

This paper introduces the polar complex for locally free sheaves on smooth varieties, establishing its cohomological equivalence and exploring its relations with other complexes and cycle theories.

Contribution

It constructs the polar complex for arbitrary locally free sheaves and proves its cohomology is canonically isomorphic to sheaf cohomology, linking it with existing cycle and Cousin complexes.

Findings

Polar complex cohomology matches sheaf cohomology.

Polar complex is a subcomplex of the Cousin complex.

Provides a smaller, quasi-isomorphic complex for sheaf cohomology.

Abstract

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.