Extended equivariant Picard complexes and homogeneous spaces

TL;DR

This paper extends the theory of equivariant Picard complexes to homogeneous spaces, providing explicit computations of UPic(X) in terms of character groups, advancing the understanding of algebraic geometry over fields of characteristic zero.

Contribution

It introduces the extended equivariant Picard complex for homogeneous spaces and computes its class in the derived category using character groups of G and H.

Findings

Computed UPic(X) for homogeneous spaces in terms of character groups

Extended the equivariant Picard complex to new classes of varieties

Provided explicit isomorphism classes in the derived category

Abstract

Let k be a field of characteristic 0 and let X be a smooth geometrically integral k-variety. In our previous paper we defined the extended Picard complex UPic(X) as a certain complex of Galois modules in degrees 0 and 1. We computed the isomorphism class of UPic(G) in the derived category of Galois modules for a connected linear k-group G. In this paper we assume that X is a homogeneous space of a connected linear k-group G with geometric stabilizer H. We compute the isomorphism class of UPic(X) in the derived category of Galois modules in terms of the character groups of G and H. The proof is based on the notion of the extended equivariant Picard complex UPic_G(X) of a G-variety X.