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Erschienen in:
2011 | OriginalPaper | Buchkapitel
Algebraic Homogeneous Spaces
verfasst von : Dmitry A. Timashev
Erschienen in: Homogeneous Spaces and Equivariant Embeddings
Verlag: Springer Berlin Heidelberg
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In this chapter,
G
denotes an
arbitrary
linear algebraic group (not supposed to be either connected nor reductive), and
H
⊆
G
a closed subgroup. We begin in §1 with the definition of an algebraic homogeneous space
G
/
H
as a geometric quotient, and prove its quasiprojectivity. We also prove some elementary facts on tangent vectors and
G
-equivariant automorphisms of
G
/
H
. In §2, we describe the structure of
G
-fibrations over
G
/
H
and compute Pic(
G
/
H
). Some related representation theory is discussed there: induction, multiplicities, the structure of
$\Bbbk[G]$
. Basic classes of homogeneous spaces are considered in §3. We prove that
G
/
H
is projective if and only if
H
is parabolic, and consider criteria of affinity of
G
/
H
. Quasiaffine
G
/
H
correspond to observable
H
, which may be defined by several equivalent conditions (see Theorem 3.12).