Let M be a connected n-dimensional complex projective manifold and
consider an Hermitian ample holomorphic line bundle (L; hL) on M. Suppose
that the unique compatible covariant derivative ▽L on L has curvature
-2πiΩ where Ω
is a Kähler form. Let G be a compact connected Lie group
and μ: G x M → M a holomorphic Hamiltonian action on (M; Ω
). Let \frac g
be the Lie algebra of G, and denote by Φ : M → g* the moment map.
¶
Let us also assume that the action of G on M linearizes to a holomorphic
action on L; given that the action is Hamiltonian, the obstruction for this is
of topological nature [GS1]. We may then also assume that the Hermitian
structure hL of L, and consequently the connection as well, are G-invariant.
Therefore for every k ∈ N there is an induced linear representation of G
on the space H0(M;L⊗k) of global holomorphic sections of
L⊗k. This representation
is unitary with respect to the natural Hermitian structure of
H0(M;L⊗k) (associated to Ω
and hL in the standard manner). We may
thus decompose H0(M;L⊗k)
equivariantly according to the irreducible representations
of G.
¶ The subject of this paper is the local and global asymptotic behaviour of
certain linear series defined in terms this decomposition. Namely, we shall
first consider the asymptotic behaviour as k →+ ∞ of the linear subseries of
H0(M;L⊗k)
associated to a single irreducible representation, and then of the
linear subseries associated to a whole ladder of irreducible representations.
To this end, we shall estimate the asymptoptic growth, in an appropriate
local sense, of these linear series on some loci in M defined in terms of the moment map Φ.