## Journal of Symplectic Geometry

- J. Symplectic Geom.
- Volume 2, Number 1 (2003), 133-175.

### Moment Maps and Equivariant Szegö Kernels

#### Abstract

Let *M* be a connected *n*-dimensional complex projective manifold and
consider an Hermitian ample holomorphic line bundle *(L; h*_{L}*)* 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 *h*_{L} 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 *H*^{0}*(M;L*^{⊗k}*)* of global holomorphic sections of
*L*^{⊗k}. This representation
is unitary with respect to the natural Hermitian structure of
*H*^{0}*(M;L*^{⊗k}*)* (associated to Ω
and *h*_{L} in the standard manner). We may
thus decompose *H*^{0}*(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
*H*^{0}*(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.

#### Article information

**Source**

J. Symplectic Geom., Volume 2, Number 1 (2003), 133-175.

**Dates**

First available in Project Euclid: 30 June 2004

**Permanent link to this document**

https://projecteuclid.org/euclid.jsg/1088600550

**Mathematical Reviews number (MathSciNet)**

MR2128391

**Zentralblatt MATH identifier**

1063.53084

#### Citation

Paoletti, Roberto. Moment Maps and Equivariant Szegö Kernels. J. Symplectic Geom. 2 (2003), no. 1, 133--175. https://projecteuclid.org/euclid.jsg/1088600550