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# Journal of Symplectic Geometry

## Volume 2 (2004)

### Number 1

### Moment Maps and Equivariant Szegö Kernels

Pages: 133 – 175

DOI: http://dx.doi.org/10.4310/JSG.2004.v2.n1.a5

#### Author

#### 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.