Scaling limits for equivariant Szego kernels

Suppose that the compact and connected Lie group G acts holomorphically on the irreducible complex projective manifold M, and that the action linearizes to the Hermitian ample line bundle L on M. Assume that 0 is a regular value of the associated moment map. The spaces of global holomorphic sections of powers of L may be decomposed over the finite dimensional irreducible representations of G. In this paper, we study how the holomorphic sections in each equivariant piece asymptotically concentrate along the zero locus of the moment map. In the special case where G acts freely on the zero locus of the moment map, this relates the scaling limits of the Szego kernel of the quotient to the scaling limits of the invariant part of the Szego kernel of (M,L).Comment: exposition improved, examples and references adde

The Szego kernel of a symplectic quotient

Suppose a compact Lie group acts on a polarized complex projective manifold (M,L). Under favorable circumstances, the Hilbert-Mumford quotient for the action of the complexified group may be described as a symplectic quotient (or reduction). This paper addresses some metric questions arising from this identification, by analyzing the relationship between the Szego kernel of the pair (M,L) and that of the quotient.Comment: Typos corrected, some improvements in expositio

On the surjectivity of Wahl maps on a general curve

This paper explores the geometric meaning of the failure of certain kinds of Wahl maps to surject on a general curve. Sufficient conditions for surjectivity are given. An approach used by Voisin to study canonical Wahl maps is applied in this direction.Comment: 18 pages amste

Lower order asymptotics for Szeg\"{o} and Toeplitz kernels under Hamiltonian circle actions

We consider a natural variant of Berezin-Toeplitz quantization of compact K\"{a}hler manifolds, in the presence of a Hamiltonian circle action lifting to the quantizing line bundle. Assuming that the moment map is positive, we study the diagonal asymptotics of the associated Szeg\"{o} and Toeplitz operators, and specifically their relation to the moment map and to the geometry of a certain symplectic quotient. When the underlying action is trivial and the moment map is taken to be identically equal to one, this scheme coincides with the usual Berezin-Toeplitz quantization. This continues previous work on near-diagonal scaling asymptotics of equivariant Szeg\"{o} kernels in the presence of Hamiltonian torus actions.Comment: Reference added, minor introductory change

Szego kernels, Toeplitz operators, and equivariant fixed point formulae

Let $\gamma$ be an automorphism of a polarized complex projective manifold $(M,L)$. Then $\gamma$ induces an automorphism $\gamma_k$ of the space of global holomorphic sections of the $k$-th tensor power of $L$, for every $k=1,2,...$; for $k\gg 0$, the Lefschetz fixed point formula expresses the trace of $\gamma_k$ in terms of fixed point data. More generally, one may consider the composition of $\gamma_k$ with the Toeplitz operator associated to some smooth function on $M$. Still more generally, in the presence of the compatible action of a compact and connected Lie group preserving $(M,L,\gamma)$, one may consider induced linear maps on the equivariant summands associated to the irreducible representations of $G$. In this paper, under familiar assumptions in the theory of symplectic reductions, we show that the traces of these maps admit an asymptotic expansion as $k\to +\infty$, and compute its leading term.Comment: statement and proof simplified, exposition improved, references adde

Free pencils on divisors

Let X be a smooth projective variety defined over an algebraically closed field, and let Y in X be a reduced and irreducible ample divisor in X. We give a numerical sufficient condition for a base point free pencil on $Y$ to be the restriction of a base point free pencil on $X$. This result is then extended to families of pencils and to morphisms to arbitrary smooth curves. Serrano had already studied this problem in the case n=2 and 3, and Reider had then attacked it in the case $n=2$ using vector bundle methods based on Bogomolov's instability theorem on a surface (char(k)=0). The argument given here is based on Bogomolov's theorem on an n-dimensional variety, and on its recent adaptations to the setting of prime charachterstic (due to Shepherd-Barron and Moriwaki).Comment: 18 pages, amslate

Seshadri constants, gonality of space curves and restriction of stable bundles

We define the Seshadri constant of a space curve and consider ways to estimate it. We then show that it governs the gonality of the curve. We use an argument based on Bogomolov's instability theorem on a threefold. The same methods are then applied to the study of the behaviour of a stable vector bundle on P^3 under restriction to curves and surfaces.Comment: 36 pages, amslate

Local scaling asymptotics in phase space and time in Berezin-Toeplitz quantization

This paper deals with the local semiclassical asymptotics of a quantum evolution operator in the Berezin-Toeplitz scheme, when both time and phase space variables are subject to appropriate scalings in the neighborhood of the graph of the underlying classical dynamics. Global consequences are then drawn regarding the scaling asymptotics of the trace of the quantum evolution as a function of time

A local Szeg\"o-type theorem in Toeplitz quantization

A Szeg\"o-type theorem for Toeplitz operators was proved by Boutet de Monvel and Guillemin for general Toeplitz structures. We give a local version of this result in the setting of positive line bundles on compact symplectic manifolds

Asymptotics of Szeg\"{o} kernels under Hamiltonian torus actions

Let $X$ be the circle bundle associated to a positive line bundle on a complex projective (or, more generally, compact symplectic) manifold. The Tian-Zelditch expansion on $X$ may be seen as a local manifestation of the decomposition of the (generalized) Hardy space $H(X)$ into isotypes for the $S^1$-action. More generally, given a compatible action of a compact Lie group, and under general assumptions guaranteeing finite dimensionality of isotypes, we may look for asymptotic expansions locally reflecting the equivariant decomposition of $H(X)$ over the irreducible representations of the group. We focus here on the case of compact tori.Comment: exposition improve