Florian T. Pokorny
Assistant Professor, School of Computer Science and Communication, KTH Royal Institute of TechnologyThe Bergman Kernel on Toric Kähler Manifolds
PhD thesis, The University of Edinburgh, 2011
Abstract
Let $(L,h)\to (X, \omega)$ be a compact toric polarized Kähler manifold of
complex dimension $n$. For each $k\in \mathbb{N}$, the fibre-wise Hermitian metric
$h^k$ on $L^k$ induces a natural inner product on the vector space $\mathcal{C}^{\infty}(X,
L^k)$ of smooth global sections of $L^k$ by integration with respect to the
volume form $\frac{\omega^n}{n!}$. The orthogonal projection
$P_k:\mathcal{C}^{\infty}(X, L^k)\to H^0(X, L^k)$ onto the space $H^0(X, L^k)$ of global
holomorphic sections of $L^k$ is represented by an integral kernel $B_k$ which
is called the Bergman kernel (with parameter $k\in \mathbb{N}$). The restriction $\rho_k:X\to
\mathbb{R}$ of the norm of $B_k$ to the diagonal in $X\times X$ is called the density
function of $B_k$.
On a dense subset of $X$, we describe a method for computing the coefficients of
the asymptotic expansion of $\rho_k$ as $k\to \infty$ in this toric
setting. We also provide a direct proof of a result which illuminates the
off-diagonal decay behaviour of toric Bergman kernels.
We fix a parameter $l\in \mathbb{N}$ and consider the projection $P_{l,k}$ from
$\mathcal{C}^{\infty}(X, L^k)$ onto those global holomorphic sections of $L^k$ that
vanish to order at least $lk$ along some toric submanifold of $X$. There exists an
associated toric partial Bergman kernel $B_{l, k}$ giving rise to a toric partial density function
$\rho_{l, k}:X\to \mathbb{R}$.
For such toric partial density functions,
we determine new asymptotic expansions over
certain subsets of $X$ as $k\to \infty$. Euler-Maclaurin sums and Laplace's method are utilized as important
tools for this. We discuss the case of a polarization of $\mathbb{CP}^n$ in detail
and also investigate the non-compact Bargmann-Fock model with imposed vanishing
at the origin.
We then discuss the relationship between the slope inequality and the asymptotics
of Bergman kernels with vanishing and study how a version of Song and Zelditch's
toric localization of sums result generalizes to arbitrary polarized Kähler
manifolds.
Finally, we construct families of induced metrics on blow-ups of
polarized Kähler manifolds. We relate those metrics to partial density
functions and study their properties for a specific blow-up of $\mathbb{C}^n$ and $\mathbb{CP}^n$
in more detail.
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Download this publicationBibtex
@phdthesis{pokorny2011a,
title = {The Bergman Kernel on Toric K{\"a}hler Manifolds},
author = {Pokorny, Florian T.},
year = {2011},
school = {The University of Edinburgh},
doi = {1842/5301},
url = {http://www.csc.kth.se/~fpokorny/static/publications/FTPokornyPhD.pdf},
}