# Florian T. Pokorny

Senior Researcher, KTH Royal Institute of Technology# The 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.

## Files

Download this publication## Bibtex

```
@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},
}
```