Florian T. Pokorny
Assistant Professor, School of Computer Science and Communication, KTH Royal Institute of TechnologyToric partial density functions and stability of toric varieties
In Mathematische Annalen, 2013
Abstract
Let $(L, h)\to (X, \omega)$ denote a polarized toric Kähler manifold. Fix a toric submanifold $Y$ and denote by
$\hat{\rho}_{tk}:X\to \mathbb{R}$ the partial density function corresponding to the partial Bergman kernel projecting
smooth sections of $L^k$ onto holomorphic sections of $L^k$ that vanish to order at least $tk$ along $Y$, for fixed
$t>0$ such that $tk\in \mathbb{N}$. We prove the existence of a distributional expansion of $\hat{\rho}_{tk}$ as $k\to
\infty$, including the identification of the coefficient of $k^{n-1}$ as a distribution on $X$. This expansion is used
to give a direct proof that if $\omega$ has constant scalar curvature, then $(X, L)$ must be slope semi-stable with
respect to $Y$. Similar results are also obtained for more general partial density functions. These results have
analogous applications to the study of toric K-stability of toric varieties.
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@article{pokorny2013d,
title={Toric partial density functions and stability of toric varieties},
author = {Pokorny, Florian T. and Singer, Michael},
journal={Mathematische Annalen},
year = {2013},
publisher={Springer Berlin Heidelberg},
issn={0025-5831},
doi={10.1007/s00208-013-0978-2},
keywords={32Q15; 32A25; 14M25; 53D20; 53C21; 58Jxx},
url = {http://dx.doi.org/10.1007/s00208-013-0978-2},
urlpreprint = {http://arxiv.org/abs/1111.5259v3}
}