Toric partial density functions and stability of toric varieties

Florian T. Pokorny, Michael Singer
In Mathematische Annalen, 2013


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 = {}, urlpreprint = {} }

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