TY - JOUR

T1 - Toric partial density functions and stability of toric varieties

AU - Pokorny, Florian T.

AU - Singer, Michael

PY - 2014/4

Y1 - 2014/4

N2 - Let $(L, h)\to (X, \omega)$ denote a polarized toric K\"ahler manifold. Fix a toric submanifold $Y$ and denote by $\hat{\rho}_{lk}:X\to \mathbb{R}$ 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 $lk$ along $Y$, for fixed $l>0$ such that $lk\in \mathbb{N}$. We prove the existence of a distributional expansion of $\hat{\rho}_{lk}$ up to order $k^{n-2}$ 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 $Y$. More generally, it is shown that under the same hypotheses, $(X,L)$ must be slope semi-stable with respect to any closed toric subscheme $Z$ of $X$. In many cases, moreover, $(X,L)$ will be slope stable with respect to $Z$.

AB - Let $(L, h)\to (X, \omega)$ denote a polarized toric K\"ahler manifold. Fix a toric submanifold $Y$ and denote by $\hat{\rho}_{lk}:X\to \mathbb{R}$ 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 $lk$ along $Y$, for fixed $l>0$ such that $lk\in \mathbb{N}$. We prove the existence of a distributional expansion of $\hat{\rho}_{lk}$ up to order $k^{n-2}$ 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 $Y$. More generally, it is shown that under the same hypotheses, $(X,L)$ must be slope semi-stable with respect to any closed toric subscheme $Z$ of $X$. In many cases, moreover, $(X,L)$ will be slope stable with respect to $Z$.

U2 - 10.1007/s00208-013-0978-2

DO - 10.1007/s00208-013-0978-2

M3 - Article

VL - 358

SP - 879

EP - 923

JO - Mathematische annalen

JF - Mathematische annalen

SN - 0025-5831

IS - 3-4

ER -