Off-diagonal decay of toric Bergman kernels

We study the off-diagonal decay of Bergman kernels ${\mathrm{\Pi }}_{h^k}(z,w)$ and Berezin kernels $P_{h^k}(z,w)$ for ample invariant line bundles over compact toric projective \kahler manifolds of dimension $m$. When the metric is real analytic, $P_{h^k}(z,w)\simeq k^m\exp -kD(z,w)$ where $D(z,w)$ is the diastasis. When the metric is only $C^{\mathrm{\infty }}$ this asymptotic cannot hold for all $(z,w)$ since the diastasis is not even defined for all $(z,w)$ close to the diagonal. We prove that for general $C^{\mathrm{\infty }}$ metrics, $P_{h^k}(z,w) \simeq k^m \exp - k D(z,w)$aslongas$w$liesonthe${\mathbb R}+^m$-orbit of $z$, and for general $(z,w)$, $\limsup{k \to \infty} \frac{1}{k} \log P_{h^k}(z,w) \leq - D(z^,w^)$where$D(z, w^*)$isthediastasisbetween$z$ and the translate of $w$ by $(S^1{)}^m$ to the ${\mathbb{R}}_{+}^m$ orbit of $z$, complementary to Mike Christ’s negative results (arXiv:1308.5644).