Off-Diagonal Decay of Toric Bergman Kernels

Abstract

We study the off-diagonal decay of Bergman kernels ${\Pi_{h^k}(z,w)}$ and Berezin kernels ${P_{h^k}(z,w)}$ for ample invariant line bundles over compact toric projective kähler manifolds of dimension m. When the metric is real analytic, ${P_{h^k}(z,w) \simeq k^m {\rm exp} - k D(z,w)}$ where ${D(z,w)}$ is the diastasis. When the metric is only ${C^{\infty}}$ this asymptotic cannot hold for all ${(z,w)}$ since the diastasis is not even defined for all ${(z,w)}$ close to the diagonal. Our main result is that for general toric ${C^{\infty}}$ metrics, ${P_{h^k}(z,w) \simeq k^m {\rm exp} - k D(z,w)}$ as long as w lies on the ${\mathbb{R}_+^m}$-orbit of z, and for general ${(z,w)}$, ${{\rm lim\,sup}_{k \to \infty} \frac{1}{k} {\rm log} P_{h^k}(z,w) \,\leq\, - D(z^*,w^*)}$ where ${D(z, w^*)}$ is the diastasis between z and the translate of w by ${(S^1)^m}$ to the ${\mathbb{R}_+^m}$ orbit of z. These results are complementary to Mike Christ’s negative results showing that ${P_{h^k}(z,w)}$ does not have off-diagonal exponential decay at “speed” k if ${(z,w)}$ lies on the same ${(S^1)^m}$-orbit.