Off-Diagonal Decay of Toric Bergman Kernels

We study the off-diagonal decay of Bergman kernels Πhk(z,w) and Berezin kernels Phk(z,w) for ample invariant line bundles over compact toric projective kähler manifolds of dimension m. When the metric is real analytic, Phk(z,w)≃kmexp-kD(z,w) where D(z, w) is the diastasis. When the metric is only C^∞ 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^∞ metrics, Phk(z,w)≃kmexp-kD(z,w) as long as w lies on the R+m-orbit of z, and for general (z, w) , limsupk→∞1klogPhk(z,w)≤-D(z∗,w∗) where D(z, w^∗) is the diastasis between z and the translate of w by (S1)m to the R+m orbit of z. These results are complementary to Mike Christ’s negative results showing that Phk(z,w) does not have off-diagonal exponential decay at “speed” k if (z, w) lies on the same (S1)m-orbit.