TY - JOUR

T1 - Off-Diagonal Decay of Toric Bergman Kernels

AU - Zelditch, Steve

N1 - Funding Information:
Research partially supported by NSF Grant DMS-1541126.
Publisher Copyright:
© 2016, Springer Science+Business Media Dordrecht.

PY - 2016/12/1

Y1 - 2016/12/1

N2 - 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.

AB - 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.

KW - Bergman kernel

KW - line bundle

KW - toric Kaehler manifold

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U2 - 10.1007/s11005-016-0888-9

DO - 10.1007/s11005-016-0888-9

M3 - Article

AN - SCOPUS:84991669509

SN - 0377-9017

VL - 106

SP - 1849

EP - 1864

JO - Letters in Mathematical Physics

JF - Letters in Mathematical Physics

IS - 12

ER -