Off-diagonal Asymptotic Properties of Bergman Kernels Associated to Analytic Kähler Potentials

Abstract

We prove a new off-diagonal asymptotic of the Bergman kernels associated to tensor powers of a positive line bundle on a compact Kähler manifold. We show that if the Kähler potential is real analytic, then the Bergman kernel accepts a complete asymptotic expansion in a neighborhood of the diagonal of shrinking size k − 1 4 . These improve the earlier results in the subject for smooth potentials, where an expansion exists in a k − 1 2 neighborhood of the diagonal. We obtain our results by finding upper bounds of the form C m m! 2 for the Bergman coefficients bm(x,ȳ), which is an interesting problem on its own. We find such upper bounds using the method of [BeBeSj08]. We also show that sharpening these upper bounds would improve the rate of shrinking neighborhoods of the diagonal x = y in our results. In the special case of metrics with local constant holomorphic sectional curvatures, we obtain off-diagonal asymptotic in a fixed (as k → ∞) neighborhood of the diagonal, which recovers a result of Berman [Ber08] (see Remark 3.5 of [Ber08] for higher dimensions). In this case, we also find an explicit formula for the Bergman kernel mod O(e −kδ ).Near the diagonal, i.e. in a 1 √ k -neighborhood of the diagonal, one has a scaling asymptotic expansion for the Bergman kernel (see [ShZe02,MaMa07,MaMa13,LuSh15,HeKeSeXu16]). For d(x, y) ≫ log k k , where d is the Riemannian distance induced by ω, no useful asymptotics are known. However, there are off-diagonal upper bounds of Agmon type (1.2) |K k (x, y)| h k ≤ Ck n e −c √ kd(x,y) ,

Asymptotic properties of Bergman kernels for potentials with Gevrey regularity

“…This article generalizes the results in [HeLuXu18] to the setting of Gevrey classes. To be precise, we prove an asymptotic expansion in a k − 1 2 + 1 4a+4ε neighborhood of the diagonal for any ε > 0 if the metric h is in the Gevrey a (a > 1) class.…”

“…it is less than the principal term k n e kψ (x,ȳ) in size, which holds only in a neighborhodd d(x, y) ≤ C log k k in general. In the case that h is real analytic, this is valid in a larger neighborhood d(x, y) ≤ k −1/4 [HeLuXu18]. In a recent preprint [RoSjNg18], this is further improved to a fixed neighborhood independent of k.…”

On a property of Bergman kernels when the Kähler potential is analytic

“…The proofs follow from Theorem 1.3 in a non-trivial way. However these conclusions have already been carried out in [HeLuXu18] (Remark 1.7).…”

“…Our proof is based on a linear recursive formula of L. Charles (See [Cha00] and also equation (20) of [Cha03]) for b m . In our work with Z. Lu [HeLuXu18], we used a linear recursive formula of [BeBeSj08] and only obtained the bounds of size m! 2 for the Bergman coefficients.…”