A Remark on Weighted Bergman Kernels on Orbifolds

In this note, we explain that Ross–Thomas’ result [4, Theorem 1.7] on the weighted Bergman kernels on orbifolds can be directly deduced from our previous result [1]. This result plays an important role in the companion paper [5] to prove an orbifold version of Donaldson theorem. In two very interesting papers [4, 5], Ross–Thomas describe a notion of ampleness for line bundles on Kähler orbifolds with cyclic quotient singularities which is related to embeddings in weighted projective spaces. They then apply [4, Theorem 1.7] to prove an orbifold version of Donaldson theorem [5]. Namely, the existence of an orbifold Kähler metric with constant scalar curvature implies certain stability condition for the orbifold. In these papers, the result [4, Theorem 1.7] on the asymptotic expansion of Bergman kernels plays a crucial role. In this note, we explain how to directly derive Ross–Thomas’ result [4, Theorem 1.7] from Dai–Liu–Ma [1, (5.25)], provided Ross–Thomas condition [4, (1.8)] on ci holds. Since in [1, Section 5], we state our results for general symplectic orbifolds, in what follows, we will just use the version from [2, Theorem 5.4.11], where Ma– Marinescu wrote them in detail for Kähler orbifolds. We will use freely the notation in [2, Section 5.4]. We assume also the auxiliary vector bundle E therein is C. Let (X, J, ω) be a compact n-dimensional Kähler orbifold with complex structure J , and with singular set Xsing. Let (L, h) be a holomorphic Hermitian proper orbifold line bundle on X. Let ∇L be the holomorphic Hermitian connections on (L, h) with curvature R = (∇L)2. We assume that (L, hL,∇L) is a prequantum line bundle, i.e., R = −2π√−1ω. (0.1) Let g = ω(·, J ·) be the Riemannian metric on X induced by ω. Let ∇TX be the Levi–Civita connection on (X, g). We denote by R = (∇TX)2 the curvature, by r the scalar curvature of ∇TX . For x ∈ X, set d(x,Xsing) := infy∈Xsing d(x, y) the distance from x to Xsing. For p ∈ N, the Bergman kernel Pp(x, x′) (x, x′ ∈ X) is the smooth kernel of the orthogonal projection from C∞(X,Lp) onto H(X,L), with respect to the Riemannian volume form dvX(x′). Theorem 0.1 ([1, Theorem 1.4], [2, Theorem 5.4.10]). There exist smooth coefficients br(x) ∈ C∞(X) which are polynomials in R and its derivatives with order 2r−2 Received by the editors August 23, 2011.