A remark on Weighted Bergman kernels on orbifolds

Abstract

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 Kahler 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 Kahler 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 Kahler orbifolds. We will use freely the notation in (2, Section 5.4). We assume also the auxiliary vector bundle E therein is C.