We investigate the frame set of regular multivariate Gaussian Gabor frames using methods from Kähler geometry such as Hörmander’s \({\overline{\partial }}\)-\(L^2\) estimate with singular weight, Demailly’s Calabi–Yau method for Kähler currents and a Kähler-variant generalization of the symplectic embedding theorem of McDuff–Polterovich for ellipsoids. Our approach is based on the well-known link between sets of interpolation for the Bargmann-Fock space and the frame set of multivariate Gaussian Gabor frames. We state sufficient conditions in terms of a certain extremal type Seshadri constant of the complex torus associated to a lattice to be a set of interpolation for the Bargmann-Fock space, and give also a condition in terms of the generalized Buser-Sarnak invariant of the lattice. In particular, we obtain an effective Gaussian Gabor frame criterion in terms of the covolume for almost all lattices, which is the first general covolume criterion in multivariate Gaussian Gabor frame theory. The recent Berndtsson–Lempert method and the Ohsawa–Takegoshi extension theorem also allow us to give explicit estimates for the frame bounds in terms of certain Robin constant. In the one-dimensional case we obtain a sharp estimate of the Robin constant using Faltings’ theta metric formula for the Arakelov Green functions.

我们使用 Kähler 几何中的方法研究规则多元高斯 Gabor 框架的框架集，例如 Hörmander 的\({\overline{\partial }}\) - \(L^2\)奇异权重估计、Demailly 的 Calabi-Yau 方法凯勒电流和椭球体 McDuff-Polterovich 辛嵌入定理的凯勒变体推广。我们的方法基于 Bargmann-Fock 空间的插值集与多元高斯 Gabor 框架的框架集之间众所周知的联系。我们根据与格子相关的复环面的某个极值类型 Seshadri 常数来陈述作为 Bargmann-Fock 空间的一组插值的充分条件，并且还根据格子。特别是，我们获得了几乎所有格的余体积方面的有效高斯Gabor框架准则，这是多元高斯Gabor框架理论中的第一个通用余体积准则。最近的 Berndtsson-Lempert 方法和 Ohsawa-Takegoshi 扩展定理也允许我们根据某些 Robin 常数给出框架边界的明确估计。在一维情况下，我们使用 Arakelov Green 函数的法尔廷斯 theta 度量公式获得了 Robin 常数的精确估计。