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We investigate the frame set of regular multivariate Gaussian Gabor framesusing methods from K\"ahler geometry such as H\"ormander's $\dbar$-$L^2$estimate with singular weight, Demailly's Calabi--Yau method for K\"ahlercurrents and a K\"ahler-variant generalization of the symplectic embeddingtheorem of McDuff--Polterovich for ellipsoids. Our approach is based on thewell-known link between sets of interpolation for the Bargmann-Fock space andthe frame set of multivariate Gaussian Gabor frames. We state sufficientconditions in terms of a certain extremal type Seshadri constant of the complextorus associated to a lattice to be a set of interpolation for theBargmann-Fock space, and give also a condition in terms of the generalizedBuser-Sarnak invariant of the lattice. In particular, we obtain an effectiveGaussian Gabor frame criterion in terms of the covolume for almost alllattices, which is the first general covolume criterion in multivariateGaussian Gabor frame theory. The recent Berndtsson--Lempert method and theOhsawa--Takegoshi extension theorem also allow us to give explicit estimatesfor the frame bounds in terms of certain Robin constant. In the one-dimensionalcase we obtain a sharp estimate of the Robin constant using Faltings' thetametric formula for the Arakelov Green functions.

Gaussian Gabor frames, Seshadri constants and generalized Buser--Sarnak invariants