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We investigate the frame set of regular multivariate Gaussian Gabor frames using methods from Kahler geometry such as Hormanders $dbar$-method, the Ohsawa--Takegoshi extension theorem and a Kahler-variant 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. Our results on Gaussian Gabor frames are in terms of the Sehsadri constant and the generalized Buser-Sarnak invariant of the associated symplectic dual lattice. The theory of Hormander estimates and the Ohsawa--Takegoshi extension theorem allow us to give estimates for the frame bounds in terms of the Buser-Sarnack invariant and in the one-dimensional case these bounds are sharp thanks to Faltings work on Green functions in Arakelov theory.
We show that a general $n$-dimensional polarized abelian variety $(A,L)$ of a given polarization type and satisfying $ h^0(A, L) geq dfrac{8^n}{2} cdot dfrac{n^n}{n !}$ is projectively normal. In the process, we also obtain a sharp lower bound for th
Using complex methods combined with Baires Theorem we show that one-sided extendability, extendability and real analyticity are rare phenomena on various spaces of functions in the topological sense. These considerations led us to introduce the p-con
We introduce Seshadri constants for line bundles in a relative setting. They generalize the classical Seshadri constants of line bundles on projective varieties and their extension to vector bundles studied by Beltrametti-Schneider-Sommese and Hacon.
We develop a local positivity theory for movable curves on projective varieties similar to the classical Seshadri constants of nef divisors. We give analogues of the Seshadri ampleness criterion, of a characterization of the augmented base locus of a
We show that for an entire function $varphi$ belonging to the Fock space ${mathscr F}^2(mathbb{C}^n)$ on the complex Euclidean space $mathbb{C}^n$, the integral operator begin{eqnarray*} S_{varphi}F(z)=int_{mathbb{C}^n} F(w) e^{z cdotbar{w}} varphi(z