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Register a new userBuser-Sarnak invariant and projective normality of abelian varieties
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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 the volume of a purely one-dimensional complex analytic subvariety in a geodesic tubular neighborhood of a subtorus of a compact complex torus.
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In this paper, we will give a precise formula to compute delta invariants of projective bundles and projective cones of Fano type.
We generalise Flo{}ystads theorem on the existence of monads on the projective space to a larger set of projective varieties. We consider a variety $X$, a line bundle $L$ on $X$, and a base-point-free linear system of sections of $L$ giving a morphism to the projective space whose image is either arithmetically Cohen-Macaulay (ACM), or linearly normal and not contained in a quadric. We give necessary and sufficient conditions on integers $a$, $b$, and $c$ for a monad of type [ 0to(L^vee)^atomathcal{O}_{X}^{,b}to L^cto0 ] to exist. We show that under certain conditions there exists a monad whose cohomology sheaf is simple. We furthermore characterise low-rank vector bundles that are the cohomology sheaf of some monad as above. Finally, we obtain an irreducible family of monads over the projective space and make a description on how the same method could be used on an ACM smooth projective variety $X$. We establish the existence of a coarse moduli space of low-rank vector bundles over an odd-dimensional $X$ and show that in one case this moduli space is irreducible.
Euler-symmetric projective varieties are nondegenerate projective varieties admitting many C*-actions of Euler type. They are quasi-homogeneous and uniquely determined by their fundamental forms at a general point. We show that Euler-symmetric projective varieties can be classified by symbol systems, a class of algebraic objects modeled on the systems of fundamental forms at general points of projective varieties. We study relations between the algebraic properties of symbol systems and the geometric properties of Euler-symmetric projective varieties. We describe also the relation between Euler-symmetric projective varieties of dimension n and equivariant compactifications of the vector group G_a^n.
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 describe the birational and the biregular theory of cyclic and Abelian coverings between real varieties.
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