Kähler packings of projective complex manifolds

We show that the multipoint Seshadri constant determines the maximum possible radii of embeddings of Kähler balls and vice versa.

Simple cyclic covers of the plane and the Seshadri constants of some general hypersurfaces in weighted projective space

Let $$X \subset {\mathbb P}(1,1,1,m)$$ X ⊂ P ( 1 , 1 , 1 , m ) be a general hypersurface of degree md for some for $$d\ge 2$$ d ≥ 2 and $$m\ge 3$$ m ≥ 3 . We prove that the Seshadri constant $$\varepsilon ( {\mathcal O}_X(1), x)$$ ε ( O X ( 1 ) , x ) at a general point $$x\in X$$ x ∈ X lies in the interval $$\left[ \sqrt{d}- \frac{d}{m}, \sqrt{d}\right] $$ d - d m , d and thus approaches the possibly irrational number $$\sqrt{d}$$ d as m grows. The main step is a detailed study of the case where X is a simple cyclic cover of the plane.

Positivity of vector bundles on homogeneous varieties

We study the following question: Given a vector bundle on a projective variety [Formula: see text] such that the restriction of [Formula: see text] to every closed curve [Formula: see text] is ample, under what conditions [Formula: see text] is ample? We first consider the case of an abelian variety [Formula: see text]. If [Formula: see text] is a line bundle on [Formula: see text], then we answer the question in the affirmative. When [Formula: see text] is of higher rank, we show that the answer is affirmative under some conditions on [Formula: see text]. We then study the case of [Formula: see text], where [Formula: see text] is a reductive complex affine algebraic group, and [Formula: see text] is a parabolic subgroup of [Formula: see text]. In this case, we show that the answer to our question is affirmative if [Formula: see text] is [Formula: see text]-equivariant, where [Formula: see text] is a fixed maximal torus. Finally, we compute the Seshadri constant for such vector bundles defined on [Formula: see text].

Applications of the Ohsawa–Takegoshi Extension Theorem to Direct Image Problems

In the 1st part of the paper, we study a Fujita-type conjecture by Popa and Schnell and obtain an effective bound on the generic global generation of direct images of twisted pluricanonical bundles. We also point out the relation between the Seshadri constant and the optimal bound. In the 2nd part, we give an affirmative answer to a question by Demailly, Peternell, and Schneider in a more general setting. As byproducts of our proofs, we extend a result by Fujino and Gongyo on images of weak Fano manifolds to the Kawamata log terminal settings and refine a theorem by Broustet and Pacienza on the rational connectedness of the image.

Some remarks on the Gromov width of homogeneous Hodge manifolds

We provide an upper bound for the Gromov width of compact homogeneous Hodge manifolds (M, ω) with b2(M) = 1. As an application we obtain an upper bound on the Seshadri constant ϵ(L) where L is the ample line bundle on M such that [Formula: see text].

Symplectic Capacities of Toric Manifolds and Related Results

In this paper we give concrete estimations for the pseudo symplectic capacities of toric manifolds in combinatorial data. Some examples are given to show that our estimates can compute their pseudo symplectic capacities. As applications we also estimate the symplectic capacities of the polygon spaces. Other related results are impacts of symplectic blow-up on symplectic capacities, symplectic packings in symplectic toric manifolds, the Seshadri constant of an ample line bundle on toric manifolds, and symplectic capacities of symplectic manifolds withS1-action.