Henrik Seppänen
Let $\mathscr{L} \rightarrow X$ be an ample line bundle over a nonsingular complex projective variety $X$. We construct an admissable flag $X_0 \subseteq X_1 \subseteq...\subseteq X_n=X$ of subvarieties for which the associated Okounkov body for $\mathscr{L}$ is a rational polytope.
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This is a survey article on Newton-Okounkov bodies in projective geometry focusing on the relationship between positivity of divisors and Newton-Okounkov bodies.
We study topological properties of functions on Okounkov bodies as introduced by Boucksom-Chen and Witt-Nyström. We note that they are continuous over the whole Okounkov body whenever the body is polyhedral, on the other hand, we exhibit an example that shows that continuity along the boundary does not hold in general.
The main goal of this article is to construct "arithmetic Okounkov bodies" for an arbitrary pseudo-effective (1,1)-class $α$ on a Kähler manifold. Firstly, using Boucksom's divisorial Zariski decompositions for pseudo-effective (1,1)-classes on compact Kähler manifolds, we prove the differentiability of volumes of big classes for Kähler manifolds on which modified nef cones and nef cones coincide; this includes Kähler surfaces. We then apply our differentiability results to prove Demailly's transcendental Morse inequality for these particular classes of Kähler manifolds. In the second part, we…
The purpose of the paper is to initiate the development of the theory of Newton Okounkov bodies of curve classes. Our denition is based on making a fundamental property of NewtonOkounkov bodies hold also in the curve case: the volume of the NewtonOkounkov body of a curve is a volume-type function of the original curve. This construction allows us to conjecture a new relation between NewtonOkounkov bodies, we prove it in certain cases.
Transcendental Okounkov bodies
本文研究了紧致Kähler流形上超越大(1,1)类的体积问题，通过凸体实现，并探讨了Okounkov体与toric退化之间的联系。
Flats in the space of Kähler metrics and Okounkov bodies
该研究依托Witt Nystrom定义的Okounkov体广义勒让德变换与Schwer–Lytchak成果，探究极化复射影流形上连续多重次调和度量空间平坦子空间存在的充分条件。
We present an explicit calculation of an Okounkov body associated to an algebraic variety. This is used to derive a formula for transfinite diameter on the variety. We relate this formula to a recent result of D. Witt Nystrom.
We associate to a test configuration of an ample line bundle a filtration of the section ring of the line bundle. Using the recent work of Boucksom-Chen we get a concave function on the Okounkov body whose law with respect to Lebesgue measure determines the asymptotic distribution of the weights of the test configuration. We show that this is a generalization of a well-known result in toric geometry. As an application, we prove that the pushforward of the Lebesgue measure on the Okounkov body is equal to a Duistermaat-Heckman measure of a certain deformation of the manifold. Via the Duisteraat…
Given a projective manifold $X$ equipped with an ample line bundle $L$, we show how to embed certain torus-invariant domains $D \subseteq\mathbb{C}^n$ into $X$ so that the Euclidean Kähler form on $D$ extends to a Kähler form on X lying in the first Chern class of $L$. This is done using Okounkov bodies $Δ(L)$, and the image of $D$ under the standard moment map will approximate $Δ(L)$. This means that the volume of $D$ can be made to approximate the Kähler volume of $X$ arbitrarily well. As a special case we can let $D$ be an ellipsoid. We also have similar results when $L$ is just big.
Partial Okounkov bodies and Duistermaat--Heckman measures of non-Archimedean metrics
本文构造光滑射影簇上 Hermite 伪有效线丛的偏 Okounkov 体，证明其为度量奇性的普适不变量，并据此推广 Boucksom–Chen 定理，建立非 Archimedees 有限能量度量的 Duistermaat–Heckman 测度。
Multipoint Okounkov bodies
本文提出了一种从射影流形上的大线丛及多点数据构造 N 个 Okounkov 体的新方法，用以编码体积、多点 Seshadri 常数及 Kähler 填充等几何特征，并详细考察了 Toric 流形与曲面。
We propose a new construction which associates to any ample (or big) line bundle $L$ on a projective manifold $X$ a canonical growth condition (i.e. a choice of a psh function well-defined up to a bounded term) on the tangent space $T_p X$ of any given point $p$. We prove that it encodes such classical invariants as the volume and the Seshadri constant. Even stronger, it allows you to recover all the infinitesimal Okounkov bodies of $L$ at $p$. The construction is inspired by toric geometry and the theory of Okounkov bodies; in the toric case the growth condition is "equivalent" to the moment …
We study asymptotic invariants of linear series on surfaces with the help of Newton-Okounkov polygons. Our primary aim is to understand local positivity of line bundles in terms of convex geometry. We work out characterizations of ample and nef line bundles in terms of their Newton-Okounkov bodies, treating the infinitesimal case as well. One of the main results is a description of moving Seshadri constants via infinitesimal Newton-Okounkov polygons. As an illustration of our ideas we reprove results of Ein-Lazarsfeld on Seshadri constants on surfaces.
Let $G$ be a complex simple Lie group, and let $U \subseteq G$ be a maximal compact subgroup. Assume that $G$ admits a homogenous space $X=G/Q=U/K$ which is a compact Hermitian symmetric space. Let $\mathscr{L} \rightarrow X$ be the ample line bundle which generates the Picard group of $X$. In this paper we study the restrictions to $K$ of the family $(H^0(X, \mathscr{L}^k))_{k \in \N}$ of irreducible $G$-representations. We describe explicitly the moment polytopes for the moment maps $X \rightarrow \fk^*$ associated to positive integer multiples of the Kostant-Kirillov symplectic form on $X$,…
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