Okounkov Bodies Research Articles

Positivity of Line Bundles and Newton-Okounkov Bodies

The purpose of this paper is to describe asymptotic base loci of line bundles on projective varieties in terms of Newton-Okounkov bodies. As a result, we obtain equivalent characterizations of ampleness and nefness via convex geometry.

Envelopes of positive metrics with prescribed singularities

We investigate envelopes of positive metrics with a prescribed singularity type. First we generalise work of Berman to this setting, proving C 1,1 regularity of such envelopes, showing their Monge–Ampère measure is supported on a certain “equilibrium set” and connecting with the asymptotics of the partial Bergman functions coming from multiplier ideals. We investigate how these envelopes behave on certain products, and how they relate to the Legendre transform of a test curve of singularity types in the context of geodesic rays in the space of Kähler potentials. Finally we consider the associated exhaustion function of these equilibrium sets, connecting it both to the Legendre transform and to the geometry of the Okounkov body.

Newton–Okounkov bodies sprouting on the valuative tree

Given a smooth projective algebraic surface X, a point O in X and a big divisor D on X, we consider the set of all Newton-Okounkov bodies of D with respect to valuations of the field of rational functions of X centred at O, or, equivalently, with respect to a flag (E,p) which is infinitely near to O, in the sense that there is a sequence of blowups mapping the smooth, irreducible rational curve E to O. The main objective of this paper is to start a systematic study of the variation of these infinitesimal Newton-Okounkov bodies as (E, p) varies, focusing on the case X = P2.

On K-stability and the volume functions of ℚ-Fano varieties: Table 1.

We introduce a new effective stability named "divisorial stability" for Fano manifolds which is weaker than K-stability and is stronger than slope stability along divisors. We show that we can test divisorial stability via the volume function. As a corollary, we prove that the first coordinate of the barycenter of the Okounkov body of the anticanonical divisor is not bigger than one for any K\"ahler-Einstein Fano manifold. In particular, for toric Fano manifolds, the existence of K\"ahler-Einstein metrics is equivalent to divisorial semistability. Moreover, we find many non-K\"ahler-Einstein Fano manifolds of dimension three.

Local positivity in terms of Newton–Okounkov bodies

In recent years, the study of Newton–Okounkov bodies on normal varieties has become a central subject in the asymptotic theory of linear series, after its introduction by Lazarsfeld–Mustaţă and Kaveh–Khovanskii. One reason for this is that they encode all numerical equivalence information of divisor classes (by work of Jow). At the same time, they can be seen as local positivity invariants, and Küronya–Lozovanu have studied them in depth from this point of view.We determine what information is encoded by the set of all Newton–Okounkov bodies of a big divisor with respect to flags centered at a fixed point of a surface, by showing that it determines and is determined by the numerical equivalence class of the divisor up to negative components in the Zariski decomposition that do not go through the fixed point.

Newton-Okounkov bodies of Bott-Samelson varieties and Grossberg-Karshon twisted cubes

We describe, under certain conditions, the Newton-Okounkov body of a Bott-Samelson variety as a lattice polytope defined by an explicit list of inequalities. The valuation that we use to define the Newton-Okounkov body is different from that used previously in the literature. The polytope that arises is a special case of the Grossberg-Karshon twisted cubes studied by Grossberg and Karshon in connection to character formulae for irreducible $G$-representations and also studied previously by the authors in relation to certain toric varieties associated to Bott-Samelson varieties.

Okounkov bodies for ample line bundles with applications to multiplicities for group representations

Let $$\mathscr {L} \rightarrow X$$ be an ample line bundle over a complex normal projective variety X. We construct a flag $$X_0 \subseteq X_1 \subseteq \cdots \subseteq X_n=X$$ of subvarieties for which the associated Okounkov body for $$\mathscr {L}$$ is a rational simplex. In the case when X is a homogeneous surface, and the pseudoeffective cone of X is rational polyhedral, we also show that the global Okounkov body is a rational polyhedral cone for a generic choice of a flag of subvarieties. Finally, we provide an application to the asymptotic study of group representations.

FAVOURABLE MODULES: FILTRATIONS, POLYTOPES, NEWTON–OKOUNKOV BODIES AND FLAT DEGENERATIONS

We introduce the notion of a favourable module for a complex unipotent algebraic group, whose properties are governed by the combinatorics of an associated polytope. We describe two filtrations of the module, one given by the total degree on the PBW basis of the corresponding Lie algebra, the other by fixing a homogeneous monomial order on the PBW basis. In the favourable case a basis of the module is parametrized by the lattice points of a normal polytope. The filtrations induce at degenerations of the corresponding ag variety to its abelianized version and to a toric variety, the special fibres of the degenerations being projectively normal and arithmetically Cohen-Macaulay. The polytope itself can be recovered as a Newton-Okounkov body. We conclude the paper by giving classes of examples for favourable modules.

Seshadi constants via functions on Newton–Okounkov bodies

The aim of this note is to establish a somewhat surprising connection between functions on Newton–Okounkov bodies and Seshadri constants of line bundles on algebraic surfaces.

Newton–Okounkov bodies and complexity functions

We show that quite universally the holonomicity of the complexity function of a divisor does not predict whether the Newton–Okounkov body is polyhedral for every choice of a flag.

NEWTON–OKOUNKOV POLYTOPES OF FLAG VARIETIES

We compute the Newton--Okounkov bodies of line bundles on the complete flag variety of GL_n for a geometric valuation coming from a flag of translated Schubert subvarieties. The Schubert subvarieties correspond to the terminal subwords in the decomposition (s_1)(s_2s_1)(s_3s_2s_1)(...)(s_{n-1}...s_1) of the longest element in the Weyl group. The resulting Newton--Okounkov bodies coincide with the Feigin--Fourier--Littelmann--Vinberg polytopes in type A.

Okounkov bodies and Zariski decompositions on surfaces

The purpose of this paper is to investigate the close relation between Okounkov bodies and Zariski decompositions of pseudoeffective divisors on smooth projective surfaces. Firstly, we completely determine the limiting Okounkov bodies on such surfaces, and give applications to Nakayama constants and Seshadri constants. Secondly, we study how the shapes of Okounkov bodies change as we vary the divisors in the big cone.

On the polyhedrality of global Okounkov bodies

Abstract We prove that the existence of a finite Minkowski basis for Okounkov bodies on a smooth projective variety with respect to an admissible flag implies the rational polyhedrality of the global Okounkov body. As an application of this general result, we deduce that the global Okounkov body of a surface with finitely generated pseudo-effective cone with respect to a general flag is rational polyhedral. We give an alternative proof for this fact which recovers the generators more explicitly. We also prove the rational polyhedrality of global Okounkov bodies in the case of certain homogeneous 3-folds using inductive methods.

Restricted volumes of effective divisors

We study the restricted volume of effective divisors, its properties and the relationship with the related notion of reduced volume, defined via multiplier ideals, and with the asymptotic intersection number. We build upon the fundamental work of Lazarsfeld and Mustata relating the restricted volume of big divisors to the volume of the associated Okounkov body. We extend their constructions and results to the case of effective divisors, recovering some results of Kaveh and Khovanskii, proving a Fujita-type approximation in this larger setting and studying the restricted volume function. In order to relate the reduced volume and the asymptotic intersection number we investigate a boundedness property of asymptotic multiplier ideals and prove it holds, for instance, for finitely generated divisors. In this way we obtain also a complete picture for the canonical divisor of an arbitrary smooth projective variety and for nef divisors on varieties of dimension at most 3.

Crystal bases and Newton–Okounkov bodies

Let G be a connected reductive algebraic group. We prove that the string parametrization of a crystal basis for a finite dimensional irreducible representation of G extends to a natural valuation on the field of rational functions on the flag variety G/B, which is a highest term valuation corresponding to a coordinate system on a Bott-Samelson variety. This shows that the string polytopes associated to irreducible representations, can be realized as Newton-Okounkov bodies for the flag variety. This is closely related to an earlier result of A. Okounkov for the Gelfand-Cetlin polytopes of the symplectic group. As a corollary we recover a multiplicativity property of the canonical basis due to P. Caldero. We generalize the results to spherical varieties. From these the existence of SAGBI bases for the homogeneous coordinate rings of flag and spherical varieties, as well as their toric degenerations follow, recovering previous results of Caldero, Alexeev-Brion and the author.

Newton-Okounkov polyhedra for character varieties and configuration spaces

We construct families of Newton-Okounkov bodies for the free group character varieties and configuration spaces of any connected reductive group. We use this construction to give a proof that these spaces are Cohen-Macaulay.

MINKOWSKI DECOMPOSITION AND GENERATORS OF THE MOVING CONE FOR TORIC VARIETIES:

We prove that for smooth projective toric varieties, the Okounkov body of a |$T$|-invariant pseudo-effective divisor with respect to a |$T$|-invariant flag decomposes as a finite Minkowski sum of indecomposable polytopes, and that the set of these polytopes corresponds to a finite Minkowski basis whose elements span the extremal rays in the secondary fan. In fact, the Minkowski basis does not depend on the choice of the |$T$|-invariant flag. Moreover, we present an algorithm that computes the Minkowski basis.

Volumes of arithmetic Okounkov bodies

This paper is an improvement and simplification of Yuan [Yu]. In particular,by a simpler method, it proves the identity in [Yu, Theorem A] withouttaking the limit p → ∞. The method also simplifies the proofs of Moriwaki[Mo], where more general arithmetic linear series were treated.Recall that [Yu] explored a way to construct arithmetic Okounkov bodiesfrom an arithmetic line bundle, inspired by the idea of Okounkov [Ok1, Ok2]and Lazarsfeld–Musta¸ta [LM] in the geometric case. Theorem A of [Yu]asserts that the volumes of the Okounkov bodies approximate the volume ofthe arithmetic line bundle. The main result of this paper asserts an exactidentity before taking the limit.On the other hand, Boucksom–Chen [BC] initiated a different way toconstruct Okounkov bodies in the arithmetic setting. From the proof ofthe main result in this paper, it is easy to recognize the similarity of theconstructions in [BC] and in [Yu]. We will make a comparison at the end ofthis paper.In the following, we recall the construction of [Yu] and state our mainresult. We use exactly the same notations as in [Yu] throughout this paper.1

Okounkov Bodies and Applications

The theory of Newton–Okounkov bodies, also called Okounkov bodies, is a relatively new connection between algebraic geometry and convex geometry. It generalizes the well-known and extremely rich correspondence between geometry of toric varieties and combinatorics of convex integral polytopes. Following a successful MFO Mini-workshop on this topic in August 2011, the MFO Half-Workshop 1422b, “Okounkov bodies and applications”, held in May 2014, explored the development of this area in recent years, with particular attention to applications and relationships to other areas such as number theory and tropical geometry.

Numerical and Enumerative Results on Okounkov Bodies

In this note we focus on three independent problems on Okounkov bodies for projective varieties. The main goal is to present a geometric version of the classical Fujita Approximation Theorem, a Jow-type theorem and a cardinality formulae for Minkowski bases on a certain class of smooth projective surfaces.