# Okounkov bodies associated to pseudoeffective divisors

@article{Choi2018OkounkovBA, title={Okounkov bodies associated to pseudoeffective divisors}, author={Sung Rak Choi and Yoonsuk Hyun and Jinhyung Park and Joonyeong Won}, journal={Journal of the London Mathematical Society}, year={2018}, volume={97} }

An Okounkov body is a convex subset in Euclidean space associated to a big divisor on a smooth projective variety with respect to an admissible flag. In this paper, we introduce two convex bodies associated to pseudoeffective divisors, called the valuative Okounkov bodies and the limiting Okounkov bodies, and show that these convex bodies reflect the asymptotic properties of pseudoeffective divisors as in the case with big divisors. Our results extend the works of Lazarsfeld–Mustaţă and Kaveh…

## 13 Citations

Local Numerical Equivalences and Okounkov Bodies in Higher Dimensions

- Mathematics
- 2018

We continue to explore the numerical nature of the Okounkov bodies focusing on the local behaviors near given points. More precisely, we show that the set of Okounkov bodies of a pseudoeffective…

Extended Okounkov bodies and multi-point Seshadri constants

- Mathematics
- 2017

Based on the work of Okounkov, Lazarsfeld-Mustata and Kaveh-Khovanskii independently associated a convex body, called the Okounkov body, to a big divisor on a smooth projective variety with respect…

Algebraic reverse Khovanskii--Teissier inequality via Okounkov bodies

- Mathematics
- 2021

Let X be a projective variety of dimension n over an algebraically closed field of arbitrary characteristic and let A,B,C be nef divisors on X. We show that for any integer 1 ≤ k ≤ n− 1, (B · A) · (A…

Concave transforms of filtrations and rationality of Seshadri constants

- Mathematics
- 2019

We show that the subgraph of the concave transform of a multiplicative filtration on a section ring is the Newton--Okounkov body of a certain semigroup, and if the filtration is induced by a…

A Product Formula for Volumes of Divisors Via Okounkov Bodies

- MathematicsInternational Mathematics Research Notices
- 2018

We generalize Kawamata’s product formula for volumes of canonical divisors to arbitrary divisors using Okounkov bodies.

Final Remarks and Open Problems Appendix A : Subvarieties of the Siegel Modular Variety Appendix B : Extending of the Torelli Map to Toroidal Compactifications of the Siegel Modular Variety Appendix C : Singular Modular Forms Appendix

- Mathematics
- 2018

In this article, we discuss and survey the recent progress towards the Schottky problem, and make some comments on the relations between the André-Oort conjecture, Okounkov convex bodies, Coleman’s…

Geometric aspects of Newton-Okounkov bodies

- Mathematics
- 2017

This is a survey article on Newton-Okounkov bodies in projective geometry focusing on the relationship between positivity of divisors and Newton-Okounkov bodies.

A G ] 1 N ov 2 02 1 COMPARING NUMERICAL IITAKA DIMENSIONS AGAIN

- Mathematics
- 2021

To seek for the useful numerical analogues to the Iitaka dimension, various numerical Iitaka dimensions have been defined from a number of different perspectives. It has been accepted that all the…

Comparing numerical Iitaka dimensions again

- Mathematics
- 2021

To seek for the useful numerical analogues to the Iitaka dimension, various numerical Iitaka dimensions have been defined from a number of different perspectives. It has been accepted that all the…

Restricted volumes on Kähler manifolds

- MathematicsAnnales de la Faculté des sciences de Toulouse : Mathématiques
- 2022

We study numerical restricted volumes of (1,1) classes on compact Kahler manifolds, as introduced by Boucksom. Inspired by work of Ein-Lazarsfeld-Mustata-Nakamaye-Popa on restricted volumes of line…

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