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Sung Rak Choi

Publications and source records attributed to Sung Rak Choi.

At least 19 recordsLinked to original sources
arXiv2026

Variation of cones of divisors in a family of varieties -- Fano type case

We investigate the relationship between the Fano type property on fibers over a Zariski dense subset and the global Fano type property. We establish the invariance of Néron-Severi spaces, nef cones, effective cones, movable cones, and Mori chamber decompositions for a family of Fano type varieties after a generically finite base change. Additionally, we show the uniform behavior of the minimal model program for this family. These results are applied to the boundedness problem of Fano type varieties.

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arXiv2026

On the Morrison-Kawamata dream space and its applications

We develop the theory of Morrison-Kawamata dream spaces, which axiomatizes varieties (not necessarily of Calabi-Yau type) that satisfy the Morrison-Kawamata cone conjecture. Using this theory, we establish the generic deformation invariance of various cones and apply it to the boundedness problem of algebraic varieties.

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arXiv2025

Adjoint asymptotic multiplier ideal sheaves associated to potential triples

In this paper, we explore the geometry of potential triples $(X,Δ,D)$, which by definition consists of a pair $(X,Δ)$ and an $\mathbb{R}$-Cartier pseudoeffective divisor $D$ on $X$. We define and study the asymptotic multiplier ideal sheaf $\mathcal{J}(X,Δ,\lVert D\rVert)$ associated to a potential triple $(X,Δ,D)$. As a first main result, when $D$ is big, we prove that the condition $\mathcal{J}(X,Δ,\lVert D\rVert)=\mathcal{O}_{X}$ is equivalent to the triple $(X,Δ,D)$ being potentially klt, which is a klt analog of the pair $(X,Δ)$. We also study the closed set defined by the ideal sheaf $\mathcal{J}(X,Δ,\lVert D\rVert)$ and prove a Nadel type cohomology vanishing theorem for $\mathcal{J}(X,Δ,\lVert D\rVert)$. As an application of the main result, we prove that we can run the $(K_X+Δ+D)$-MMP with scaling of an ample divisor for a pklt triple $(X,Δ,D)$.

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arXiv2025

Anticanonical divisor with good asymptotic base loci

In this paper, we give a characterization of Fano type varieties in terms of the asymptotic base loci of $-(K_X+Δ)$. We also show that for a potentially lc pair $(X,Δ)$, if no plc centers are contained in the augmented base locus $\mathbf{B}_{+}(-(K_X+Δ))$, then $(X,Δ)$ has a good $-(K_X+Δ)$-minimal model. This gives an analogous result of Birkar--Hu on the existence of good minimal models.

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arXiv2025

A valuative approach to the anticanonical minimal model program

In this paper, we show that the log canonical threshold of a potentially klt triple can be computed by a quasi-monomial valuation. The notion of potential triples provides a larger and more flexible framework to work with than that of generalized pairs. Our main result can be considered as an extension to the result of Xu on klt pairs. As an application of the main result, we show that we can run the MMP on any potentially klt triples and $-(K_X+Δ)$-MMP on the potentially klt pairs.

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arXiv2025

On minimal model program and Zariski decomposition of potential triples

In this paper, we investigate properties of potential triples $(X,Δ,D)$ which consists of a pair $(X,Δ)$ and a pseudoeffective $\mathbb{R}$-Cartier divisor $D$. In particular, we show that if $D$ admits a birational Zariski decomposition, then one can associate a generalized pair structure to the potential triple $(X,Δ,D)$. Moreover, we can run the generalized MMP on $(K_X+Δ+D)$ as special cases. As an application, we also show that for a pklt pair $(X,Δ)$, if $-(K_X+Δ)$ admits a birational Zariski decomposition with $\mathrm{NQC}$ positive part, then there exists a $-(K_X+Δ)$-minimal model.

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arXiv2023

ACC of plc thresholds

In this paper, we define potential log canonical threshold and prove that the set of those thresholds satisfies the ascending chain condition (ACC). We also consider collections of sequences of Fano type varieties and we study their basic properties including boundedness.

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arXiv2021

On subadditivity of Okounkov bodies for algebraic fiber spaces

The purpose of this paper is to establish a subadditivity theorem of Okounkov bodies for algebraic fiber spaces. As applications, we obtain a product formula of the restricted canonical volumes for algebraic fiber spaces and a sufficient condition for an algebraic fiber space to be birationally isotrivial in terms of Okounkov bodies when a general fiber is of general type. Furthermore, we also prove the subadditivity of the numerical Iitaka dimensions for algebraic fiber spaces, and this confirms some numerical variants of the Iitaka conjecture. We hope that our results would provide a new approach toward the Iitaka conjecture.

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arXiv2021

Okounkov bodies associated to abundant divisors and Iitaka fibrations

The aim of this paper is to study the Okounkov bodies associated to abundant divisors. As a main result, we prove that the valuative Okounkov bodies of an abundant divisor encode all the numerical properties. We apply this result to recover the asymptotic base loci of an abundant divisor from the valuative Okounkov bodies. We also give a criterion of when the valuative and limiting Okounkov bodies of an abundant divisor coincide by comparing their Euclidean volumes. To obtain these results, we prove some variants of Fujita's approximations for Okounkov bodies using Iitaka fibrations.

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arXiv2021

Comparing numerical Iitaka dimensions again

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 known numerical Iitaka dimensions coincide with each other until the recent discovery of a counterexample constructed by Lesieutre. In this paper, we prove that many of them still coincide with the numerical Iitaka dimension introduced by Boucksom-Demailly-Păun-Peternell. On the other hand, we show that some other numerical Iitaka dimensions introduced by Nakayama and Lehmann can be arbitrarily larger than the rest of numerical Iitaka dimensions. We also study some properties of abundant divisors.

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arXiv2020

Local numerical equivalences and Okounkov bodies in higher dimensions

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 divisor with respect to admissible flags centered at a fixed point determines the local numerical equivalence class of divisors which is defined in terms of refined divisorial Zariski decompositions. Our results extend Roé's work on surfaces to higher dimensional varieties although our proof is essentially different in nature.

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arXiv2018

Okounkov bodies associated to pseudoeffective divisors

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-Khovanskii. For this purpose, we define and study special subvarieties, called the Nakayama subvarieties and the positive volume subvarieties, associated to pseudoeffective divisors.

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arXiv2017

Asymptotic base loci via Okounkov bodies

An Okounkov body is a convex subset of Euclidean space associated to a divisor on a smooth projective variety with respect to an admissible flag. In this paper, we recover the asymptotic base loci from the Okounkov bodies by studying various asymptotic invariants such as the asymptotic valuations and the moving Seshadri constants. Consequently, we obtain the nefness and ampleness criteria of divisors in terms of the Okounkov bodies. Furthermore, we compute the divisorial Zariski decomposition by the Okounkov bodies, and find upper and lower bounds for moving Seshadri constants given by the size of simplexes contained in the Okounkov bodies.

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arXiv2017

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.

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arXiv2017

Okounkov bodies associated to pseudoeffective divisors II

We first prove some basic properties of Okounkov bodies, and give a characterization of Nakayama and positive volume subvarieties of a pseudoeffective divisor in terms of Okounkov bodies. Next, we show that each valuative and limiting Okounkov bodies of a pseudoeffective divisor which admits the birational good Zariski decomposition is a rational polytope with respect to some admissible flag. This is an extension of the result of Anderson-Küronya-Lozovanu about the rational polyhedrality of Okounkov bodies of big divisors with finitely generated section rings.

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