Numerical analogues of the Kodaira dimension and the Abundance Conjecture

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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“…The numerical Iitaka dimension

κ_{ν} false(D false)

actually coincides with many other invariants defined with

D

. For details, see and .…”

Section: Preliminariesmentioning

confidence: 62%

Okounkov bodies associated to pseudoeffective divisors

Choi

Hyun

Park

et al. 2018

J. London Math. Soc.

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“…The equivalence of this definition to Definition 2.2 was claimed in [Leh13], see the subsequent corrections in [E16,Les19]. Still, by the results in [Leh13, Sec.…”

Section: Preliminariesmentioning

confidence: 78%

Towards classification of codimension 1 foliations on threefolds of general type

Aleksei¹

2021

Preprint

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We aim to classify codimension 1 foliations F with canonical singularities and ν(K F ) < 3 on threefolds of general type. We prove a classification result for foliations satisfying these conditions and having non-trivial algebraic part. We also describe purely transcendental foliations F with the canonical class K F being not big on manifolds of general type in any dimension, assuming that F is non-singular in codimension 2.

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“…This is a measure of positivity of an R-Cartier R-divisor that lies on the boundary of the pseudoeffective cone. However, this definition looks slightly different from the one that appeared in the literature ([Nak04], [Leh13] and [Eck16]). We shall prove in proposition (2.13) that the definition is well-defined, i.e.…”

Section: σ-Dimensionmentioning

confidence: 90%

Numerical dimension and locally ample curves

Lau¹

2017

Preprint

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In the paper [Lau16], it was shown that the restriction of a pseudoeffective divisor D to a subvariety Y with nef normal bundle is pseudoeffective. Assuming the normal bundle is ample and that D|Y is not big, we prove that the numerical dimension of D is bounded above by that of its restriction, i.e. κσ(D) ≤ κσ(D|Y ). The main motivation is to study the cycle classes of "positive" curves: we show that the cycle class of a curve with ample normal bundle lies in the interior of the cone of curves, and the cycle class of an ample curve lies in the interior of the cone of movable curves. We do not impose any condition on the singularities on the curve or the ambient variety. For locally complete intersection curves in a smooth projective variety, this is the main result of Ottem [Ott16]. The main tool in this paper is the theory of q-ample divisors.

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Numerical analogues of the Kodaira dimension and the Abundance Conjecture

Cited by 20 publications

References 15 publications

Okounkov bodies associated to pseudoeffective divisors

Okounkov bodies associated to pseudoeffective divisors

Towards classification of codimension 1 foliations on threefolds of general type

Numerical dimension and locally ample curves

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