Zariski Decomposition Research Articles

Divisorial Zariski decomposition and some properties of full mass currents

Letbe a big class on a compact Kahler manifold. We prove that a decomposition � = �1 +�2 into the sum of a modified nef class �1 and a pseudoeffective class �2 is the divisorial Zariski decomposition ofif and only if vol(�) = vol(�1). We deduce from this result some properties of full mass currents.

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.

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-Kuronya-Lozovanu about the rational polyhedrality of Okounkov bodies of big divisors with finitely generated section rings.

Transcendental Morse inequality and generalised Okounkov bodies

The main goal of this article is to construct arithmetic Okounkov for an arbitrary pseudo-effective (1,1)-class $\alpha$ on a Kahler manifold. Firstly, using Boucksom's divisorial Zariski decompositions for pseudo-effective (1,1)-classes on compact Kahler manifolds, we prove the differentiability of volumes of big classes for Kahler manifolds on which modified nef cones and nef cones coincide; this includes Kahler surfaces. We then apply our differentiability results to prove Demailly's transcendental Morse inequality for these particular classes of Kahler manifolds. In the second part, we construct the convex body $\Delta(\alpha)$ for any big class $\alpha$ with respect to a fixed flag by using positive currents, and prove that this newly defined convex body coincides with the Okounkov body when $\alpha\in {\rm NS}_{\mathbb{R}}(X)$; such convex sets $\Delta(\alpha)$ will be called generalized Okounkov bodies. As an application we prove that any rational point in the interior of Okounkov bodies is valuative. Next we give a complete characterisation of generalized Okounkov bodies on surfaces, and show that the generalized Okounkov bodies behave very similarly to original Okounkov bodies. By the differentiability formula, we can relate the standard Euclidean volume of $\Delta(\alpha)$ in $\mathbb{R}^2$ to the volume of a big class $\alpha$, as defined by Boucksom; this solves a problem raised by Lazarsfeld in the case of surfaces. Finally, we study the behavior of the generalized Okounkov bodies on the boundary of the big cone, which are characterized by numerical dimension.

Zariski decompositions of numerical cycle classes

We construct a Zariski decomposition for cycle classes of arbitrary codimension. This decomposition is an analogue of well-known constructions for divisors. Examples illustrate how Zariski decompositions of cycle classes reflect the geometry of the underlying space. We also analyze the birational behavior of Zariski decompositions, leading to a Fujita approximation-type result for curve classes.

Moriwaki divisors and the augmented base loci of divisors on the moduli space of curves

We study the cone of Moriwaki divisors on \bar{M}_g by means of augmented base loci. Using a result of Moriwaki, we prove that an R-divisor D satisfies the strict Moriwaki inequalities if and only if the augmented base locus of D is contained in the boundary of \bar{M}_g. Then we draw some interesting consequences on the Zariski decomposition of divisors on \bar{M}_g, on the minimal model program of \bar{M}_g and on the log canonical models \bar{M}_g(\alpha).

Convexity and Zariski decomposition structure

This is the first part of our work on Zariski decomposition structures, where we study Zariski decompositions using Legendre–Fenchel type transforms. In this way we define a Zariski decomposition for curve classes. This decomposition enables us to develop the theory of the volume function for curves defined by the second named author, yielding some fundamental positivity results for curve classes. For varieties with special structures, the Zariski decomposition for curve classes admits an interesting geometric interpretation.

On Zariski Decomposition with and without Support

We study Zariski decomposition with support in a negative definite cycle, a variation of Zariski decomposition introduced by Miyaoka [4]: given a negative definite cycle $G$, any $\mathbb{Q}$-divisor $D$ decomposes into the sum of a $G$-nef and a rigid $\mathbb{Q}$-divisor. We prove that such a decomposition actually exists for an arbitrary $\mathbb{Q}$-divisor. Moreover, we show that, under the hypothesis that $D$ is pseudo-effective, we can drop the assumption of $G$ being negative definite, and obtain decompositions of $D$ with respect to arbitrary cycles. Our methods are inspired by a work of Bauer [1], in which he gives a simpler proof of Zariski's original result [5], and by adapting his proof to other cases, we are able to provide an alternative approach to this circle of ideas.

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.

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.

A pathology of asymptotic multiplicity in the relative setting

We point out an example of a projective family $\pi : X \to S$, a $\pi$-pseudoeffective divisor $D$ on $X$, and a subvariety $V \subset X$ for which the asymptotic multiplicity $\sigma_V(D;X/S)$ is infinite. This shows that the divisorial Zariski decomposition is not always defined for pseudoeffective divisors in the relative setting.

Arithmetic positivity on toric varieties

We continue the study of the arithmetic geometry of toric varieties started by J. Burgos Gil, P. Philippon, and M. Sombra in 2011. In this text, we study the positivity properties of metrized R \mathbb {R} -divisors in the toric setting. For a toric metrized R \mathbb {R} -divisor, we give formulae for its arithmetic volume and its χ \chi -arithmetic volume, and we characterize when it is arithmetically ample, nef, big or pseudo-effective, in terms of combinatorial data. As an application, we prove a higher-dimensional analogue of Dirichlet’s unit theorem for toric varieties, we give a characterization for the existence of a Zariski decomposition of a toric metrized R \mathbb {R} -divisor, and we prove a toric arithmetic Fujita approximation theorem.

On the boundedness of the denominators in the Zariski decomposition on surfaces

Abstract Zariski decompositions play an important role in the theory of algebraic surfaces. For making geometric use of the decomposition of a given divisor, one needs to pass to a multiple of the divisor in order to clear denominators. It is therefore an intriguing question whether the surface has a “universal denominator” that can be used to simultaneously clear denominators in all Zariski decompositions on the surface. We prove in this paper that, somewhat surprisingly, this condition of bounded Zariski denominators is equivalent to the bounded negativity of curves that is addressed in the Bounded Negativity Conjecture. Furthermore, we provide explicit bounds for Zariski denominators and negativity of curves in terms of each other.

On images of Mori dream spaces

The purpose of this paper is to study the geometry of images of morphisms from Mori dream spaces. First we prove that a variety which admits a surjective morphism from a Mori dream space is again a Mori dream space. Secondly we introduce a natural fan structure on the effective cone of Mori dream spaces. We show that it encodes the information of Zariski decompositions, which in turn is equivalent to the information of the variation of GIT quotients of their Cox rings. Finally we show that under a surjective morphism between Mori dream spaces, the fan of the target space coincides with the restriction of the fan of the source.

Stability and Singularities of Relative Hypersurfaces

We study relative hypersurfaces over curves, and prove an instability condition for the fibres. This gives an upper bound on the log canonical threshold of the relative hypersurface. We compare these results with the information that can be derived from Nakayama's Zariski decomposition of effective divisors on relative projective bundles.

Minimal singular metrics of a line bundle admitting no Zariski decomposition

We give a concrete expression of a minimal singular metric on a big line bundle on a compact Kahler manifold which is the total space of a toric bundle over a complex torus. In this class of manifolds, Nakayama constructed examples which have line bundles admitting no Zariski decomposition even after modifications. As an application, we discuss the Zariski closedness of non-nef loci.

On the existence of a weak Zariski decomposition on projectivized vector bundles

A pseudoeffective divisor is said to have a weak Zariski decomposition if it can be, up to a birational transformation, numerically written as the sum of a nef and an effective divisor. In this paper we consider the problem of the existence of a weak Zariski decomposition for each pseudoeffective divisor on a variety \(X= \mathbb {P}(\fancyscript{E})\), where \(\fancyscript{E}\) is a vector bundle on a smooth complex projective variety \(Z\) of Picard number one. We prove the existence of such a decomposition in a number of meaningful situations.

A Nadel vanishing theorem for metrics with minimal singularities on big line bundles

The purpose of this paper is to establish a Nadel vanishing theorem for big line bundles with multiplier ideal sheaves of singular metrics admitting an analytic Zariski decomposition (such as, metrics with minimal singularities and Siu's metrics). For this purpose, we apply the theory of harmonic integrals and generalize Enoki's proof of Kollár's injectivity theorem. Moreover we investigate the asymptotic behavior of harmonic forms with respect to a family of regularized metrics.

On the Numerical Dimension of Pseudo-Effective Divisors in Positive Characteristic

Let $X$ be a smooth projective variety over an algebraically closed field of positive characteristic. We prove that if $D$ is a pseudo-effective ${\bf R}$-divisor on $X$ which is not numerically equivalent to the negative part in its divisorial Zariski decomposition, then the numerical dimension of $D$ is positive. In characteristic zero, this was proved by Nakayama using vanishing theorems.

The diminished base locus is not always closed

AbstractWe exhibit a pseudoeffective$\mathbb{R}$-divisor${D}_{\lambda }$on the blow-up of${\mathbb{P}}^{3}$at nine very general points which lies in the closed movable cone and has negative intersections with a set of curves whose union is Zariski dense. It follows that the diminished base locus${\boldsymbol{B}}_{-}({D}_{\lambda })={\bigcup }_{A\,\text{ample}}\boldsymbol{B}({D}_{\lambda }+A)$is not closed and that${D}_{\lambda }$does not admit a Zariski decomposition in even a very weak sense. By a similar method, we construct an$\mathbb{R}$-divisor on the family of blow-ups of${\mathbb{P}}^{2}$at ten distinct points, which is nef on a very general fiber but fails to be nef over countably many prime divisors in the base.