Zariski Decomposition Research Articles

Cohomology Chambers on Complex Surfaces and Elliptically Fibered Calabi–Yau Three-Folds

We determine several classes of smooth complex projective surfaces on which Zariski decomposition can be combined with vanishing theorems to yield cohomology formulae for all line bundles. The obtained formulae express cohomologies in terms of divisor class intersections, and are adapted to the decomposition of the effective cone into Zariski chambers. In particular, we show this occurs on generalised del Pezzo surfaces, toric surfaces, and K3 surfaces. In the second part we use these surface results to derive formulae for all line bundle cohomology on a simple class of elliptically fibered Calabi–Yau three-folds. Computing such quantities is a crucial step in deriving the massless spectrum in string compactifications.

Non-Archimedean Green’s functions and Zariski decompositions

Non-Archimedean Green’s functions and Zariski decompositions

An integral version of Zariski decompositions on normal surfaces

An integral version of Zariski decompositions on normal surfaces

Waldschmidt constants in projective spaces

In this note we introduce a Waldschmidt decomposition of divisors which might be viewed as a generalization of Zariski decomposition based on the effectivity rather than the nefness of divisors. As an immediate application we prove a recursive formula providing new effective lower bounds on Waldschmidt constants of very general points in projective spaces. We use these bounds in order to verify Demailly's conjecture in a number of new cases.

Relative Nakayama–Zariski Decomposition and Minimal Models of Generalized Pairs

Relative Nakayama–Zariski Decomposition and Minimal Models of Generalized Pairs

On the Termination of Flips for Log Canonical Generalized Pairs

We prove the termination of flips for pseudo-effective NQC log canonical generalized pairs of dimension 4. As main ingredients, we verify the termination of flips for 3-dimensional NQC log canonical generalized pairs, and show that the termination of flips for pseudo-effective NQC log canonical generalized pairs which admit NQC weak Zariski decompositions follows from the termination of flips in lower dimensions.

Special MMP for log canonical generalised pairs (with an appendix joint with Xiaowei Jiang)

We show that minimal models of mathbb {Q}-factorial NQC log canonical generalised pairs exist, assuming the existence of minimal models of smooth varieties. More generally, we prove that on a mathbb {Q}-factorial NQC log canonical generalised pair (X,B+M) we can run an MMP with scaling of an ample divisor which terminates, assuming that it admits an NQC weak Zariski decomposition or that K_X+B+M is not pseudoeffective. As a consequence, we establish several existence results for minimal models and Mori fibre spaces.

Boucksom–Zariski and Weyl Chambers on Irreducible Holomorphic Symplectic Manifolds

Abstract Inspired by the work of Bauer, Küronya, and Szemberg, we provide for the big cone of a projective irreducible holomorphic symplectic (IHS) manifold a decomposition into chambers (which we describe in detail), in each of which the support of the negative part of the divisorial Zariski decomposition is constant. We see how the obtained decomposition of the big cone allows to describe the volume function. Moreover, similarly to the case of surfaces, we see that the big cone of a projective IHS manifold admits a decomposition into simple Weyl chambers, which we compare to that induced by the divisorial Zariski decomposition. To conclude, we determine the structure of the pseudo-effective cone.

Weak Zariski decompositions and log terminal models for generalized pairs

Weak Zariski decompositions and log terminal models for generalized pairs

Restricted volumes on Kähler manifolds

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 bundles on projective manifolds, we pose a natural conjecture to the effect that irreducible components of the non-Kahler locus of a big class should have vanishing numerical restricted volume. We prove this conjecture when the class has a Zariski decomposition, and give several applications.

Intersection theory of nefb-divisor classes

We prove that any nef$b$-divisor class on a projective variety defined over an algebraically closed field of characteristic zero is a decreasing limit of nef Cartier classes. Building on this technical result, we construct an intersection theory of nef$b$-divisors, and prove several variants of the Hodge index theorem inspired by the work of Dinh and Sibony. We show that any big and basepoint-free curve class is a power of a nef$b$-divisor, and relate this statement to the Zariski decomposition of curves classes introduced by Lehmann and Xiao. Our construction allows us to relate various Banach spaces contained in the space of$b$-divisors which were defined in our previous work.

On the Mori theory and Newton–Okounkov bodies of Bott–Samelson varieties

We prove that on a Bott–Samelson variety X every movable divisor is nef. This enables us to consider Zariski decompositions of effective divisors, which in turn yields a description of the Mori chamber decomposition of the effective cone. This amounts to information on all possible birational morphisms from X. Applying this result, we prove the rational polyhedrality of the global Newton–Okounkov cone of a Bott–Samelson variety with respect to the so called ‘horizontal’ flag. In fact, we prove the stronger property of the finite generation of the corresponding global value semigroup.

Higher dimensional elliptic fibrations and Zariski decompositions

We study the existence and properties of birationally equivalent models for elliptically fibered varieties. In particular these have either the structure of Mori fiber spaces or, assuming some standard conjectures, minimal models with a Zariski decomposition compatible with the elliptic fibration. We prove relations between the birational invariants of the elliptically fibered variety, the base of the fibration and of its Jacobian.

Pseudo-effective cones of projective bundles and weak Zariski decomposition

In this article, we consider the projective bundle $\mathbb{P}_X(E)$ over a smooth complex projective variety $X$, where $E$ is a semistable bundle on $X$ with $c_2(End(E)) =0$. We give a necessary and sufficient condition to get the equality $ Nef^1\bigl(\mathbb{P}_X(E)\bigr) = \overline{Eff}^1\bigl(\mathbb{P}_X(E)\bigr)$ of nef cone and pseudoeffective cone of divisors in $\mathbb{P}_X(E)$. As an application of our result, we show the equality of nef and pseudoeffective cones of divisors of projective bundles over some special varieties. In particular, we show that weak Zariski decomposition exists on these projective bundles. We also show that a semistable bundle $E$ of rank $r \geq 2$ with $c_2\bigl(End(E)\bigr) = 0$ on a smooth complex projective variety of Picard number 1 is $k$-homogeneous i.e. $\overline{Eff}^k\bigl(\mathbb{P}_X(E)\bigr) = Nef^k\bigl(\mathbb{P}_{X}(E)\bigr)$ for all $1 \leq k < r$. Finally, we show that weak Zariski decomposition exists for a fibre product $\mathbb{P}_C(E)\times_C\mathbb{P}(E')$ over a smooth projective curve $C$.

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 [R] on surfaces to higher-dimensional varieties although our proof is essentially different in nature.

On weak Zariski decompositions and termination of flips

On weak Zariski decompositions and termination of flips

Positivity functions for curves on algebraic varieties

This is the second part of our work on Zariski decomposition structures, where we compare two different volume type functions for curve classes. The first function is the polar transform of the volume for ample divisor classes. The second function captures the asymptotic geometry of curves analogously to the volume function for divisors. We prove that the two functions coincide, generalizing Zariski's classical result for surfaces to all varieties. Our result confirms the log concavity conjecture of the first named author for weighted mobility of curve classes in an unexpected way, via Legendre-Fenchel type transforms. We also give a number of applications to birational geometry, including a refined structure theorem for the movable cone of curves.

A note on a smooth projective surface with Picard number 2

AbstractWe characterize the integral Zariski decomposition of a smooth projective surface with Picard number 2 to partially solve a problem of B. Harbourne, P. Pokora, and H. Tutaj‐Gasinska [Electron. Res. Announc. Math. Sci. 22 (2015), 103–108].

On strong vanishing property and plurigenera of isolated singularities

We introduce a new concept of strong vanishing property and investigate this property for minimally elliptic singularities. A comparison theorem about strong vanishing property is also obtained. We use Zariski decomposition to give a formula of plurigenera for Gorenstein singularities and use it to test whether a divisor is f-nef.

Orthogonality of divisorial Zariski decompositions for classes with volume zero

We show that the orthogonality conjecture for divisorial Zariski decompositions on compact Kahler manifolds holds for pseudoeffective (1,1) classes with volume zero.