Using currents with minimal singularities, we construct pointwise minimal multiplicities for a real pseudo-effective $(1,1)$-class $α$ on a compact complex $n$-fold
$X$, which are the local obstructions to the numerical effectivity of $α$. The negative part of $α$ is then defined as the real effective divisor $N(α)$ whose multiplicity along a prime divisor $D$ is just the generic multiplicity of $α$ along $D$, and we get in that way a divisorial Zariski decomposition of $α$ into the sum of a class $Z(α)$ which is nef in codimension 1 and the class of its negative part $N(α)$, which is exceptional in the sense that it is very rigidly embedded in $X$. The positive parts $Z(α)$ generate a modified nef cone, and the pseudo-effective cone is shown to be locally polyhedral away from the modified nef cone, with extremal rays generated by exceptional divisors. We then treat the case of a surface and a hyper-Kähler manifold in some detail: under the intersection form (resp. the
Beauville-Bogomolov form), we characterize the modified nef cone and the exceptional divisors; our divisorial Zariski decomposition is orthogonal, and is thus a rational decomposition, which fact accounts for the usual existence statement of a Zariski decomposition on a projective surface, which is thus extended to the hyper-Kähler case. Finally, we explain how the divisorial Zariski decomposition of (the first Chern class of) a big line bundle on a projective manifold can be characterized in terms of the asymptotics of the linear series $|kL|$ as $k\to\infty$.
Based on a recent work by Thomas Bauer reproving the existence of Zariski decompositions for surfaces, we construct a b-divisorial analogue of Zariski decomposition in all dimensions.
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.
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.
We prove that termination of lower dimensional flips for generalized klt pairs implies termination of flips for log canonical generalized pairs with a weak Zariski decomposition. Moreover, we prove that the existence of weak Zariski decompositions for pseudo-effective generalized klt pairs implies the existence of minimal models for such pairs.
We first introduce a weak type of Zariski decomposition in higher dimensions: an $\R$-Cartier divisor has a weak Zariski decomposition if birationally and in a numerical sense it can be written as the sum of a nef and an effective $\R$-Cartier divisor. We then prove that there is a very basic relation between Zariski decompositions and log minimal models which has long been expected: we prove that assuming the log minimal model program in dimension $d-1$, a lc pair $(X/Z,B)$ of dimension $d$ has a log minimal model if and only if $K_X+B$ has a weak Zariski decomposition$/Z$.
In this note we consider the problem of integrality of Zariski decompositions for pseudoeffective integral divisors on algebraic surfaces. We show that while sometimes integrality of Zariski decompositions forces all negative curves to be $(-1)$-curves, there are examples where this is not true.
We show that the existence of a birational weak Zariski decomposition for a pseudo-effective generalized polarized lc pair is equivalent to the existence of a generalized polarized log terminal model.
The big cone of every K3 surface admits two natural chamber decompositions: the decomposition into Zariski chambers, and the decomposition into simple Weyl chambers. In the present paper we compare these two decompositions and we study their mutual relationship: First, we give a numerical criterion for the two decompositions to coincide. Secondly, we study the mutual inclusions of Zariski and simple Weyl chambers. Finally, we establish the fact that -- even though the decompositions themselves may differ -- the number of Zariski chambers always equals the number of simple Weyl chambers.
We continue our study of the relation between log minimal models and various types of Zariski decompositions. Let $(X,B)$ be a projective log canonical pair. We will show that $(X,B)$ has a log minimal model if either $K_X+B$ birationally has a Nakayama-Zariski decomposition with nef positive part, or that $K_X+B$ is big and birationally it has a Fujita or CKM Zariski decomposition. Along the way we introduce polarized pairs $(X,B+P)$ where $(X,B)$ is a usual projective pair and $P$ is nef, and study the birational geometry of such pairs.
In this note we give a quick and simple proof of the existence (and uniqueness) of Zariski decompositions on surfaces. While Zariski's original proof employs a rather sophisticated procedure to construct the negative part of the decomposition, the present approach is based on the idea that the positive part can be constructed from a maximality condition. It may also be useful that this approach yields a practical algorithm for the computation of the positive part.
We prove that, on a smooth threefold, pseudoeffective divisors with closed and one-dimensional diminished base locus have birationally a Fujita-Zariski decomposition.
We study the divisorial Zariski decomposition on varieties whose first Chern class is zero. We first prove that any exceptional divisor is contractible (up to a birational map that is an isomorphism in codimension one). We then characterize prime exceptional divisors on holomorphic symplectic manifolds.
In this paper we study Zariski Decomposition with support in a negative definite cycle, a variation introduced by Y. Miyaoka. We provide two extensions of the original statement, which was originally meant for effective $\Q$-divisors: we can either state it for any $\Q$-divisor, or we can take the support to be in any cycle. Ultimately, we present a new approach to Zariski Decomposition of pseudo-effective $\Q$-divisors, which consists in iterating Zariski Decomposition with support.
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 N…
We study graded rings associated to big divisors on LC pairs whose difference with the log-canonical divisor is nef. For divisors that are positive enough at the LC centers of the pair, we prove the finite generation of such rings if the pair is DLT or the dimension is low, given that a Zariski decomposition exists.
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.
We show that any pseudo-effective divisor on a normal surface decomposes uniquely into its "integral positive" part and "integral negative" part, which is an integral analog of Zariski decompositions. By using this decomposition, we give three applications: a vanishing theorem of divisors on surfaces (a generalization of Kawamata-Viehweg and Miyaoka vanishing theorems), Reider-type theorems of adjoint linear systems on surfaces (including a log version and a relative version of the original one) and extension theorems of morphisms defined on curves on surfaces (generalizations of Serrano and P…
On the Boucksom-Zariski decomposition for irreducible symplectic varieties and bounded negativity
Institute of Mathematics of the Polish Academy of Sciences（波兰科学院数学研究所）；Dipartimento di Matematica, Università degli studi di Bologna（博洛尼亚大学数学系）；Institut Elie Cartan de Lorraine, Université de Lorraine et CNRS（洛林埃利嘉当研究所，洛林大学与法国国家科学研究中心）；Department of Mathematics, University of the National Education Commission Krakow（克拉科夫国民教育委员会大学数学系）
Let $X$ be a nonsingular complex projective surface. The Weyl and Zariski chambers give two interesting decompositions of the big cone of $X$. We study these two decompositions and determine when a Weyl chamber is contained in the interior of a Zariski chamber and vice versa. We also determine when a Weyl chamber can intersect non-trivially with a Zariski chamber.
We show that the orthogonality conjecture for divisorial Zariski decompositions on compact Kahler manifolds holds for pseudoeffective (1,1) classes with volume zero.
In this paper, we establish the Zariski decompositions of arithmetic R-divisors of continuous type on arithmetic surfaces and investigate several properties. We also develop the general theory of arithmetic R-divisors on arithmetic varieties.
The big cone of every smooth projective surface $X$ admits the natural decomposition into Zariski chambers. The purpose of this note is to give a simple criterion for the interiors of all Zariski chambers on $X$ to be numerically determined Weyl chambers. Such a criterion generalizes the results of Bauer-Funke on K3 surfaces to arbitrary smooth projective surfaces. In the last section, we study the relation between decompositions of the big cone and elliptic fibrations on Enriques surfaces.
In a 1962 paper, Zariski introduced the decomposition theory that now bears his name. Although it arose in the context of algebraic geometry and deals with the configuration of curves on an algebraic surface, we have recently observed that the essential concept is purely within the realm of linear algebra. In this paper, we formulate Zariski decomposition as a theorem in linear algebra and present a linear algebraic proof. We also sketch the geometric context in which Zariski first introduced his 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…
Let $α$ be a big class on a compact Kähler manifold. We prove that a decomposition $α=α_1+α_2$ into the sum of a modified nef class $α_1$ and a pseudoeffective class $α_2$ is the divisorial Zariski decomposition of $α$ if and only if $\operatorname{vol}(α)=\operatorname{vol}(α_1)$. We deduce from this result some properties of full mass currents.
We give a concrete expression of a minimal singular metric of a big line bundle on a compact Kähler 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 any proper modifications. As an application, we discuss the Zariski-closedness of non-nef loci and the openness conjecture of Demailly and Kollár in this class.
For a compact hyperkähler manifold X, we show certain Zariski decomposition for every pseudo-effective R-divisor, and give a sufficient condition for X to be bimeromorphic to a (holomorphic) Lagrangian fibration. We also prove that any sequence of D-flops between projective hyperkähler manifolds terminates after finite steps.
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.
We give a relation between the existence of a Zariski decomposition and the behavior of the restricted volume of a big divisor on a smooth (complex) projective variety. Moreover, we give an analytic description of the restricted volume in the line of Boucksom's work. It enables us to define the restricted volume of a transcendental class on a compact Kähler manifold in natural way. The relation can be extended to a transcendental class.
Zariski chambers provide a natural decomposition of the big cone of an algebraic surface into rational locally polyhedral subcones that are interesting from the point of view of linear series. In the present paper we present an algorithm that allows to effectively determine Zariski chambers when the negative curves on the surface are known. We show how the algorithm can be used to compute the number of chambers on Del Pezzo surfaces.
We explain the relationship between $α_{1}...α_{q}$ (standard cohomology product) and $(α_{1}...α_{q})$ (mobile intersection product) of pseudo-effective classes $α_{1},...,α_{q}$ on a compact K$\ddot{a}$hler manifold. We also show how to use this relationship for proving some holomorphic Morse inequalities. Then we prove a result concerning the direct image of Lelong numbers under a modification in dimension 3, deriving a continuity property for the Lelong numbers of the wedge of $(1,1)-$currents.
We prove a general Zariski-van Kampen-Lefschetz type theorem for higher homotopy groups of generic and nongeneric pencils on singular open complex spaces.
This paper is an enhancement of the previous note "Explicit computations of Zariski decompositions on P_Z^1". In this paper, we observe several properties of a certain kind of an arithmetic divisor D on the n-dimensional projective space over Z and give the exact form of the Zariski decomposition of D on the projective line over Z. Further, we show that, if n>=2 and D is big and non-nef, then, for any birational morphism f: X --> P^n_Z of projective, generically smooth and normal arithmetic varieties, we can not expect a suitable Zariski decomposition of f^*(D). We also give a concrete constru…
We prove two theorems on the locally finite decompositions of the cones of divisors by the cones which correspond to canonical and minimal models. We introduce the concept of the numerical linear systems in order to simplify the argument on the Zariski decompositions.
This paper gives an extension of the classical Zariski-van Kampen theorem describing the fundamental groups of the complements of plane singular curves by generators and relations. It provides a procedure for computation of the first non-trivial higher homotopy groups of the complements of singular projective hypersurfaces in terms of the homotopy variation operators introduced here.
We prove the Lipman-Zariski conjecture for complex surface singularities with $p_g - g - b \le 2$. Here $p_g$ is the geometric genus, $g$ is the sum of the genera of the exceptional curves and $b$ is the first Betti number of the dual graph. This improves on a previous result of the second author. As an application, we show that a compact complex surface with locally free tangent sheaf is smooth as soon as it admits two generically linearly independent twisted vector fields and its canonical sheaf has at most two global sections.
We 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].
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 Ca…
We exhibit a pseudoeffective R-divisor D_λon the blow-up of 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 B_-(D_λ) = \bigcup_{A ample}} B(D_λ+A) is not closed and that D_λdoes not admit a Zariski decomposition in even a very weak sense. By a similar method, we construct an R-divisor on the family of blow-ups of 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.
For line arrangements in P^2 with nice combinatorics (in particular, for those which are nodal away the line at infinity), we prove that the combinatorics contains the same information as the fundamental group together with the meridianal basis of the abelianization. We consider higher dimensional analogs of the above situation. For these analogs, we give purely combinatorial complete descriptions of the following topological invariants (over an arbitrary field): the twisted homology of the complement, with arbitrary rank one coefficients; the homology of the associated Milnor fiber and Alexan…
We use methods of algebraic geometry to find new, effective methods for detecting the identifiability of symmetric tensors. In particular, for ternary symmetric tensors T of degree 7, we use the analysis of the Hilbert function of a finite projective set, and the Cayley-Bacharach property, to prove that, when the Kruskal's rank of a decomposition of T are maximal (a condition which holds outside a Zariski closed set of measure 0), then the tensor T is identifiable, i.e. the decomposition is unique, even if the rank lies beyond the range of application of both the Kruskal's and the reshaped Kru…
Based on Cynk-Hulek method we construct complex Calabi-Yau varieties of arbitrary dimensions using elliptic curves with automorphism of order 6. Also we give formulas for Hodge numbers of varieties obtained from that construction. We shall generalize result of Katsura and Schütt to obtain arbitrarily dimensional Calabi-Yau manifolds which are Zariski in any characteristic $p\not\equiv 1\pmod{12}.$
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 body 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.
In 1933, van Kampen described the fundamental groups of the complements of plane complex projective algebraic curves. Recently, Chéniot-Libgober proved an analogue of this result for higher homotopy groups of the complements of complex projective hypersurfaces with isolated singularities. Their description is in terms of some "homotopical variation operators". We generalize here the notion of "homotopical variation" to (singular) quasi-projective varieties. This is a first step for further generalizations of van Kampen's theorem. A conjecture, with a first approach, is stated in the special ca…
We prove a product decomposition of the Zariski closure of the jet lifts of a holomorphic map f from C into a semi-abelian variety A, provided that f is of finite order. On the other hand, by giving an example of such a map f into a three dimensional abelian variety we show that this product decomposition does not hold in general; there was a gap in the proofs of some articles which claimed that such a product decomposition is always true.
