相关论文: Transcendental Morse Inequality and Generalized Ok…

We show that the volume of transcendental big $(1,1)$-classes on compact K\"ahler manifolds can be realized by convex bodies, thus answering questions of Lazarsfeld-Musta\c{t}\u{a} and Deng. In our approach we use an approximation process…

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,…

Let $\{\alpha\}$ and $\{\beta\}$ be nef cohomology classes of bidegree $(1,\,1)$ on a compact $n$-dimensional K\"ahler manifold $X$ such that the difference of intersection numbers $\{\alpha\}^n - n\,\{\alpha\}^{n-1}.\,\{\beta\}$ is…

We prove a conjecture of Boucksom-Demailly-P\u{a}un-Peternell, namely that on a projective manifold $X$ the cone of pseudoeffective classes in $H^{1,1}_{\mathbb{R}}(X)$ is dual to the cone of movable classes in $H^{n-1,n-1}_{\mathbb{R}}(X)$…

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…

Given a projective manifold $X$ equipped with an ample line bundle $L$, we show how to embed certain torus-invariant domains $D \subseteq\mathbb{C}^n$ into $X$ so that the Euclidean K\"ahler form on $D$ extends to a K\"ahler form on X lying…

Transcendental holomorphic Morse inequalities aim at characterizing the positivity of transcendental cohomology classes of type $(1,1)$. In this paper, we prove a weak version of Demailly's conjecture on transcendental Morse inequalities on…

Towards the boundary of the big cone, Newton-Okounkov bodies do not vary continuously and in fact the body of a boundary class is not well defined. Using the global Okounkov body one can nonetheless define a numerical invariant, the…

Let $X$ be a projective variety of dimension $n$ over an algebraically closed field of arbitrary characteristic and let $A, B, C$ be nef divisors on $X$. We show that for any integer $1\leq k\leq n-1$, $$ (B^k\cdot A^{n-k})\cdot (A^k\cdot…

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…

The purpose of the paper is to initiate the development of the theory of Newton Okounkov bodies of curve classes. Our denition is based on making a fundamental property of NewtonOkounkov bodies hold also in the curve case: the volume of the…

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…

Based on the work of Okounkov (\cite{Ok96}, \cite{Ok03}), Lazarsfeld and Musta\c t\u a (\cite{LM08}) and Kaveh and Khovanskii (\cite{KK08}) have independently associated a convex body, called the Okounkov body, to a big divisor on a smooth…

Let $L$ be a big holomorphic line bundle on a complex projective manifold $X.$ We show how to associate a convex function on the Okounkov body of $L$ to any continuous metric $\psi$ on $L.$ We will call this the Chebyshev transform of…

A Minkowski class is a closed subset of the space of convex bodies in Euclidean space Rn which is closed under Minkowski addition and non-negative dilatations. A convex body in Rn is universal if the expansion of its support function in…

Using $\delta$-invariants and Newton--Okounkov bodies, we derive the optimal volume upper bound for K\"ahler manifolds with positive Ricci curvature, from which we get a new characterization of the complex projective space.

We associate to a filtration of a graded linear series of a big line bundle a concave function on the Okounkov body whose law with respect to Lebesgue's measure describes the asymptotic distribution of the jumps of the filtration. As a…

Generalizing the notion of Newton polytope, we define the Newton-Okounkov body, respectively, for semigroups of integral points, graded algebras, and linear series on varieties. We prove that any semigroup in the lattice Z^n is…

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…

We prove that the existence of a finite Minkowski base for Okounkov bodies on a smooth projective variety with respect to an admissible flag implies rational polyhedrality of the global Okounkov body. As an application of this general…