Birational positivity in dimension 4
type: Item
instance of: scholarly article
title: Birational positivity in dimension 4
zbMATH Open document ID: 1326.14093
author: Behrouz Taji
publication date: 2015-01-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/1304.6686
full work available at URL: http://www.numdam.org/item
review text: The article under review studies the relationship between the Kodaira dimension \(\kappa (X)\) and a more general notion \(\kappa^+ (X)\) introduced by \textit{F. Campana} [J. Algebr. Geom. 4, No. 3, 487--502 (1995; Zbl 0845.32027)]. Let \(X\) be a projective manifold of dimension \(n\) and let \(L\) be a holomorphic line bundle on \(X\). The Iitaka dimension of \(L\) is defined as the largest number \(k\in\mathbb{N}\) such that \[ \varlimsup_{m\rightarrow +\infty} \frac{h^0 (X, L^{\otimes m})}{m^k} > 0 . \] The Kodaira dimension \(\kappa (X) := \kappa (K_X)\), and \(\kappa^+ (X)\) is defined by \[ \kappa^+ (X) := \max \{ \kappa (\det \mathcal{F}) | \mathcal{F} \text{ is a coherent subsheaf of }\Omega_X ^p, \text{ for some }p\} . \] It was conjectured by Campana [loc. cit.] that \(\kappa (X) =\kappa^+ (X)\) when \(\kappa (X) \geq 0\). In the same article, Campana proved that, if the good minimal model conjecture holds for dimension \(n\), then we have \(\kappa (X) =\kappa^+ (X)\). The main result of the article under review is concerned with replacing the abundance conjecture for dimension \(n\) with the abundance conjecture in lower dimension. More precisely, the main theorem of the article is: Let \(X\) be a projective manifold of dimension \(n\). Assume that the good minimal model conjecture holds for terminal varieties with zero Kodaira dimension up to dimension \(n-m\). If \(\kappa (X) \geq m-1\) then \(\kappa (X) =\kappa ^+ (X)\). One of the key ingredient of the proof is a result of \textit{F. Campana} and \textit{T. Peternell} [Bull. Soc. Math. Fr. 139, No. 1, 41--74 (2011; Zbl 1218.14030), with an appendix by Matei Toma] about the pseudo-effectivity of quotients of \(\Omega_X ^p\) for non-uniruled projective manifolds. As an application, we know that for \(4\)-dimensional projective manifolds with \(\kappa (X) \geq 0\), we have \(\kappa (X) =\kappa ^+ (X)\). As another interesting application, combining this result with a previous result of Campana, we know…
Mathematics Subject Classification ID: 14J35
Mathematics Subject Classification ID: 14E30
zbMATH DE Number: 6387272
zbMATH Keywords: Kodaira dimension
zbMATH Keywords: varieties of Kodaira dimension zero
zbMATH Keywords: minimal model theory
MaRDI profile type: Publication
OpenAlex ID: W2333752357
arXiv ID: 1304.6686
cites work: The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension
cites work: Connexité rationnelle des variétés de Fano
cites work: Q4838948
cites work: Orbifolds, special varieties and classification theory.
cites work: Orbifoldes géométriques spéciales et classification biméromorphe des variétés kählériennes compactes
cites work: Geometric stability of the cotangent bundle and the universal cover of a projective manifold
cites work: Subsheaves of the cotangent bundle
cites work: Families of rationally connected varieties
cites work: Q3136464
cites work: Q5287412
cites work: Q4212603
cites work: Q3792786
cites work: Q3808216
cites work: Q5655476
cites work: Calabi's conjecture and some new results in algebraic geometry
DOI: 10.5802/AIF.2845
published in: Annales de l’institut Fourier
Recommended article: BIRATIONAL PROPERTIES OF A SURFACE OF DEGREE 4 IN $ \mathrm{P}^4_k$
Recommended article: Characterization of the 4-canonical birationality of algebraic threefolds
Recommended article: Birational geometry of rational quartic surfaces
Recommended article: Birational isomorphisms of four-dimensional quintics
Recommended article: Birational geometry of quadrics
Recommended article: On the birational geometry of Fano 4-folds
Recommended article: Characterization of the 4-canonical birationality of algebraic threefolds. II
Recommended article: On an open problem of characterizing the birationality of \(\mathrm{4K}\)
Recommended article: Birational Geometry of the Space of Complete Quadrics
description: scientific article