Orbifolds, special varieties and classification theory.
type: Item
instance of: scholarly article
title: Orbifolds, special varieties and classification theory.
zbMATH Open document ID: 1062.14014
publication date: 2004-12-20T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/116120
review text: In this comprehensive treatise, the author develops the foundations of a substantially new approach towards a better understanding of the geometry, arithmetic, and classification of compact Kähler manifolds. His approach, grown out of diverse foregoing attempts by himself and other researchers in the field, appears now to be remarkably general, conceptionally natural, and methodologically powerful. Actually, the present work is an expanded version of the author's recent electronic paper ``Special Varieties and Classification Theory'' [\texttt{http://arXiv.org/abs/math.AG/0110051}], an overview of which was published in [\textit{F. Campana}, Acta Appl. Math., 75, 1--3, 29--49 (2003; Zbl 1059.14047)]. The new approach to the general problem of classification of algebraic varieties is based on the following novel concepts and constructions. (1) The orbifold base of a fibration. A fibration \(X@>f>> Y\) is here a surjective meromorphic map with irreducible generic fibres between irreducible compact complex analytic spaces \(X\) and \(Y\). By means of the \(\mathbb{Q}\)-divisor of multiple fibres, the base space \(Y\) is given the structure of an orbifold, and it is in view of this particular structure that \(Y\) is then called the orbifold base of the fibration \(f\). As an orbifold, \(Y\) comes with special orbifold invariants (e.g., Kodaira dimension, canonical ring, etc.) which are thoroughly analyzed. (2) Special fibrations and special manifolds (or orbifolds). The author defines special manifolds as being the compact Kähler manifolds with no meromorphic map onto an orbifold of general type. In this context, he proves that rationally connected Kähler manifolds or Kähler manifolds with zero Kodaira dimension, respectively, are special manifolds. (3) The core of a manifold. For any manifold \(X\) the author shows that there is a unique fibration \(c_X: X\to C(X)\) such that the general fibre \(F\) of \(c_X\) is special and, moreover, its orbifold base is either of gen…
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 14E05
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14J40
Mathematics Subject Classification ID: 32J27
Mathematics Subject Classification ID: 32Q15
Mathematics Subject Classification ID: 32Q45
Mathematics Subject Classification ID: 32Q57
zbMATH DE Number: 2123576
zbMATH Keywords: Kodaira dimension
zbMATH Keywords: Kähler manifolds
zbMATH Keywords: rational connectedness
zbMATH Keywords: fibrations
zbMATH Keywords: varieties of general type
zbMATH Keywords: hyperbolic manifolds
zbMATH Keywords: arithmetic geometry
zbMATH Keywords: birational geometry
author: Frédéric Campana
reviewed by: Werner Kleinert
MaRDI profile type: Publication
OpenAlex ID: W1257863412
cites work: Solvable fundamental groups of algebraic varieties and K\"ahler manifolds
cites work: Lang maps and Harris's conjecture
cites work: Q3134931
cites work: Algebraic surfaces holomorphically dominable by \(\mathbb{C}^2\)
cites work: Q5749404
cites work: Density of rational points on elliptic \(K3\) surfaces
cites work: Q4650871
cites work: Q4097533
cites work: Q4027233
cites work: HOLOMORPHIC TENSORS AND VECTOR BUNDLES ON PROJECTIVE VARIETIES
cites work: Q4422840
cites work: Q4946476
cites work: Green-Lazarsfeld sets and resolvable quotients of Kähler groups
cites work: Q4941083
cites work: Special varieties and classification theory: An overview
cites work: Orbifolds, special varieties and classification theory: appendix.
cites work: Reduction algébrique d'un morphisme faiblement Kählerien propre et applications
cites work: Coreduction algébrique d'un espace analytique faiblement Kählerien compact
cites work: Q3749320
cites work: An application of twistor theory to the non-hyperbolicity of certain compact symplectic Kähler manifolds.
cites work: Connexité rationnelle des variétés de Fano
cites work: Remarques sur le revêtement universel des variétés kählériennes compactes
cites work: Q4838948
cites work: Q4267940
cites work: Mordell-Weil groups of elliptic curves over C(t) with pg=O or 1
cites work: Q4888847
cites work: Q3475400
cites work: Hyperbolicity of generic surfaces of high degree in projective 3-space
cites work: On the Equations zm = F (x, y ) and Axp + Byq = Czr
cites work: Algebraic approximations of holomorphic maps from Stein domains to projective manifolds
cites work: Q4285666
cites work: Q4880508
cites work: Higher-dimensional algebraic geometry
cites work: Q3937520
cites work: Q4289665
cites work: Q4280963
cites work: On Kähler fiber spaces over curves
cites work: Q4322758
cites work: Families of rationally connected varieties
cites work: Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer Strukturen
cites work: Berichtigung zu der Arbeit 'Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer Strukturen'
cites work: Mordells Vermutung über rationale Punkte auf algebraischen Kurven und Funktionenkörper
cites work: Jetmetriken und hyperbolische Geometrie. (On jet metrics and hyperbolic geometry)
cites work: Q4486426
cites work: Rational points on quartics
cites work: Q3215358
cites work: Q3825176
cites work: Q5287412
cites work: Q4212603
cites work: Rational curves on quasi-projective surfaces
cites work: Meromorphic mappings onto compact complex spaces of general type

Orbifolds, special varieties and classification theory. (cites work)
cites work: Q3207118

Orbifolds, special varieties and classification theory. (cites work)
cites work: On Bloch's conjecture

Orbifolds, special varieties and classification theory. (cites work)
cites work: Q3924325

Orbifolds, special varieties and classification theory. (cites work)
cites work: Q4722961

Orbifolds, special varieties and classification theory. (cites work)
cites work: Q4941090

Orbifolds, special varieties and classification theory. (cites work)
cites work: Intrinsic distances, measures and geometric function theory

Orbifolds, special varieties and classification theory. (cites work)
cites work: Higher direct images of dualizing sheaves. I

Orbifolds, special varieties and classification theory. (cites work)
cites work: Shafarevich maps and plurigenera of algebraic varieties

Orbifolds, special varieties and classification theory. (cites work)
cites work: Q4859907

Orbifolds, special varieties and classification theory. (cites work)
cites work: Q4393442

Orbifolds, special varieties and classification theory. (cites work)
cites work: Rational Points of Abelian Varieties Over Function Fields

Orbifolds, special varieties and classification theory. (cites work)
cites work: Semicontinuity of Kodaira dimension

Orbifolds, special varieties and classification theory. (cites work)
cites work: Hyperbolic and Diophantine analysis

Orbifolds, special varieties and classification theory. (cites work)
cites work: Number theory III: Diophantine geometry

Orbifolds, special varieties and classification theory. (cites work)
cites work: Q4172306

Orbifolds, special varieties and classification theory. (cites work)
cites work: Rational points of algebraic curves over function fields

Orbifolds, special varieties and classification theory. (cites work)
cites work: A finiteness property of varieties of general type

Orbifolds, special varieties and classification theory. (cites work)
cites work: On the Kodaira dimension of minimal threefolds

Orbifolds, special varieties and classification theory. (cites work)
cites work: Q5639789

Orbifolds, special varieties and classification theory. (cites work)
cites work: Flip Theorem and the Existence of Minimal Models for 3-Folds

Orbifolds, special varieties and classification theory. (cites work)
cites work: Factorization of semisimple discrete representations of Kähler groups

Orbifolds, special varieties and classification theory. (cites work)
cites work: Geometric height inequality on varieties with ample cotangent bundles

Orbifolds, special varieties and classification theory. (cites work)
cites work: Q3996764

Orbifolds, special varieties and classification theory. (cites work)
cites work: Holomorphic mappings into closed Riemann surfaces

Orbifolds, special varieties and classification theory. (cites work)
cites work: On holomorphic curves in algebraic varieties with ample irregularity

Orbifolds, special varieties and classification theory. (cites work)
cites work: ALGEBRAIC CURVES OVER FUNCTION FIELDS. I

Orbifolds, special varieties and classification theory. (cites work)
cites work: On the numerically effective compact Kähler manifolds of Ricci class

Orbifolds, special varieties and classification theory. (cites work)
cites work: Q5655476

Orbifolds, special varieties and classification theory. (cites work)
cites work: Homotopy groups of pullbacks of varieties

Orbifolds, special varieties and classification theory. (cites work)
cites work: Q5635540

Orbifolds, special varieties and classification theory. (cites work)
cites work: Complex-analyticity of harmonic maps, vanishing and Lefschetz theorems

Orbifolds, special varieties and classification theory. (cites work)
cites work: Invariance of plurigenera

Orbifolds, special varieties and classification theory. (cites work)
cites work: Subspaces of moduli spaces of rank one local systems

Orbifolds, special varieties and classification theory. (cites work)
cites work: Classification theory of algebraic varieties and compact complex spaces. Notes written in collaboration with P. Cherenack

Orbifolds, special varieties and classification theory. (cites work)
cites work: Die Additivität der Kodaira Dimension für projektive Faserräume über Varietäten des allgemeinen Typs.

Orbifolds, special varieties and classification theory. (cites work)
cites work: Q3658114

Orbifolds, special varieties and classification theory. (cites work)
cites work: Q4735262

Orbifolds, special varieties and classification theory. (cites work)
cites work: On projective manifolds with nef anticanonical bundles.

Orbifolds, special varieties and classification theory. (cites work)
cites work: Representations of fundamental groups of algebraic varieties

Orbifolds, special varieties and classification theory. (DOI)
DOI: 10.5802/AIF.2027

Orbifolds, special varieties and classification theory. (published in)
published in: Annales de l’institut Fourier