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” [http://arXiv.org/abs/math.AG/0110051], an overview of which was published in [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 general type or a point. The fibration \(c_X\) is called the core of the manifold \(X\). The crucial fact is that the core has a canonical and functorial decomposition as a tower of vibrations with generic orbifold fibres which are either weakly rationally connected or of general type. The core thus gives a very simple synthetic view of the global structure of any manifold \(X\) entirely unified from the viewpoints of geometry, positivity of cotangent sheaves, topology, hyperbolicity, and arithmetic, as the author shows in great detail. The main technical ingredient in the proofs and constructions, which to a large extent also hold for complex analytic spaces, is the author’s orbifold version of S. Iitaka’s classical “\(C_{n,m}\)-additivity conjecture”. This result provides a very powerful tool, in particular with a view towards further applications to fibrations where the classical results from birational algebraic geometry are of little use only.
Another fundamental result of the treatise under review is the orbifold version of the extension theorem by S. Kobayashi and T. Ochiai [Invent. Math. 31, 7–16 (1975; Zbl 0331.32020)], which permits the author to solve a special case of the so-called “Conjecture \(\text{III}_H\)”, asserting that special manifolds are exactly the ones admitting a vanishing Kobayashi pseudo-metric.
The text of this utmost important and pioneering treatise is organized in 9 chapters, each of which is divided into several sections. Here is a brief table of contents:
1. Orbifold base of a fibration; 2. Special fibrations and general type fibrations; 3. The core; 4. Orbifold additivity; 5. Geometric consequences of additivity; 6. The decomposition of the core; 7. The fundamental group; 8. An orbifold generalization of the Kobayashi-Ochiai extension theorem; 9. Relationships with arithmetics and hyperbolicity.
The exposition is extremely thorough, comprehensive, detailed, rigorous and lucid. The author has illustrated his novel approach by numerous examples and remarks, and various conjectures have been woven in. These conjectures connect the various concepts and results presented here in a vivid manner, thereby providing concrete hints and plans for further research within this promising framework.
There is an appendix to the paper under review, published as a separate article in the same volume [Ann. Inst. Fourier 54, No. 3, 631–665 (2004; Zbl 1062.14015)]. This appendix provides some additional material on meromorphic quotients, as it was used in the course of the present treatise.