# zbMATH — the first resource for mathematics

Orbifolds, special varieties and classification theory. (English) Zbl 1062.14014
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.

##### MSC:
 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 14E05 Rational and birational maps 14G05 Rational points 14J40 $$n$$-folds ($$n>4$$) 32J27 Compact Kähler manifolds: generalizations, classification 32Q15 Kähler manifolds 32Q45 Hyperbolic and Kobayashi hyperbolic manifolds 32Q57 Classification theorems for complex manifolds
Full Text:
##### References:
  Solvable fundamental groups of algebraic varieties and Kähler manifolds, Comp. Math., 116, 173-193, (1999) · Zbl 0971.14020  Lang’s map and harris conjecture, Isr. J. Math., 101, 85-91, (1997) · Zbl 0933.14002  On the fibers of analytic mappings, Complex Analysis and Geometry, 45-102, (1993), Plenum Press · Zbl 0792.13005  Algebraic surfaces holomorphically dominable by $$\cal C^2,$$ Inv. Math., 139, 3, 617-659, (2000) · Zbl 0967.14025  Compact Complex Surfaces, Band 4, (1984), Springer Verlag · Zbl 0718.14023  Density of rational points on elliptic K3 surfaces, Asian Math. J., 4, 351-368, (2000) · Zbl 0983.14008  Special Elliptic Fibrations · Zbl 1069.14009  Espace analytique réduit des cycles analytiques complexes compacts d’un espace analytique de dimension finie, LNM, 482, 1-158, (1975) · Zbl 0331.32008  Annulation du $$H^1$$ pour LES fibrés en droites plats, Proceedings Bayreuth 1989, 1507, 1-15, (1992) · Zbl 0792.14006  Sur LES systèmes de fonctions uniformes satisfaisant l’équation d’une variété algébrique dont l’irrégularité dépasse la dimension, J. Math. Pures et Appliquées, 5, 19-66, (1926) · JFM 52.0373.04  Holomorphic tensors and vector bundles on projective varieties, Math. USSR. Izv., 13, 499-555, (1979) · Zbl 0439.14002  Courbes entières dans LES surfaces algébriques complexes, Séminaire Bourbaki, Exposé 881, (2000) · Zbl 1031.14014  Complex threefolds with non-trivial holomorphic 2-forms, J. Alg. Geom., 9, 223-264, (2000) · Zbl 0994.32016  Kähler threefolds covered by $$\cal C^3, (1999)$$  Ensembles de Green-lazarsfeld et quotients résolubles des groupes de Kähler, J. Alg. Geom., 10, 599-622, (2001) · Zbl 1072.14512  $$\cal G$$-connectedness of compact Kähler manifolds. I., Contemp. Math., 241, 85-96, (1999) · Zbl 0965.32021  Special Varieties and Classification Theory  Special varieties and classification theory: an overview, Acta Applicandae Mathematicae, 75, 29-49, (2003) · Zbl 1059.14047  Orbifolds, special varieties and classification theory: appendix., Ann. Inst. Fourier, 54, 3, 631-665, (2004) · Zbl 1062.14015  Réduction algébrique d’un morphisme faiblement Kählérien propre et applications, Math. Ann., 256, 157-189, (1980) · Zbl 0461.32010  Coréduction algébrique d’un espace analytique faiblement Kählérien compact, Inv. Math., 63, 187-223, (1981) · Zbl 0436.32024  Réduction d’Albanese d’un morphisme faiblement Kählérien propre et applications I, II, Comp. Math., 54, 373-416, (1985) · Zbl 0609.32008  An application of twistor theory to the non-hyperbolicity of certain compact symplectic Kähler manifolds, J. Reine. Angew. Math, 425, 1-7, (1992) · Zbl 0738.53037  Connexité rationnelle des variétés de Fano, Ann. Sc. ENS., 25, 539-545, (1992) · Zbl 0783.14022  Remarques sur le revêtement universel des variétés kählériennes compactes, Bull. S.M.F, 122, 2, 255-284, (1994) · Zbl 0810.32013  Fundamental group and positivity properties of cotangent bundles of compact Kähler manifolds, J. Alg. Geom., 4, 487-505, (1995) · Zbl 0845.32027  Negativity of compact curves in infinite étale covers of projective surfaces, J. Alg. Geom., 7, 673-693, (1998) · Zbl 0951.14009  Mordell-Weil groups of elliptic curves over $$C(t)$$ with $$p_g=0,$$ or $$1,$$ Duke Math. J., 49, 677-689, (1982) · Zbl 0503.14018  Double fibers and double covers: paucity in rational points, Acta Arithm., 79, 113-135, (1997) · Zbl 0863.14011  Arithmétique des variétés rationnelles et problèmes birationnels, Proc. ICM Berkeley, 641-653, (1986) · Zbl 0698.14060  Hyperbolicity of generic hypersurfaces in the projective 3-space, Amer. J. Math., 122, 515-546, (2000) · Zbl 0966.32014  On the equations $$z^m=F(x,y)$$ and $$A\,x^p+B\,y^q=C\,z^r,$$ Bull. London Math. Soc., 27, 513-543, (1995) · Zbl 0838.11023  Algebraic approximations of holomorphic maps from Stein domains to projective manifolds, Duke Math. J., 76, 2, 333-363, (1994) · Zbl 0861.32006  Kähler manifolds with numerically effective Ricci class, Comp. Math., 89, 2, 217-240, (1993) · Zbl 0884.32023  Compact Kähler manifolds with Hermitian semipositive anticanonical bundle, Comp. Math., 101, 217-224, (1996) · Zbl 1008.32008  Higher-Dimensional Algebraic Geometry, (2001), Springer Verlag · Zbl 0978.14001  Classification des variétés de dimension 3 et plus, Séminaire Bourbaki, Exposé 568, (198081) · Zbl 0481.14013  Smooth Four-Manifolds and Complex Surfaces, 27, (1994), Springer Verlag · Zbl 0817.14017  Log abundance for surfaces, Flips And Abundance for Algebraic Threefolds, 211, 127-137, (1992), Soc. Math. de France · Zbl 0807.14029  On Kähler fibre spaces over curves, J. Math. Soc. Jap., 30, 779-794, (1978) · Zbl 0393.14006  The general case of S. Lang’s conjecture, The Barsotti Symposium, 175-182, (1994), Academic Press, Cambridge Mass · Zbl 0823.14009  On the douady space of a complex space in $$\cal C,$$ Publ. RIMS, (1982)  Families of rationally connected varieties, (2001) · Zbl 1092.14063  Ein theorem der analytischen garbentheorie und die modulräume komplexer strukturen, Publ. Math. IHES, 5, 1-64, (1960) · Zbl 0100.08001  Berichtigung zu der arbeit ’ein theorem der analytischen garbentheorie und die modulräume komplexer strukturen’, Publ. Math., Inst. Hautes Étud. Sci., 16, 131-132, (1963) · Zbl 0113.29103  Mordell’s vermutung über rationale punkte auf algebraischen kurven und funktionenk\"orpern, Publ. Math. IHES, 25, 131-149, (1965) · Zbl 0137.40503  Jetmetriken und hyperbolische geometrie, Math. Z., 200, 149-168, (1989) · Zbl 0664.32020  Diophantine Geometry: an Introduction, 201, (2000), Springer-Verlag · Zbl 0948.11023  Rational points on quartics, Duke Math. J., 104, 477-500, (2000) · Zbl 0982.14013  Genera and classification of algebraic varieties, Sugaku, 24, 14-27, (1972)  On Algebraic Varieties whose Universal covering Manifolds are Complex Affine $$\cal C^3$$-space, (1973), Tokyo Kunikuniya · Zbl 0271.14015  Introduction to the minimal model problem, Adv. Studies in Pure Mathematics, 10, 283-360, (1987) · Zbl 0672.14006  Rationally connected varieties, J. Alg. Geom., 1, 3, 429-448, (1992) · Zbl 0780.14026  Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, 134, (1998) · Zbl 0926.14003  Rational curves on quasi-projective surfaces, Memoirs AMS, 669, (1999) · Zbl 0955.14031  Meromorphic mappings into compact complex spaces of general type, Inv. Math., 31, 7-16, (1975) · Zbl 0331.32020  On a characterization of an abelian variety, Comp. Math., 41, 355-359, (1980) · Zbl 0417.14033  On Bloch’s conjecture, Inv. Math., 57, 97-100, (1980) · Zbl 0569.32012  Characterization of abelian varieties, Comp. Math., 253-276, (1981) · Zbl 0471.14022  Brownian motion and harmonic functions on covering manifolds. an entropy approach., Sov. Math. Dokl., 33, 812-816, (1986) · Zbl 0615.60074  Pluricanonical forms, Contemp. Math., 241, 193-209, (1999) · Zbl 0972.14005  Intrinsic distances, measures and geometric function theory, Bull. AMS, 82, 357-416, (1976) · Zbl 0346.32031  Higher direct image sheaves of dualising sheaves, Ann. Math., 123, 11-42, (1986) · Zbl 0598.14015  Shafarevitch maps and plurigenera of algebraic varieties, Inv. Math., 113, 177-215, (1993) · Zbl 0819.14006  Rational curves on Algebraic varieties, 32, (1996), Springer Verlag · Zbl 0877.14012  Hyperbolic complex spaces, 318, (1998), Springer Verlag · Zbl 0917.32019  Rational points of abelian varieties over function fields, Amer. J. Math., 81, 95-118, (1959) · Zbl 0099.16103  Semi-continuity of Kodaira dimension, Bull. Amer. Math. Soc., 81, 459-460, (1975) · Zbl 0308.14002  Hyperbolic and Diophantine analysis, Bull. AMS, 14, 159-205, (1986) · Zbl 0602.14019  Number Theory III: Diophantine Geometry, vol. 60, (1991), Springer Verlag · Zbl 0744.14012  Compactness of the Chow scheme : applications to automorphisms and deformations of Kähler manifolds, 670, 140-186, (1975) · Zbl 0391.32018  Multiply marked Riemann surfaces and the Kobayashi pseudometric on algebraic manifolds, Preprint, (2001)  Rational points of algebraic curves over function fields, Izv. Akad. Nauk. SSSR, 27, 737-756, (1963) · Zbl 0178.55102  A finiteness property of varieties of general type, Math. Ann., 262, 101-123, (1983) · Zbl 0438.14011  On the Kodaira dimension of minimal threefolds, Math. Ann., 281, 325-332, (1988) · Zbl 0625.14023  Algebraic varieties and compact complex spaces, Actes du Congrès International des Mathématiciens, Nice, Vol. 2, 643-648, (1970) · Zbl 0232.14004  Flip theorem and the existence of minimal models for threefolds, J. AMS, 1, 117-253, (1988) · Zbl 0649.14023  Factorisation of semi-simple discrete representations of Kähler groups, Inv. Math., 110, 557-614, (1992) · Zbl 0823.53051  Geometric height inequality on varieties with ample cotangent bundle, J. Alg. Geom., 4, 2, 385-396, (1995) · Zbl 0873.14029  Branched Coverings and Algebraic Functions, 161, (1987), Longman Sc. and Tech. · Zbl 0706.14017  Invariance of Plurigenera of algebraic Varieties, (1998)  Holomorphic mappings into closed Riemann surfaces, Hiroshima Math. J., 6, 281-291, (1976) · Zbl 0338.32017  On holomorphic curves in algebraic varieties with ample irregularity, Inv. Math., 26, 83-96, (1976) · Zbl 0374.32006  Algebraic curves over function fields, Math. USSR Izv., 2, 1145-1170, (1968) · Zbl 0188.53003  Sur LES variétés Kählériennes compactes à classe de Ricci nef, Bull. Sci. Math., 122, 83-92, (1998) · Zbl 0946.53037  Oral communication, (2001)  Flat modules in algebraic geometry, Comp. Math., 24, 11-31, (1972) · Zbl 0244.14001  Homotopy of pullback of varieties, Nagoya Math. J., 102, 79-90, (1986) · Zbl 0564.14010  Applications of algebraic geometry to number theory, Proc. Symp. Pure Math., XX, 1-52, (1969) · Zbl 0228.14001  Séminaire: Géometrie Analytique (Deuxième partie), (1982)  3-fold log-flips, Izv. Akad. Nauk. Ser. Mat., 56, 105-203, (1992)  Extension of Pluricanonical Sections with Plurisubharmonic Weights, (2003), Springer Verlag  Complex analyticity of harmonic maps, J. Diff. Geom., 17, 55-138, (1982) · Zbl 0497.32025  Invariance of plurigenera, Inventiones Math., 134, 661-673, (1998) · Zbl 0955.32017  Subspaces of moduli spaces of rank one local systems, Ann. Sc. ENS., 26, 361-401, (1993) · Zbl 0798.14005  Classification Theory of Algebraic Varieties and Compact Complex Manifolds, 439, (1975), Springer Verlag · Zbl 0299.14007  Die additivität der Kodaira dimension für projektive faserräume über varietäten des allgemeinen typs, J. Reine Angew. Math., 330, 132-142, (1982) · Zbl 0466.14009  Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces, Vol. I, 329-353, (1983), North-Holland · Zbl 0513.14019  Vanishing theorems and and positivity of algebraic fibre spaces, J. Proc. Int. Congr. Math. Berkeley, (1986) · Zbl 0685.14013  Intrinsic Pseudovolume Forms, (2002)  On projective manifolds with nef anticanonical bundles, J. Reine. Angew. Math., 478, 57-60, (1996) · Zbl 0855.14007  Factorisation theorems for representations of fundamental groups of algebraic varieties, (1997)  Representations of fundamental groups of algebraic varieties, 1708, (1999), Springer Verlag · Zbl 0987.14014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.