Orbifolds, special varieties and classification theory

Campana, Frédéric

Annales de l'Institut Fourier, Tome 54 (2004) no. 3, pp. 499-630.

Le présent article décrit à l’aide de fibrations fonctorielles intrinsèques, la structure géométrique (et conjecturalement la pseudométrique de Kobayashi ainsi que l’arithmétique dans le cas projectif) des variétés Kählériennes compactes. Les variétés spéciales sont tout d’abord définies comme étant les variétés Kählériennes compactes ne possédant pas d’application méromorphe surjective sur une orbifolde de type général, la structure d’orbifolde de la base provenant du diviseur des fibres multiples. On montre que les variétés Kählériennes compactes qui sont, soit rationnellement connexes, soit à dimension de Kodaira nulle sont spéciales. Nous construisons ensuite fonctoriellement, pour toute telle variété $X$, l’ unique fibration $c_X:X\to C\left(X\right)$ (le coeur de $X$) dont les fibres sont spéciales et dont la base orbifolde est soit de type général, soit un point (ce dernier cas se produisant si et seulement si $X$ est spéciale). Le coeur de $X$ est ensuite canoniquement décomposé comme une tour de fibrations à fibres soit $\kappa$-rationnellement engendrées (une version faible de la connéxité rationnelle), soit à dimension de Kodaira nulle. En particulier, les variétés spéciales sont donc de telles tours de fibrations. L’ingrédient technique essentiel des démonstrations est une version orbifolde de la conjecture $C_{n,m}$ d’Iitaka, établie lorsque la base orbifolde est de type général. Le coeur de $X$ permet de donner une description qualitative conjecturale très simple de la pseudométrique de Kobayashi de $X$ et de la distribution de ses points $K$-rationnels (si $X$ est projective, définie sur le corps $K$ de type fini sur $\mathbb{Q}$), description se réduisant à celle de Lang lorsque $X$ est de type général.

This article gives a description, by means of functorial intrinsic fibrations, of the geometric structure (and conjecturally also of the Kobayashi pseudometric, as well as of the arithmetic in the projective case) of compact Kähler manifolds. We first define special manifolds as being the compact Kähler manifolds with no meromorphic map onto an orbifold of general type, the orbifold structure on the base being given by the divisor of multiple fibres. We next show that rationally connected Kähler manifolds or Kähler manifolds with zero Kodaira dimension are special. For any $X$, we then construct the unique functorial fibration $c_X:X\to C\left(X\right)$ (called its core), such that its general fibre is special, and its orbifold base is either of general type, or a point (the last case occuring if and only if $X$ is special). We next show that the core has a canonical and functorial decomposition as a tower of fibrations with generic (orbifold) fibres either $\kappa$-rationally generated (a weak version of rational connectedness), or with zero Kodaira dimension. In particular, special manifolds are thus canonically towers of such fibrations. The main technical ingredient in the proofs is an orbifold version of Iitaka’s $C_{n,m}$ additivity conjecture, proved here when the orbifold base is of general type. The core of $X$ also gives a very simple conjectural qualitative of description of both the Kobayashi pseudometric and the distribution of its $K$-rational points (if $X$ is projective), description which reduces to Lang’s conjectures when $X$ is of general type.

Classification : 14C30, 14D10, 14E05, 14G05, 14J40, 32J27, 32Q15, 32Q57
Keywords: canonical bundle, Kodaira dimension, orbifold, Kähler manifold, rational connectedness, fibration, Albanese map, Kobayashi pseudometric, rational point.
Mot clés : fibré canonique, dimension de Kodaira, orbifolde, variété Kählérienne compacte, connexité rationnelle, fibration, morphisme d'Albanese, pseudométrique de Kobayashi, points rationnels.

[A-N99] D. Arapura; M. Nori Solvable Fundamental Groups of Algebraic Varieties and Kähler Manifolds, Comp. Math., Volume 116 (1999), pp. 173-193 | MR | Zbl

[Ab97] D. Abramovich Lang's map and Harris conjecture, Isr. J. Math., Volume 101 (1997), pp. 85-91 | MR | Zbl

[B-F 93] J. Bingener; H. Flenner; V. Ancona and A. Silva Eds. On the fibers of analytic mappings, Complex Analysis and Geometry (1993), pp. 45-102 | Zbl

[B-L00] G. Buzzard; S. Lu Algebraic surfaces holomorphically dominable by ${𝒞}^2$, Inv. Math., Volume 139 (2000) no. 3, pp. 617-659 | MR | Zbl

[B-P-V84] W. Barth; C. Peters; A. Van de Ven Compact Complex Surfaces, Erg. der Math. u. Ihrer Grenzgebiete. 3 Folge, Band 4, Springer Verlag, 1984 | MR | Zbl

[B-T00] F.A. Bogomolov; Y. Tschinkel Density of rational points on elliptic K3 surfaces, Asian Math. J., Volume 4 (2000), pp. 351-368 | MR | Zbl

[B-T02] F.A. Bogomolov; Y. Tschinkel Special Elliptic Fibrations (e-print, math.AG/0303044)

[Ba75] D. Barlet Espace analytique réduit des cycles analytiques complexes compacts d'un espace analytique de dimension finie, LNM, Volume 482 (1975), pp. 1-158 | MR | Zbl

[Be89] A. Beauville Annulation du $H^1$ pour les fibrés en droites plats, Proceedings Bayreuth 1989 (LNM), Volume 1507 (1992), pp. 1-15 | Zbl

[Bl26] S. Bloch Sur les systèmes de fonctions uniformes satisfaisant l'équation d'une variété algébrique dont l'irrégularité dépasse la dimension, J. Math. Pures et Appliquées, Volume 5 (1926), pp. 19-66 | JFM | Numdam

[Bo79] F.A. Bogomolov Holomorphic Tensors and vector Bundles on Projective Varieties, Math. USSR. Izv., Volume 13 (1979), pp. 499-555 | Zbl

[Br00] M. Brunella Courbes entières dans les surfaces algébriques complexes, Séminaire Bourbaki, Volume Exposé 881 (Novembre 2000) | Numdam | Zbl

[C-P00] F. Campana; T. Peternell Complex Threefolds with Non-trivial holomorphic 2-Forms, J. Alg. Geom., Volume 9 (2000), pp. 223-264 | MR | Zbl

[C-Z99] F. Campana; Q. Zhang Kähler threefolds covered by ${𝒞}^3$ (1999) (Preprint)

[Ca01] F. Campana Ensembles de Green-Lazarsfeld et quotients résolubles des groupes de Kähler, J. Alg. Geom., Volume 10 (2001), pp. 599-622 | MR | Zbl

[Ca01] F. Campana $𝒢$-connectedness of compact Kähler manifolds. I., Contemp. Math., Volume 241 (1999), pp. 85-96 | MR | Zbl

[Ca01'] F. Campana Special Varieties and Classification Theory (e-print, math AG/0110051)

[Ca03] F. Campana Special Varieties and Classification Theory: An overview, Acta Applicandae Mathematicae, Volume 75 (2003), pp. 29-49 | MR | Zbl

[Ca04] F. Campana Orbifolds, special varieties and classification theory: appendix., Ann. Inst. Fourier, Volume 54 (2004) no. 3, pp. 631-665 | Numdam | MR | Zbl

[Ca80] F. Campana Réduction algébrique d'un morphisme faiblement Kählérien propre et applications, Math. Ann., Volume 256 (1980), pp. 157-189 | MR | Zbl

[Ca81] F. Campana Coréduction algébrique d'un espace analytique faiblement Kählérien compact, Inv. Math., Volume 63 (1981), pp. 187-223 | MR | Zbl

[Ca85] F. Campana Réduction d'Albanese d'un morphisme faiblement Kählérien propre et applications I, II, Comp. Math., Volume 54 (1985), pp. 373-416 | Numdam | MR | Zbl

[Ca92] F. Campana An application of twistor theory to the non-hyperbolicity of certain compact symplectic Kähler manifolds, J. Reine. Angew. Math, Volume 425 (1992), pp. 1-7 | MR | Zbl

[Ca92'] F. Campana Connexité rationnelle des variétés de Fano, Ann. Sc. ENS., Volume 25 (1992), pp. 539-545 | Numdam | MR | Zbl

[Ca94] F. Campana Remarques sur le revêtement universel des variétés kählériennes compactes, Bull. S.M.F, Volume 122 (1994) no. 2, pp. 255-284 | Numdam | MR | Zbl

[Ca95] F. Campana Fundamental Group and Positivity Properties of Cotangent Bundles of Compact Kähler Manifolds, J. Alg. Geom., Volume 4 (1995), pp. 487-505 | MR | Zbl

[Ca98] F. Campana Negativity of compact curves in infinite étale covers of projective surfaces, J. Alg. Geom., Volume 7 (1998), pp. 673-693 | MR | Zbl

[Co82] D. Cox Mordell-Weil groups of elliptic curves over $C\left(t\right)$ with $p_g=0$, or $1$, Duke Math. J., Volume 49 (1982), pp. 677-689 | MR | Zbl

[CT-S-S97] J. L. Colliot-Thélène; A. Skorobogatov; P. Swinnerton-Dyer Double fibers and double covers: paucity in rational points, Acta Arithm., Volume 79 (1997), pp. 113-135 | MR | Zbl

[CT86] J. L. Colliot-Thélène Arithmétique des variétés rationnelles et problèmes birationnels, Proc. ICM Berkeley (1986), pp. 641-653 | MR | Zbl

[D-E00] J.-P. Demailly; J. El Goul Hyperbolicity of generic hypersurfaces in the projective 3-space, Amer. J. Math., Volume 122 (2000), pp. 515-546 | MR | Zbl

[D-G95] H. Darmon; A. Granville On the equations $z^m=F(x,y)$ and $A{x}^p+B{y}^q=C{z}^r$, Bull. London Math. Soc., Volume 27 (1995), pp. 513-543 | MR | Zbl

[D-L-S94] J.-P. Demailly; L. Lempert; B. Shiffman Algebraic approximations of holomorphic maps from Stein domains to projective manifolds, Duke Math. J., Volume 76 (1994) no. 2, pp. 333-363 | MR | Zbl

[D-P-S93] J.-P. Demailly; T. Peternell; M. Schneider Kähler manifolds with numerically effective Ricci class, Comp. Math., Volume 89 (1993) no. 2, pp. 217-240 | Numdam | MR | Zbl

[D-P-S96] J.-P. Demailly; T. Peternell; M. Schneider Compact Kähler Manifolds with hermitian semipositive anticanonical bundle, Comp. Math., Volume 101 (1996), pp. 217-224 | Numdam | MR | Zbl

[De01] O. Debarre Higher-Dimensional Algebraic Geometry, Universitext, Springer Verlag, 2001 | MR | Zbl

[Es80] H. Esnault Classification des variétés de dimension 3 et plus, Séminaire Bourbaki, Volume Exposé 568 (1980/81) | Numdam | Zbl

[F-M94] R. Friedman; J. Morgan Smooth Four-Manifolds and Complex Surfaces, Erg. der Mathem., 27, Springer Verlag, 1994 | MR | Zbl

[F-MK92] L. Fong; J. Mc; Kernan; J. Kollár Ed. Log Abundance For Surfaces, Flips And Abundance for Algebraic Threefolds (Astérisque), Volume 211 (1992), pp. 127-137 | Zbl

[F78] T. Fujita On Kähler fibre spaces over curves, J. Math. Soc. Jap., Volume 30 (1978), pp. 779-794 | MR | Zbl

[Fa94] G. Faltings; W. Messing and V. Cristante Eds The general case of S. Lang's Conjecture, The Barsotti Symposium (1994), pp. 175-182 | Zbl

[Fu82] A. Fujiki On the Douady Space of a complex space in $𝒞$, Publ. RIMS (1982)

[G-H-S01] T. Graber; M. Harris; J. Starr Families of rationally connected varieties (2001) (Preprint) | Zbl

[Gr60] H. Grauert Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer Strukturen, Publ. Math. IHES, Volume 5 (1960), pp. 1-64 | Numdam | MR | Zbl

[Gr60] H. Grauert Berichtigung zu der Arbeit 'Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer Strukturen', Publ. Math., Inst. Hautes Étud. Sci., Volume 16 (1963), pp. 131-132 | Numdam | MR | Zbl

[Gr65] H. Grauert Mordell's Vermutung über rationale Punkte auf algebraischen Kurven und Funktionenk\"orpern, Publ. Math. IHES, Volume 25 (1965), pp. 131-149 | Numdam | MR | Zbl

[Gr89] H. Grauert Jetmetriken und hyperbolische Geometrie, Math. Z., Volume 200 (1989), pp. 149-168 | MR | Zbl

[H-S00] M. Hindry; J. Silverman Diophantine Geometry: an Introduction, GTM, 201, Springer-Verlag, 2000 | MR | Zbl

[H-T00] J. Harris; Y. Tschinkel Rational points on quartics, Duke Math. J., Volume 104 (2000), pp. 477-500 | MR | Zbl

[Ii72] S. Iitaka Genera and Classification of Algebraic Varieties, Sugaku, Volume 24 (1972), pp. 14-27 | MR

[Ii73] S. Iitaka On Algebraic Varieties whose Universal covering Manifolds are Complex Affine ${𝒞}^3$-space, Number Theory, algebr. Geom., commut. Algebra in Honor of Yasuo Akizuki, Tokyo Kunikuniya, 1973 | Zbl

[K-M-M87] Y. Kawamata; K. Matsuda; K. Matsuki Introduction to the Minimal Model Problem, Adv. Studies in Pure Mathematics, Volume 10 (1987), pp. 283-360 | MR | Zbl

[K-M-M92] J. Kollár; Y. Miyaoka; S. Mori Rationally connected Varieties, J. Alg. Geom., Volume 1 (1992) no. 3, pp. 429-448 | MR | Zbl

[K-M98] J. Kollár; S. Mori Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics, Volume 134 (1998) | MR | Zbl

[K-MK97] S. Keel; S. McKiernan Rational curves on quasi-projective surfaces, Memoirs AMS, Volume 669 (1999) | Zbl

[K-O75] S. Kobayashi; T. Ochiai Meromorphic mappings into compact complex spaces of general type, Inv. Math., Volume 31 (1975), pp. 7-16 | MR | Zbl

[K-V80] Y. Kawamata; E. Viehweg On a characterization of an Abelian Variety, Comp. Math., Volume 41 (1980), pp. 355-359 | Numdam | MR | Zbl

[Ka80] Y. Kawamata On Bloch's Conjecture, Inv. Math., Volume 57 (1980), pp. 97-100 | MR | Zbl

[Ka81] Y. Kawamata Characterization of Abelian Varieties, Comp. Math. (1981), pp. 253-276 | Numdam | MR | Zbl

[Ka86] V.A. Kaimanovitch Brownian Motion and Harmonic Functions on Covering Manifolds. An Entropy Approach., Sov. Math. Dokl., Volume 33 (1986), pp. 812-816 | Zbl

[Ka99] Y. Kawamata Pluricanonical Forms, Contemp. Math., Volume 241 (1999), pp. 193-209 | MR | Zbl

[Ko76] S. Kobayashi Intrinsic distances, measures and geometric function theory, Bull. AMS, Volume 82 (1976), pp. 357-416 | MR | Zbl

[Ko86] J. Kollár Higher direct image sheaves of dualising sheaves, Ann. Math., Volume 123 (1986), pp. 11-42 | MR | Zbl

[Ko93] J. Kollár Shafarevitch maps and plurigenera of Algebraic Varieties, Inv. Math., Volume 113 (1993), pp. 177-215 | MR | Zbl

[Ko96] J. Kollár Rational curves on Algebraic varieties, Erg. der Math. und ihrer Grenz, 32, Springer Verlag, 1996 | MR | Zbl

[Ko98] S. Kobayashi Hyperbolic complex spaces, Grundlehren der Math. Wiss., 318, Springer Verlag, 1998 | MR | Zbl

[L-N59] S. Lang; A. Néron Rational points of Abelian Varieties over Function Fields, Amer. J. Math., Volume 81 (1959), pp. 95-118 | MR | Zbl

[L-S75] D. Liebermann; E. Sernesi Semi-continuity of Kodaira dimension, Bull. Amer. Math. Soc., Volume 81 (1975), pp. 459-460 | Zbl

[La86] S. Lang Hyperbolic and Diophantine Analysis, Bull. AMS, Volume 14 (1986), pp. 159-205 | MR | Zbl

[La91] S. Lang Number Theory III: Diophantine Geometry, EMS, vol. 60, Springer Verlag, 1991 | MR | Zbl

[Li75] D. Liebermann; F. Norguet Ed. Compactness of the Chow Scheme : Applications to automorphisms and deformations of Kähler Manifolds (Lecture Notes in Mathematics), Volume 670 (1975), pp. 140-186 | Zbl

[Lu01] S. Lu Multiply marked Riemann surfaces and the Kobayashi pseudometric on Algebraic manifolds, Preprint (2001)

[Ma63] Y. Manin Rational Points of Algebraic Curves over Function Fields, Izv. Akad. Nauk. SSSR, Volume 27 (1963), pp. 737-756 | MR | Zbl

[Ma83] K. Maehara A finiteness Property of Varieties of General Type, Math. Ann., Volume 262 (1983), pp. 101-123 | MR | Zbl

[Mi88] Y. Miyaoka On the Kodaira Dimension of Minimal Threefolds, Math. Ann., Volume 281 (1988), pp. 325-332 | MR | Zbl

[Mo70] B. Moishezon Algebraic Varieties and Compact Complex Spaces, Actes du Congrès International des Mathématiciens, Nice, Volume Vol. 2 (1970), pp. 643-648 | MR | Zbl

[Mo88] S. Mori Flip Theorem and the existence of Minimal Models for Threefolds, J. AMS, Volume 1 (1988), pp. 117-253 | MR | Zbl

[Mo92] N. Mok Factorisation of Semi-Simple Discrete Representations of Kähler Groups, Inv. Math., Volume 110 (1992), pp. 557-614 | Zbl

[Mo95] A. Moriwaki Geometric Height inequality on Varieties with ample cotangent bundle, J. Alg. Geom., Volume 4 (1995) no. 2, pp. 385-396 | MR | Zbl

[Na87] M. Namba Branched Coverings and Algebraic Functions, Pitman Research Notes, 161, Longman Sc. and Tech., 1987 | MR | Zbl

[Na98] N. Nakayama Invariance of Plurigenera of algebraic Varieties (1998) (RIMS Preprint) | MR

[Nog76] J. Noguchi Holomorphic mappings into closed Riemann Surfaces, Hiroshima Math. J., Volume 6 (1976), pp. 281-291 | MR | Zbl

[Oc77] T. Ochiai On holomorphic curves in algebraic varieties with ample irregularity, Inv. Math., Volume 26 (1976), pp. 83-96 | MR | Zbl

[Pa68] A. N. Parshin Algebraic Curves over Function Fields, Math. USSR Izv., Volume 2 (1968), pp. 1145-1170 | Zbl

[Pa98] M. Paun Sur les variétés Kählériennes compactes à classe de Ricci nef, Bull. Sci. Math., Volume 122 (1998), pp. 83-92 | MR | Zbl

[Pe01] E. Peyre Oral communication (July 2001)

[R 74] M. Raynaud Flat Modules in algebraic geometry, Comp. Math., Volume 24 (1972), pp. 11-31 | Numdam | MR | Zbl

[S-V86] A. Sommese; A. Van de Ven Homotopy of Pullback of Varieties, Nagoya Math. J., Volume 102 (1986), pp. 79-90 | MR | Zbl

[SD69] H. Swinnerton-Dyer Applications of Algebraic Geometry to number Theory, Proc. Symp. Pure Math., Volume XX (1969), pp. 1-52 | MR | Zbl

[SGAN 82] D. Barlet; F. Campana; C. Sabbah Séminaire: Géometrie Analytique (Deuxième partie) (1982) (Prépublication de l'Institut Elie Cartan, 5)

[Sh92] V. Shokurov 3-fold log-flips, Izv. Akad. Nauk. Ser. Mat., Volume 56 (1992), pp. 105-203 | MR

[Si02] Y.T. Siu Extension of Pluricanonical Sections with Plurisubharmonic Weights, "Complex Geometry", a collection of papers dedicated to H. Grauert., Springer Verlag, 2003

[Si82] Y.T. Siu Complex Analyticity of Harmonic Maps, J. Diff. Geom., Volume 17 (1982), pp. 55-138 | MR | Zbl

[Si98] Y.T. Siu Invariance of Plurigenera, Inventiones Math., Volume 134 (1998), pp. 661-673 | MR | Zbl

[Sim93] C. Simpson Subspaces of Moduli Spaces of Rank One Local Systems, Ann. Sc. ENS., Volume 26 (1993), pp. 361-401 | Numdam | MR | Zbl

[Ue75] K. Ueno Classification Theory of Algebraic Varieties and Compact Complex Manifolds, LNM, 439, Springer Verlag, 1975 | MR | Zbl

[Vi82] E. Viehweg Die Additivität der Kodaira Dimension für Projektive Faserräume über Varietäten des Allgemeinen Typs, J. Reine Angew. Math., Volume 330 (1982), pp. 132-142 | MR | Zbl

[Vi83] E. Viehweg Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces (Adv. Stud. Pure Math.), Volume Vol. I (1983), pp. 329-353 | Zbl

[Vi86] E. Viehweg Vanishing theorems and and positivity of Algebraic fibre spaces, J. Proc. Int. Congr. Math. Berkeley (1986) | Zbl

[Vo 02] C. Voisin Intrinsic Pseudovolume Forms (2002) (Preprint)

[Zh96] Q. Zhang On projective manifolds with nef anticanonical bundles, J. Reine. Angew. Math., Volume 478 (1996), pp. 57-60 | MR | Zbl

[Zu97] K. Zuo Factorisation theorems for representations of fundamental groups of algebraic varieties (1997) (Preprint Kaiserslautern)

[Zu97] K. Zuo Representations of fundamental groups of algebraic varieties, Lecture Notes in Mathematics, 1708, Springer Verlag, 1999 | MR | Zbl