zbMATH — the first resource for mathematics

Invariance of the \(\Gamma\)-dimension for certain Kaehlerian families of dimension 3. (Invariance de la \(\Gamma \)-dimension pour certaines familles kählériennes de dimension 3.) (French) Zbl 1197.32009
Let \(X\) be a complex projective manifold, then a famous conjecture of Shafarevich claims that the universal cover \(\tilde{X}\) admits a proper morphism onto a Stein space. So far the Shafarevich conjecture is open even for surfaces, but the fundamental work of F. Campana [Bull. Soc. Math. Fr. 122, No. 2, 255–284 (1994; Zbl 0810.32013)] and J. Kollár [Invent. Math. 113, No. 1, 177–215 (1993; Zbl 0819.14006)] shows that for every compact Kähler manifold \(X\) there exists a meromorphic map \(\gamma: X \dashrightarrow \Gamma(X)\) such that for every subvariety \(Z \subset X\) passing through a general point such that \(\pi_1(Z) \rightarrow \pi_1(X)\) has finite image, the map \(\gamma\) contracts \(Z\). The \(\Gamma\)-dimension of the variety \(X\) is defined as the dimension of the variety \(\Gamma(X)\). Kollár conjectured that the \(\Gamma\)-dimension is invariant under deformation, this conjecture has been proven by B. de Oliveira, L. Katzarkov and M. Ramachandran [Geom. Funct. Anal. 12, No. 4, 651–668 (2002; Zbl 1169.14305)] if the fundamental group \(\pi_1(X)\) is residually unipotent.
In general Kollár’s conjecture is wide open, the paper under review studies the case of surfaces and threefolds. While the surface case follows quickly from a result of Yum-Tong Siu [Discrete groups in geometry and analysis, Pap. Hon. G. D. Mostow 60th Birthday, Prog. Math. 67, 124–151 (1987; Zbl 0647.53052)], the threefold case needs much more effort. More precisely the author shows that if \(\mathcal X \rightarrow B\) is a smooth family of compact Kähler threefolds such that the fibres are not of general type, then the \(\Gamma\)-dimension is locally constant. In the case of a family of threefolds of general type the statement also holds if a conjecture of Campana on the orbifold fundamental group of certain surfaces is true (we refer to the paper for more details on this interesting problem). An important tool in the proof is a detailed study of the natural fibrations attached to the family of threefolds.

32Q30 Uniformization of complex manifolds
14J30 \(3\)-folds
32J27 Compact Kähler manifolds: generalizations, classification
Full Text: DOI
[1] Alexeev, V., Mori, S.: Bounding singular surfaces of general type. In: Algebra, Arithmetic and Geometry with Applications (West Lafayette, IN, 2000), pp. 143–174. Springer, Berlin (2004) · Zbl 1103.14021
[2] Barlet, D.: Espace analytique réduit des cycles analytiques complexes compacts d’un espace analytique complexe de dimension finie. In: Fonctions de plusieurs variables complexes, II (Sém. François Norguet, 1974–1975), Lecture Notes in Math., vol. 482, pp. 1–158. Springer, Berlin (1975)
[3] Barth, W.P., Hulek, K., Peters, C.A.M., Van de Ven, A.: Compact complex surfaces, 2nd edn. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 4, Springer-Verlag, Berlin (2004) · Zbl 1036.14016
[4] Blanchard A.: Sur les variétés analytiques complexes. Ann. Sci. Ecole Norm. Sup. 73(3), 157–202 (1956) · Zbl 0073.37503
[5] Bogomolov, F.A.: Unstable vector bundles and curves on surfaces. In: Proceedings of the International Congress of Mathematicians (Helsinki, 1978) (Helsinki), Acad. Sci. Fennica, pp. 517–524 (1980) · Zbl 0485.14004
[6] Campana F.: Remarques sur le revêtement universel des variétés kählériennes compactes. Bull. Soc. Math. France 122(2), 255–284 (1994) · Zbl 0810.32013
[7] Campana F.: Fundamental group and positivity of cotangent bundles of compact Kähler manifolds. J. Algebraic Geom. 4(3), 487–502 (1995) · Zbl 0845.32027
[8] Campana, F.: Orbifoldes à première classe de Chern nulle. The Fano Conference, Univ. Torino, Turin, pp. 339–351 (2004) · Zbl 1068.53051
[9] Campana F.: Orbifolds, special varieties and classification theory. Ann. Inst. Fourier (Grenoble) 54(3), 499–630 (2004) · Zbl 1062.14014
[10] Campana F.: Orbifolds, special varieties and classification theory: an appendix. Ann. Inst. Fourier (Grenoble) 54(3), 631–665 (2004) · Zbl 1062.14015
[11] Campana, F.: Orbifoldes spéciales et classifications biméromorphes des variétés kählériennes compactes. Preprint arXiv:0705.0737 (2007)
[12] Catanese F.: Moduli and classification of irregular Kaehler manifolds (and algebraic varieties) with Albanese general type fibrations. Invent. Math. 104(2), 263–289 (1991) · Zbl 0743.32025 · doi:10.1007/BF01245076
[13] Campana, F., Zhang, Q.: Compact Kähler threefolds of \(\pi\)1-general type. Recent progress in arithmetic and algebraic geometry, Contemp. Math., vol. 386, Amer. Math. Soc., Providence, RI, pp. 1–12 (2005) · Zbl 1216.32011
[14] de Oliveira B., Katzarkov L., Ramachandran M.: Deformations of large fundamental groups. Geom. Funct. Anal. 12(4), 651–668 (2002) · Zbl 1169.14305 · doi:10.1007/s00039-002-8262-8
[15] Fong, L., McKernan, J.: Log abundance for surfaces. In: Kollar, J. (ed.), Flips and Abundance for Algebraic Threefolds, pp. 127–137. Astérisque, no. 211 (1992) · Zbl 0807.14029
[16] Friedman, R., Morgan, J.: Smooth four-manifolds and complex surface. In: Ergebnisse der Mathematik und ihrer Grenzgebiete, no. 27, Springer Verlag (1994) · Zbl 0817.14017
[17] Fujita T.: Approximating Zariski decomposition of big line bundles. Kodai Math. J. 17(1), 1–3 (1994) · Zbl 0814.14006 · doi:10.2996/kmj/1138039894
[18] Gromov M.: Kähler hyperbolicity and L 2-Hodge theory. J. Differ. Geom. 33(1), 263–292 (1991) · Zbl 0719.53042
[19] Kawamata, Y.: A product formula for volumes of varieties. Preprint arXiv0704.1014 (2007)
[20] Kollár J., Mori S.: Classification of three-dimensional flips. J. Am. Math. Soc. 5(3), 533–703 (1992) · Zbl 0773.14004 · doi:10.1090/S0894-0347-1992-1149195-9
[21] Kollár J.: Shafarevich maps and plurigenera of algebraic varieties. Invent. Math. 113(1), 177–215 (1993) · Zbl 0819.14006 · doi:10.1007/BF01244307
[22] Kollár J.: Shafarevich maps and automorphic forms. M. B. Porter Lectures, Princeton University Press, Princeton, NJ (1995) · Zbl 0871.14015
[23] Lieberman, D.I.: Compactness of the Chow scheme: applications to automorphisms and deformations of Kähler manifolds. Fonctions de plusieurs variables complexes, III (Sém. François Norguet, 1975–1977), Lecture Notes in Math., vol. 670, pp. 140–186. Springer, Berlin (1978)
[24] Shafarevich, I.R.: Basic algebraic geometry. Springer-Verlag, New York (1974). Translated from the Russian by K. A. Hirsch, Die Grundlehren der mathematischen Wissenschaften, Band 213 · Zbl 0284.14001
[25] Siu, Y.-T.: Strong rigidity for kähler manifolds and the construction of bounded holomorphic functions. In: Howe, R. (ed.), Discrete Groups and Analysis. Birkhäuser Verlag (1987)
[26] Siu Y.-T.: Invariance of plurigenera. Invent. Math. 134(3), 661–673 (1998) · Zbl 0955.32017 · doi:10.1007/s002220050276
[27] Voisin, C.: Théorie de hodge et géométrie algébrique complexe. Cours spécialisés, vol. 10, S.M.F. (2002)
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.