×

zbMATH — the first resource for mathematics

Orbifold generic semi-positivity: an application to families of canonically polarized manifolds. (Semi-positivité orbifolde : une application aux familles de variétés canoniquement polarisées.) (English. French summary) Zbl 1338.14012
Let \(f : Z \rightarrow B\) be a family of canonically polarised complex manifolds over a smooth base, i.e. a smooth morphism between quasi-projective complex manifolds such that the fibres are projective manifolds with ample canonical bundle. The family induces a natural map from the base \(B\) to a moduli space (of canonically polarised manifolds with fixed Hilbert polynomial) and we define the variation of the family \(f\) as the dimension of the image of this map. If the variation is maximal, i.e. equal to the dimension of \(B\), a conjecture of Viehweg claims that the manifold \(B\) is of log-general type.
This question has been studied intensively over the last years by S. Kebekus and S. J. Kovács [Duke Math. J. 155, No. 1, 1–33 (2010; Zbl 1208.14027); Invent. Math. 172, No. 3, 657–682 (2008; Zbl 1140.14031)] and Z. Patakfalvi [Adv. Math. 229, No. 3, 1640–1642 (2012; Zbl 1235.14031)].
In this paper the authors prove Viehweg’s conjecture in arbitrary dimension. For the proof they consider a compactification \(B \subset X\) such that the complement \(D := X \setminus B\) is a normal crossings divisor. By an important result of E. Viehweg and K. Zuo [in: Complex geometry. Collection of papers dedicated to Hans Grauert on the occasion of his 70th birthday. Berlin: Springer. 279–328 (2002; Zbl 1006.14004)] the family \(f\) induces an injective map \(L \rightarrow \Omega_X(\log D)\) where \(L\) is a big line bundle and \(\Omega_X(\log D)\) the logarithmic cotangent sheaf. The authors prove that such an injection can only exist if \(K_X+D\) is also big, i.e. the pair \((X, D)\) is of log-general type. The proof uses in a very tricky way the termination of special log MMPs due to C. Birkar et al. [J. Am. Math. Soc. 23, No. 2, 405–468 (2010; Zbl 1210.14019)] and a new semipositivity result for cotangent sheaves: if for some log pair \((X, D)\) the logarithmic canonical divisor \(K_X+D\) is pseudoeffective, then the logarithmic cotangent sheaf \(\Omega_X(\log D)\) is generically semipositive (in the sense of Miyaoka). This last result holds even more generally if \(D\) is an orbifold divisor (cf. F. Campana [Ann. Inst. Fourier 54, No. 3, 631–665 (2004; Zbl 1062.14015)]), in which case \(\Omega_X(\log D)\) should be interpreted on some appropriate finite cover on \(X\).

MSC:
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
14D22 Fine and coarse moduli spaces
14E22 Ramification problems in algebraic geometry
14E30 Minimal model program (Mori theory, extremal rays)
14J40 \(n\)-folds (\(n>4\))
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Berndtsson, Bo; Păun, Mihai, Quantitative extensions of pluricanonical forms and closed positive currents, Nagoya Math. J., 205, 25-65, (2012) · Zbl 1248.32012
[2] Birkar, Caucher; Cascini, Paolo; Hacon, Christopher D.; McKernan, James, Existence of minimal models for varieties of log general type · Zbl 1210.14019
[3] Bogomolov, Fedor; McQuillan, Michael, Rational curves on foliated varieties
[4] Bost, Jean-Benoît, Algebraic leaves of algebraic foliations over number fields, Publ. Math. Inst. Hautes Études Sci., 93, 161-221, (2001) · Zbl 1034.14010
[5] Boucksom, Sébastien; Demailly, Jean-Pierre; Paun, Mihai; Peternell, Thomas, The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension · Zbl 1267.32017
[6] Campana, Frédéric, Special orbifolds and birational classification: a survey · Zbl 1229.14011
[7] Campana, Frédéric, Orbifolds, special varieties and classification theory, Ann. Inst. Fourier (Grenoble), 54, 3, 499-630, (2004) · Zbl 1062.14014
[8] Campana, Frédéric, Orbifoldes géométriques spéciales et classification biméromorphe des variétés kählériennes compactes, J. Inst. Math. Jussieu, 10, 4, 809-934, (2011) · Zbl 1236.14039
[9] Campana, Frédéric; Guenancia, Henri; Păun, Mihai, Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields, Ann. Sci. Éc. Norm. Supér. (4), 46, 6, 879-916, (2013) · Zbl 1310.32029
[10] Campana, Frédéric; Păun, M., A differential-geometric approach for the generic semi-positivity of orbifold tensor bundles
[11] Campana, Frédéric; Peternell, Thomas, Geometric stability of the cotangent bundle and the universal cover of a projective manifold, Bull. Soc. Math. France, 139, 1, 41-74, (2011) · Zbl 1218.14030
[12] Demailly, Jean-Pierre, Algebraic geometry—Santa Cruz 1995, 62, Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials, 285-360, (1997), Amer. Math. Soc., Providence, RI · Zbl 0919.32014
[13] Demailly, Jean-Pierre, Holomorphic Morse inequalities and the Green-Griffiths-lang conjecture, Pure Appl. Math. Q., 7, 4, 1165-1207, (2011) · Zbl 1316.32014
[14] Esnault, Hélène; Viehweg, Eckart, Lectures on vanishing theorems, 20, vi+164 pp., (1992), Birkhäuser Verlag, Basel · Zbl 0779.14003
[15] Hartshorne, Robin, Cohomological dimension of algebraic varieties, Ann. of Math. (2), 88, 403-450, (1968) · Zbl 0169.23302
[16] Höring, A., On a conjecture of Beltrametti and Sommese · Zbl 1253.14007
[17] Jabbusch, Kelly; Kebekus, Stefan, Positive sheaves of differentials coming from coarse moduli spaces · Zbl 1253.14009
[18] Jabbusch, Kelly; Kebekus, Stefan, Families over special base manifolds and a conjecture of campana, Math. Z., 269, 3-4, 847-878, (2011) · Zbl 1238.14024
[19] Kawamata, Yujiro, Birational algebraic geometry (Baltimore, MD, 1996), 207, Subadjunction of log canonical divisors for a subvariety of codimension \(2, 79-88, (1997),\) Amer. Math. Soc., Providence, RI · Zbl 0901.14004
[20] Kawamata, Yujiro; Matsuda, Katsumi; Matsuki, Kenji, Algebraic geometry, Sendai, 1985, 10, Introduction to the minimal model problem, 283-360, (1987), North-Holland, Amsterdam · Zbl 0672.14006
[21] Kebekus, Stefan, Differential forms on singular spaces, the minimal program, and hyperbolicity of moduli · Zbl 1322.14055
[22] Kebekus, Stefan; Kovács, Sándor J., Families of canonically polarized varieties over surfaces, Invent. Math., 172, 3, 657-682, (2008) · Zbl 1140.14031
[23] Kebekus, Stefan; Kovács, Sándor J., The structure of surfaces and threefolds mapping to the moduli stack of canonically polarized varieties, Duke Math. J., 155, 1, 1-33, (2010) · Zbl 1208.14027
[24] Kebekus, Stefan; Solá Conde, Luis; Toma, Matei, Rationally connected foliations after Bogomolov and mcquillan, J. Algebraic Geom., 16, 1, 65-81, (2007) · Zbl 1120.14011
[25] Kollár, János, Lectures on resolution of singularities, 166, vi+208 pp., (2007), Princeton University Press, Princeton, NJ · Zbl 1113.14013
[26] Langer, A., Logarithmic orbifold Euler numbers of surfaces with applications · Zbl 1052.14037
[27] Miyaoka, Yoichi, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 46, Deformations of a morphism along a foliation and applications, 245-268, (1987), Amer. Math. Soc., Providence, RI · Zbl 0659.14008
[28] Paršin, A. N., Algebraic curves over function fields, Dokl. Akad. Nauk SSSR, 183, 524-526, (1968) · Zbl 0176.50903
[29] Patakfalvi, Zsolt, Viehweg hyperbolicity conjecture is true over compact bases · Zbl 1235.14031
[30] Taji, Behrouz, The isotriviality of families of canonically-polarised manifolds over a special quasi-projective base · Zbl 1427.14031
[31] Viehweg, Eckart; Zuo, Kang, Complex geometry (Göttingen, 2000), Base spaces of non-isotrivial families of smooth minimal models, 279-328, (2002), Springer, Berlin · Zbl 1006.14004
[32] Yano, K.; Bochner, S., Curvature and Betti numbers, ix+190 pp., (1953), Princeton University Press, Princeton, N. J. · Zbl 0051.39402
[33] Yau, Shing Tung, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math., 31, 3, 339-411, (1978) · Zbl 0369.53059
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.