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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)
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