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 \(X\).
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 \(X\).
Reviewer: Andreas Höring (La Roquette-sur-Var)
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) |
Keywords:
logarithmic cotangent bundle; orbifold cotangent bundle; generic semipositivity; Viehweg’s conjecture; canonically polarized manifoldsCitations:
Zbl 1208.14027; Zbl 1140.14031; Zbl 1235.14031; Zbl 1006.14004; Zbl 1210.14019; Zbl 1062.14015References:
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