## Viehweg’s hyperbolicity conjecture for families with maximal variation.(English)Zbl 1375.14043

Let $$f^\circ: Y^\circ \rightarrow X^\circ$$ be a family of complex projective manifolds of general type (or more generally projective manifolds admitting a good minimal model). We say that the variation of the family is maximal if for a general point $$x \in X^0$$ there exist at most countably many $$x' \in X^0$$ such that $$Y_x$$ is birational to $$Y_{x'}$$. A famous conjecture of Viehweg claims that for a family of maximal variation the manifold $$X^0$$ is of log-general type, i.e., for a compactification $$X^0 \subset X$$ such that $$B:=X \setminus X^0$$ is a normal crossings divisor, the Kodaira dimension $$\kappa(X, K_X+B)$$ is equal to the dimension $$\dim X$$. In this paper the authors prove Viehweg’s conjecture.
This generalises results of S. Kebekus and S. J. Kovács for low-dimensional bases [Invent. Math. 172, No. 3, 657–682 (2008; Zbl 1140.14031); Duke Math. J. 155, No. 1, 1–33 (2010; Zbl 1208.14027)], but also the theorem of F. Campana and M. Păun for families of projective manifolds with ample canonical divisor [Ann. Inst. Fourier 65, No. 2, 835–861 (2015; Zbl 1338.14012)]. While these earlier results rely on the Viehweg-Zuo sheaves introduced in [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)], this paper is based on the construction of Hodge modules and associated Higgs bundles having certain positivity properties. As a consequence of these Hodge theoretic considerations the authors construct a Viehweg-Zuo sheaf $$\mathcal H \hookrightarrow (\Omega_X(\log B))^{\otimes s}$$ that is big. Combined with the recent positivity result for the logarithmic cotangent bundle $$\Omega_X(\log B)$$ by F. Campana and M. Pǎun [“Foliations with positive slopes and birational stability of orbifold cotangent bundles”, Preprint, arXiv:1508.02456] this implies the main theorem. The introduction of the paper gives a detailed account of the various technical elements that enter in the proof this important breakthrough.

### MSC:

 14D06 Fibrations, degenerations in algebraic geometry 14D07 Variation of Hodge structures (algebro-geometric aspects) 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 14E30 Minimal model program (Mori theory, extremal rays)

### Citations:

Zbl 1140.14031; Zbl 1208.14027; Zbl 1338.14012; Zbl 1006.14004
Full Text:

### References:

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