Viehwegʼs hyperbolicity conjecture is true over compact bases

Patakfalvi, Zsolt

doi:10.1016/j.aim.2011.12.013

Cited by 11 publications

(6 citation statements)

References 3 publications

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“…If D is big, then we immediately get the conclusion. Otherwise we employ the standard argument based on the pseudo-effectivity of quotients by Viehweg-Zuo type sheaves, inspired by an idea in [CP11, §2.2] (see also [Pat12]): using (ii), we have a short exact sequence 0 −→ H −→ (Ω 1 X ) ⊗s ⊗ O X (rD) −→ Q −→ 0, and passing to the saturation of H we can assume that Q is torsion-free. Using that X is not uniruled, a special case of [CP15, Theorem 1.2] says that every torsion-free quotient of (Ω 1 X ) ⊗s has pseudo-effective determinant; it follows that det Q(−rD) is pseudo-effective, which implies that det Q is pseudo-effective as well.…”

Section: Positivity For Hodge Modulesmentioning

confidence: 99%

“…The result was shown by Kebekus-Kovács when X is a surface in [KK08a], and then in [KK08b] when D = ∅ assuming the main conjectures of the minimal model program, while [KK10] contains more refined results in dimension at most three. It was then deduced unconditionally by Patakfalvi [Pat12] from the results of [CP11], when D = ∅ and when X is not uniruled. Finally, Campana-Pȃun obtained the result in general, based on their bigness criterion [CP15,Theorem 7.7] that we use here as well.…”

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confidence: 99%

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Viehweg’s hyperbolicity conjecture for families with maximal variation

Popa

Schnell

2016

Invent. math.

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Abstract. We use the theory of Hodge modules to construct Viehweg-Zuo sheaves on base spaces of families with maximal variation and fibers of general type; and, more generally, families whose geometric generic fiber has a good minimal model. Combining this with a result of Campana-Pȃun, we deduce Viehweg's hyperbolicity conjecture in this context, namely the fact that the base spaces of such families are of log general type. This is approached as part of a general problem of identifying what spaces can support Hodge theoretic objects with certain positivity properties.

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Section: Positivity For Hodge Modulesmentioning

confidence: 99%

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confidence: 99%

Viehweg’s hyperbolicity conjecture for families with maximal variation

Popa

Schnell

2016

Invent. math.

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“…Roughly speaking, Viehweg predicted that for families with maximal variation, a log smooth compactification (Y, D) of V is of log general type. The proof of the original statement of the conjecture, in the canonically polarized case, was established in important special cases in [VZ02], [KK08], [KK08b], [KK10], [Pat12], and was recently completed by Campana and Pȃun [CP15, Thm. 8.1]; for a more detailed overview of this body of work and for further references, please see [PS17, §1.2].…”

Section: Introductionmentioning

confidence: 99%

“…[KK08], [KK08b], [KK10], [Pat12], and was recently completed by Campana and Pȃun [CP15, Thm. 8.1]; for a more detailed overview of this body of work and for further references, please see [PS17, §1.2].…”

Section: Introductionmentioning

confidence: 99%

Brody hyperbolicity of base spaces of certain families of varieties

Popa

Taji²,

2019

Alg. Number Th.

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We prove that quasi-projective base spaces of smooth families of minimal varieties of general type with maximal variation do not admit Zariski dense entire curves. We deduce the fact that moduli stacks of polarized varieties of this sort are Brody hyperbolic, answering a special case of a question of Viehweg and Zuo. For two-dimensional bases, we show analogous results in the more general case of families of varieties admitting a good minimal model.

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“…Then X is hyperbolic. This is a consequence of Viehweg's conjecture for "compact" base varieties [Pat12].…”

Section: Hyperbolicitymentioning

confidence: 79%

Bounding heights uniformly in families of hyperbolic varieties

Ascher,

Javanpeykar

2016

Preprint

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We show that, assuming Vojta's height conjecture, the height of a rational point on an algebraically hyperbolic variety can be bounded "uniformly" in families. This generalizes a result of Su-Ion Ih for curves of genus at least two to higher-dimensional varieties. As an application, we show that, assuming Vojta's height conjecture, the height of a rational point on a curve of general type is uniformly bounded. Finally, we prove a similar result for smooth hyperbolic surfaces with c 2 1 > c 2 .

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Viehwegʼs hyperbolicity conjecture is true over compact bases

Cited by 11 publications

References 3 publications

Viehweg’s hyperbolicity conjecture for families with maximal variation

Viehweg’s hyperbolicity conjecture for families with maximal variation

Brody hyperbolicity of base spaces of certain families of varieties

Bounding heights uniformly in families of hyperbolic varieties

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