Hyperbolicity of geometric orbifolds

Rousseau, Erwan

doi:10.1090/s0002-9947-10-05019-1

Cited by 18 publications

(21 citation statements)

References 23 publications

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“…Examples of algebraically hyperbolic projective varieties are given in [3,4,7,9,10,30,33,34]. Also, a logarithmic analogue of algebraic hyperbolicity (for quasi-projective varieties) was introduced and studied in [5].…”

Section: Introductionmentioning

confidence: 99%

Demailly’s notion of algebraic hyperbolicity: geometricity, boundedness, moduli of maps

Javanpeykar

Kamenova

2020

Math. Z.

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Demailly's conjecture, which is a consequence of the Green-Griffiths-Lang conjecture on varieties of general type, states that an algebraically hyperbolic complex projective variety is Kobayashi hyperbolic. Our aim is to provide evidence for Demailly's conjecture by verifying several predictions it makes. We first define what an algebraically hyperbolic projective variety is, extending Demailly's definition to (not necessarily smooth) projective varieties over an arbitrary algebraically closed field of characteristic zero, and we prove that this property is stable under extensions of algebraically closed fields. Furthermore, we show that the set of (not necessarily surjective) morphisms from a projective variety Y to a projective algebraically hyperbolic variety X that map a fixed closed subvariety of Y onto a fixed closed subvariety of X is finite. As an application, we obtain that Aut(X) is finite and that every surjective endomorphism of X is an automorphism. Finally, we explore "weaker" notions of hyperbolicity related to boundedness of moduli spaces of maps, and verify similar predictions made by the Green-Griffiths-Lang conjecture on hyperbolic projective varieties.

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Section: Introductionmentioning

confidence: 99%

Demailly’s notion of algebraic hyperbolicity: geometricity, boundedness, moduli of maps

Javanpeykar

Kamenova

2020

Math. Z.

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“…Because f • h is such an orbifold map for any h : C → X, the conclusion follows In [Rou06] this result is generalised, and its proof simplified, using some further results of Mc Quillan.…”

Section: Theorem([c-pa 07])mentioning

confidence: 77%

“…and (X|∆ X ) are integral, and if moreover, in the second condition above, Rou06], answering a question in [C-W 05] …”

Section: C Orbifold Morphismsmentioning

confidence: 98%

Special orbifolds and birational classification: a survey

Campana

2010

EMS Series of Congress Reports

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“…Example 24. Following [5] (see also [33]), we see that nodal surfaces in P 3 of degree d with l nodes provide examples of orbifolds with positive orbifold second Segre number if…”

Section: Orbifold Hyperbolicitymentioning

confidence: 85%

“…We will see that "orbifold" techniques as systematically introduced by Campana ( [7], [8], see also [33]) are useful in this context.…”

Section: Introductionmentioning

confidence: 98%

On the hyperbolicity of surfaces of general type with small c ₁ ²

Roulleau

Rousseau

2012

Journal of the London Mathematical Society

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Surfaces of general type with positive second Segre number s2 := c 2 1 − c2 > 0 are known by results of Bogomolov to be algebraically quasihyperbolic i.e. with finitely many rational and elliptic curves. These results were extended by McQuillan in his proof of the Green-Griffiths conjecture for entire curves on such surfaces. In this work, we study hyperbolic properties of minimal surfaces of general type with minimal c 2 1 , known as Horikawa surfaces. In principle these surfaces should be the most difficult case for the above conjecture as illustrate the quintic surfaces in P 3 . Using orbifold techniques, we exhibit infinitely many irreducible components of the moduli of Horikawa surfaces whose very generic member has no rational curves or even is algebraically hyperbolic. Moreover, we construct explicit examples of algebraically hyperbolic and quasi-hyperbolic orbifold Horikawa surfaces.

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Hyperbolicity of geometric orbifolds

Cited by 18 publications

References 23 publications

Demailly’s notion of algebraic hyperbolicity: geometricity, boundedness, moduli of maps

Demailly’s notion of algebraic hyperbolicity: geometricity, boundedness, moduli of maps

Special orbifolds and birational classification: a survey

On the hyperbolicity of surfaces of general type with small c ₁ ²

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Hyperbolicity of geometric orbifolds

Cited by 18 publications

References 23 publications

Demailly’s notion of algebraic hyperbolicity: geometricity, boundedness, moduli of maps

Demailly’s notion of algebraic hyperbolicity: geometricity, boundedness, moduli of maps

Special orbifolds and birational classification: a survey

On the hyperbolicity of surfaces of general type with small c 1 2

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On the hyperbolicity of surfaces of general type with small c ₁ ²