Geometry and arithmetic of primary Burniat surfaces

Bauer, Ingrid; Stoll, Michael

doi:10.1002/mana.201600282

Cited by 3 publications

(2 citation statements)

References 24 publications

(63 reference statements)

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“…A one-dimensional variety X over k is Mordellic over k if and only if it is groupless over k. This follows from Faltings's theorem [27,28]. We refer to [4,5,6,17,22,29,55,58,70,71,72,73] for more examples of Mordellic varieties. Faltings proved that a closed subvariety X of an abelian variety A is Mordellic modulo its special locus (or Ueno locus) Sp(X); see [29].…”

Section: Remark 75 (Examples)mentioning

confidence: 99%

Finiteness properties of pseudo-hyperbolic varieties

Javanpeykar

Xie

2019

Preprint

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Motivated by Lang-Vojta's conjecture, we show that the set of dominant rational self-maps of an algebraic variety over a number field with only finitely many rational points in any given number field is finite by combining Amerik's theorem for dynamical systems of infinite order with properties of Prokhorov-Shramov's notion of quasi-minimal models. We also prove a similar result in the geometric setting by using again Amerik's theorem and Prokhorov-Shramov's notion of quasi-minimal model, but also Weil's regularization theorem for birational self-maps and properties of dynamical degrees. Furthermore, in the geometric setting, we obtain an analogue of Kobayashi-Ochiai's finiteness result for varieties of general type, and thereby generalize Noguchi's theorem (formerly Lang's conjecture). Our proof here relies on a deformation-theoretic result for surjective maps of normal varieties due to Hwang-Kebekus-Peternell. Contents 19 8. Dominant self-maps of pseudo-Mordellic varieties 21 9. Dominant self-maps of pseudo-1-bounded varieties 22 References 24

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Section: Remark 75 (Examples)mentioning

confidence: 99%

Finiteness properties of pseudo-hyperbolic varieties

Javanpeykar

Xie

2019

Preprint

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“…It follows from Faltings's theorem [36,37] that a one-dimensional variety X over k is arithmetically hyperbolic over k if and only if it is groupless over k. Moreover, it follows from Faltings's theorem [38] that a closed subvariety X of an abelian variety A over k is arithmetically hyperbolic over k if and only if X is groupless. We refer to [3,4,6,27,30,38,71,74,86,87,85] for more examples of arithmetically hyperbolic varieties.…”

Section: 2mentioning

confidence: 99%

Arithmetic hyperbolicity: automorphisms and persistence

Javanpeykar¹

2018

Preprint

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We verify some "arithmetic" predictions made by conjectures of Campana, Hassett-Tschinkel, Green-Griffiths, Lang, and Vojta. Firstly, we prove that every dominant endomorphism of an arithmetically hyperbolic variety over an algebraically closed field of characteristic zero is in fact an automorphism of finite order, and that the automorphism group of an arithmetically hyperbolic variety is a locally finite group. To prove these two statements we use (a mild generalization of) a theorem of Amerik on dynamical systems which in turn builds on work of Bell-Ghioca-Tucker, and combine this with a classical result of Bass-Lubotzky. Furthermore, we show that if the automorphism group of a projective variety is torsion, then it is finite. In particular, we obtain that the automorphism group of a projective arithmetically hyperbolic variety is finite, as predicted by Lang's conjectures. Next, we apply this result to verify that projective hyperkähler varieties with Picard rank at least three are not arithmetically hyperbolic. Finally, we show that arithmetic hyperbolicity is a "geometric" notion, as predicted by Green-Griffiths-Lang's conjecture, under suitable assumptions related to Demailly's notion of algebraic hyperbolicity. For instance, if k ⊂ C is an algebraically closed subfield and X is an arithmetically hyperbolic projective variety over k such that X C is Brody hyperbolic, then X remains arithmetically hyperbolic after any algebraically closed field extension of k.

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Finiteness Properties of Pseudo-Hyperbolic Varieties

Javanpeykar

Xie

2020

International Mathematics Research Notices

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Motivated by Lang–Vojta’s conjecture, we show that the set of dominant rational self-maps of an algebraic variety over a number field with only finitely many rational points in any given number field is finite by combining Amerik’s theorem for dynamical systems of infinite order with properties of Prokhorov–Shramov’s notion of quasi-minimal models. We also prove a similar result in the geometric setting by using again not only Amerik’s theorem and Prokhorov–Shramov’s notion of quasi-minimal model but also Weil’s regularization theorem for birational self-maps and properties of dynamical degrees. Furthermore, in the geometric setting, we obtain an analogue of Kobayashi–Ochiai’s finiteness result for varieties of general type and thereby generalize Noguchi’s theorem (formerly Lang’s conjecture).

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Geometry and arithmetic of primary Burniat surfaces

Cited by 3 publications

References 24 publications

Finiteness properties of pseudo-hyperbolic varieties

Finiteness properties of pseudo-hyperbolic varieties

Arithmetic hyperbolicity: automorphisms and persistence

Finiteness Properties of Pseudo-Hyperbolic Varieties

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