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Hyperbolicity and uniformity of varieties of log general type

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Kenneth Ascher, Kristin DeVleming, Amos Turchet

Algebraic Geometry · Submitted 16 Jul 2018

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We show that all subvarieties of a quasi-projective variety with almost ample log cotangent bundle are of log general type.

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arXiv ID1807.05946 (version 5)
Pith ID6MK5LUYE, Pith's permanent record of this paper and its review
CitationAmos Turchet, Kenneth Ascher, Kristin DeVleming. "Hyperbolicity and uniformity of varieties of log general type." arXiv:1807.05946 (2018). https://pith.science/paper/6MK5LUYE
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,929 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.76 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.54 · compares with prior results 0.66 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v5. Results and the main claim are underlined; .

Projective varieties with ample cotangent bundle satisfy many notions of hyperbolicity, and one goal of this paper is to discuss generalizations to quasi-projective varieties. A major hurdle is that the naive generalization fails, i.e. the log cotangent bundle is never ample. Instead, we define a notion called almost ample which roughly asks that the log cotangent is as positive as possible. We show that all subvarieties of a quasi-projective variety with almost ample log cotangent bundle are of log general type. In addition, if one assumes globally generated then we obtain that such varieties contain finitely many integral points. In another direction, we show that the Lang-Vojta conjecture implies the number of stably integral points on curves of log general type, and surfaces of log general type with almost ample log cotangent sheaf are uniformly bounded.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 Projective varieties with ample cotangent bundle satisfy many notions of hyperbolicity, and one goal of this paper is to discuss generalizations to quasi-projective varieties. background 0.97 0.13 0.04 asserted 1.00
s1 A major hurdle is that the naive generalization fails, i.e. the log cotangent bundle is never ample. background 0.49 0.24 0.03 asserted 1.00
s2 Instead, we define a notion called almost ample which roughly asks that the log cotangent is as positive as possible. contribution 0.81 0.90 0.06 asserted 0.92
s3 We show that all subvarieties of a quasi-projective variety with almost ample log cotangent bundle are of log general type. result 0.99 0.94 0.33 asserted 1.00
s4 In addition, if one assumes globally generated then we obtain that such varieties contain finitely many integral points. result 1.00 0.89 0.10 asserted 0.99
s5 In another direction, we show that the Lang-Vojta conjecture implies the number of stably integral points on curves of log general type, and surfaces of log general type with almost ample log cotangent sheaf are uniformly bounded. result 0.92 0.79 0.24 asserted 0.98

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Original abstract

Projective varieties with ample cotangent bundle satisfy many notions of hyperbolicity, and one goal of this paper is to discuss generalizations to quasi-projective varieties. A major hurdle is that the naive generalization fails, i.e. the log cotangent bundle is never ample. Instead, we define a notion called almost ample which roughly asks that the log cotangent is as positive as possible. We show that all subvarieties of a quasi-projective variety with almost ample log cotangent bundle are of log general type. In addition, if one assumes globally generated then we obtain that such varieties contain finitely many integral points. In another direction, we show that the Lang-Vojta conjecture implies the number of stably integral points on curves of log general type, and surfaces of log general type with almost ample log cotangent sheaf are uniformly bounded.

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