6533b820fe1ef96bd127a4dd
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Finiteness properties of pseudo-hyperbolic varieties
Ariyan JavanpeykarJunyi Xiesubject
Pure mathematicsDynamical systems theoryGeneral Mathematics[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS]Dynamical Systems (math.DS)Type (model theory)01 natural sciencesSurjective functionMathematics - Algebraic Geometry0103 physical sciencesFOS: MathematicsNumber Theory (math.NT)0101 mathematicsMathematics - Dynamical Systems[MATH]Mathematics [math]Algebraic Geometry (math.AG)MathematicsConjectureMathematics - Number Theory010102 general mathematicsOrder (ring theory)Algebraic varietyAlgebraic number field[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT]Regularization (physics)010307 mathematical physics[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG]description
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).
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2019-11-18 |