scholarly journals The Erdős–Ulam problem, Lang's conjecture and uniformity

2020 ◽  
Vol 52 (6) ◽  
pp. 1053-1063 ◽  
Author(s):  
Kenneth Ascher ◽  
Lucas Braune ◽  
Amos Turchet
Keyword(s):  
Lang's Conjecture ◽  
Lang’S Conjecture

Schmidt’s subspace theorem for non-subdegenerate families of hyperplanes

2021 ◽  
pp. 1-18
Author(s):  
Duc Hiep Pham
Keyword(s):  
Type Theorem ◽  
Exceptional Sets ◽  
Subspace Theorem ◽  
Lang's Conjecture ◽  
Lang’S Conjecture

In this paper, we establish a Schmidt’s subspace theorem for non-subdegenerate families of hyperplanes. In particular, our result improves the previous result on Schmidt’s subspace type theorem for the case of non-degenerate families of hyperplanes, and furthermore, also shows the sharpness of the condition of non-subdegeneracy. As a consequence, we deduce a version of Lang’s conjecture on exceptional sets in the case of complements of hyperplanes.

Going-Down Results for Ci-Fields

2006 ◽  
Vol 49 (1) ◽  
pp. 11-20
Author(s):  
Anthony J. Bevelacqua ◽  
Mark J. Motley
Keyword(s):  
Function Fields ◽  
Algebraic Extension ◽  
Real Closed Field ◽  
Closed Field ◽  
Difficult Case ◽  
Lang's Conjecture ◽  
Lang’S Conjecture

AbstractWe search for theorems that, given a Ci-field K and a subfield k of K, allow us to conclude that k is a Cj -field for some j. We give appropriate theorems in the case K = k(t) and K = k((t)). We then consider the more difficult case where K/k is an algebraic extension. Here we are able to prove some results, and make conjectures. We also point out the connection between these questions and Lang's conjecture on nonreal function fields over a real closed field.

scholarly journals A note on Lang's conjecture for quotients of bounded domains

2021 ◽  
Vol Volume 5 ◽  
Author(s):  
Sébastien Boucksom ◽  
Simone Diverio
Keyword(s):  
Plurisubharmonic Function ◽  
General Type ◽  
Universal Cover ◽  
Projective Manifold ◽  
Bounded Domains ◽  
Final Version ◽  
Lang's Conjecture ◽  
Projective Manifolds ◽  
Lang’S Conjecture ◽  
Complex Projective

It was conjectured by Lang that a complex projective manifold is Kobayashi hyperbolic if and only if it is of general type together with all of its subvarieties. We verify this conjecture for projective manifolds whose universal cover carries a bounded, strictly plurisubharmonic function. This includes in particular compact free quotients of bounded domains. Comment: 10 pages, no figures, comments are welcome. v3: following suggestions made by the referee, the exposition has been improved all along the paper, we added a variant of Theorem A which includes manifolds whose universal cover admits a bounded psh function which is strictly psh just at one point, and we added a section of examples. Final version, to appear on \'Epijournal G\'eom. Alg\'ebrique

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