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

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.

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

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

Arithmetic hyperbolicity: automorphisms and persistence

We show that if the automorphism group of a projective variety is torsion, then it is finite. Motivated by Lang’s conjecture on rational points of hyperbolic varieties, we use this to prove that a projective variety with only finitely many rational points has only finitely many automorphisms. Moreover, we investigate to what extent finiteness of S-integral points on a variety over a number field persists over finitely generated fields. To this end, we introduce the class of mildly bounded varieties and prove a general criterion for proving this persistence.

LANG'S CONJECTURE AND SHARP HEIGHT ESTIMATES FOR THE ELLIPTIC CURVES y2 = x3 + ax

For elliptic curves given by the equation Ea : y2 = x3 + ax, we establish the best-possible version of Lang's conjecture on the lower bound for the canonical height of non-torsion rational points along with best-possible upper and lower bounds for the difference between the canonical and logarithmic height.