A fibered power theorem for pairs of log general type

Abstract

Abstract. Let f : (X, D) → B be a stable family with log canonical general fiber. We prove that, after a birational modification of the base B → B, there is a morphism from a high fibered power of the family to a pair of log general type. If in addition the general fiber is openly canonical, then there is a morphism from a high fibered power of the original family to a pair openly of log general type.

Lang–Vojta conjecture over function fields for surfaces dominating $${{\mathbb {G}}}_m^2$$

“…When the log canonical divisor of the generic fiber of ρ is ample, we can choose a model of (X , D) over C that is a stable family in the sense of Kollár (see for example [3,Definition 2.7]). In this situation we show that, when D is an ample divisor on X, the divisor K X/C + D, and therefore Z , is indeed big.…”

The Lang-Vojta conjectures on projective pseudo-hyperbolic varieties

“…(3) The projective variety X is pseudo-Mordellic over k. (4) The projective variety X is pseudo-arithmetically hyperbolic over k. (5) The projective variety X is pseudo-groupless over k. (6) The projective variety X is pseudo-algebraically hyperbolic over k. (7) The projective variety X is pseudo-bounded over k. (8) The projective variety X is pseudo-1-bounded over k. (9) The projective variety X is pseudo-geometrically hyperbolic over k. (10) The projective variety X is of general type over k.…”

“…(5) The projective variety X is groupless over k. (6) The projective variety X is algebraically hyperbolic over k. (7) The projective variety X is bounded over k. (8) The projective variety X is 1-bounded over k.…”

“…The first striking consequence of Lang-Vojta's conjecture was obtained by Caporaso-Harris-Mazur [18]. Their results were further investigated by Abramovich, Ascher-Turchet, Hassett, and Voloch; see [1,2,3,4,8,40].…”

Hyperbolicity of bases of log Calabi-Yau families

“…We will not recall the basic properties of slc families, and we refer the readers to [AT16,Pat16] for further details. Let us mention that although Theorem 1.3 is only stated for the compact setting in [Den18a, Corollary A.2], i.e.…”