Complex-analytic intermediate hyperbolicity, and finiteness properties

Finiteness of non-constant maps over number fields

Motivated by the intermediate Lang conjectures on hyperbolicity and rational points, we prove new finiteness results for non-constant morphisms from a fixed variety to a fixed variety defined over a number field by applying Faltings's finiteness results to moduli spaces of maps.

Algebraic intermediate hyperbolicities

We extend Lang's conjectures to the setting of intermediate hyperbolicity and prove two new results motivated by these conjectures. More precisely, we first extend the notion of algebraic hyperbolicity (originally introduced by Demailly) to the setting of intermediate hyperbolicity and show that this property holds if the appropriate exterior power of the cotangent bundle is ample. Then, we prove that this intermediate algebraic hyperbolicity implies the finiteness of the group of birational automorphisms and of the set of surjective maps from a given projective variety. Our work answers the a…

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.

Ampleness of Schur powers of cotangent bundles and k-hyperbolicity

In this paper, we study a variation of a conjecture of Debarre on positivity of cotangent bundles of complete intersections. We establish the ampleness of Schur powers of cotangent bundles of generic complete intersections in projective manifolds, with high enough explicit codimension and multi-degrees. Our approach is naturally formulated in terms of flag bundles and allows one to reach the optimal codimension. On complex manifolds, this ampleness property implies intermediate hyperbolic properties. We give a natural application of our main result in this context.

代数线映射与对称幂的双曲性

Algebroid maps and hyperbolicity of symmetric powers

本文定义了复射影代数簇对称幂双曲性的不变量$h(X)$，利用Nevanlinna理论给出其非平凡下界，并探讨了与数域上有界度代数点有限性的关联及具体应用。

双曲性性质的标记超kähler流形模空间

Hyperbolicity properties of moduli spaces of marked hyperk{ä}hler manifolds

研究标记超kähler流形模空间的双曲性性质，证明Kobayashi伪距离消失，并推广Greb和Schwald的定理到任意维数和非紧曲线情况。