Remarks on Rational Points of Varieties Whose Cotangent Bundles Are Generated by Global Sections

Theorem A. Let A be an abelian variety over a number field k and X a subvariety of A. Then there are a finite number of translated abelian subvarieties B1, . . . , Bn over k such that Bi ⊂ X and the closure X(k) of X(k) in X is contained in ⋃ i Bi. In this short note, as applications of the above Faltings’ theorem, we will give several remarks on rational points of varieties whose cotangent bundles are generated by global sections. The first remark is the following theorem which is a slight generalization of Theorem A.