Remarks on rational points of varieties whose cotangent bundles are generated by global sections

Atsushi Moriwaki
1995 Mathematical Research Letters  
A b s t r a c t . In this short note, we will give several remarks on rational points of varieties whose cotangent bundles are generated by global sections. For example, we will show that if the sheaf of differentials Ω 1 X/k of a projective variety X over a number field k is ample and generated by global sections, then the set of k-rational points of X is finite. Theorem B. Let X be a projective variety over a number field k, A an abelian variety over k, and α : X → A a morphism over k. If α *
more » ... hism over k. If α * (Ω 1 A/k ) → Ω 1 X/k is surjective, then every irreducible component of X(k) is geometrically irreducible and isomorphic to an abelian variety. As a corollary of Theorem B, we have the following. Received November 11, 1994. 113 D e pa rt m e n t o f M at h e m at ic s , Fa c u lt y o f S c ie n c e , K y o t o U n iv e r s it y , K yo t o , 6 0 6 -0 1 , J a pa n
doi:10.4310/mrl.1995.v2.n1.a10 fatcat:byz7qaui4bhwfapgopzzujejem