Geometric height inequality on varieties with ample cotangent bundles

Abstract

Let F be a function field of one variable over an algebraically closed field of characteristic zero, X a geometrically irreducible smooth projective variety over F, and L a line bundle on X. In this note, we will prove that if the contangent bundle of X is ample and X is non-isotrivial, then there are a proper closed algebraic set Y of X and a constant A > 0 such that h_L(P) <= A d(P) + O(1) for all P \in X(\bar{F}) \ Y(\bar{F}), where h_L(P) is a geometric height of P with respect to L and d(P) is the geometric logarithmic discriminant of P. As corollary of the above height inequality, we can recover Noguchi's theorem, i.e. there is a non-empty Zariski open set U of X with U(F) = \emptyset.