Geometric Height Inequality on Varieties with Ample Cotangent Bundles

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 Ω X/F 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 hL(P ) ≤ A · d(P ) + O(1) for all P ∈ X(F̄ )\Y (F̄ ), where hL(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 [No], i.e. there is a non-empty Zariski open set U of X with U(F ) = ∅. 0. Introduction. Let k be an algebraically closed field of characteristic zero and F a finitely generated extension field of k with trans. degk F = 1. We will fix two fields k and F throughout this note. Let X be a geometrically irreducible smooth projective variety over F . Let X and C be smooth projective varieties over k, and f : X → C a k-morphism such that the function field of C is F and the generic fiber of f is X , i.e. X = X ⊗ F . In this note, X is said to be non-isotrivial if there is a non-empty open set C0 of C such that, for all t ∈ C0, the Kodaira-Spencer map ̺t : TC,t → H (Xt, TXt) is not zero. Let F̄ be the algebraic closure of F . For a point P ∈ X(F̄ ), let us denote by ∆P the corresponding integral curve on X . We fix a line bundle L on X . Let L be a line bundle on X with L ⊗ F = L. The pair (f : X → C,L) is called a model of (X,L). A geometric height hL(P ) of P with respect to L is defined by hL(P ) = (L ·∆P ) [F (P ) : F ] . Typeset by AMS-TEX 1