Hodge Index Theorem for Arithmetic Cycles of Codimension One

For example, K. Künnemann [Ku] proved that if X is a projective space, then the conjecture is true. Here we fix a notation. We say a Hermitian line bundle (H, k) on X is arithmetically ample if (1) H is f -ample, (2) the Chern form c1(H∞, k∞) is positive definite on the infinite fiber X∞, and (3) there is a positive integer m0 such that, for any integer m ≥ m0, H(X, H) is generated by the set { s ∈ H(X, H) | ‖s∞‖sup < 1 } . Note that by virtue of [Zh], the third condition can be replaced by a numerical condition : (3)’ for every irreducible horizontal subvariety Y (i.e. Y is flat over Spec(Z)), the height ĉ1( (H, k)|Y ) Y of Y is positive. In this note, we would like to prove the following partial answer of the above conjecture for general regular arithmetic varieties.