Relative Ampleness and Nefness

The aim of this chapter is to discuss some strong relative positivity conditions on adelic line bundles, such as relative ampleness and nefness. In Sects. 6.1 and 6.2, we introduce a numerical invariant, asymptotic minimal slope, to measure the relative positivity. In Sect. 6.3, we define the relative ampleness of an adelic line bundle and discuss its properties. In particular, we establish a lower bound for the arithmetic intersection number in terms of asymptotic minimal slopes. In Sect. 6.4, we extend by continuity the function of asymptotic minimal slope to the cone of relatively nef adelic line bundles and generalize the lower bound for the arithmetic intersection number in this setting. In Sect. 6.5, we prove a generalized Hodge index theorem that gives a bigness criterion of relatively nef adelic line bundles in terms of the positivity of the arithmetic self-intersection number. In Sect. 6.6 we prove the non-decreasing property of the asymptotic minimal slope by the pull-back by a projective morphism. This property is useful to provide lower bounds of the asymptotic minimal slope. In Sect. 6.7 we compare the asymptotic minimal slope of a generically big and relatively nef adelic line bundle to normalized height of the arithmetic variety with respect to the adelic line bundle, by using the arithmetic Hilbert-Samuel formula.