Generically Big and Pseudo-effective Adelic Line Bundles

The purpose of this chapter is to study weak relative positivity conditions of adelic line bundle. In Sect. 8.1, we first extend the arithmetic intersection product by allowing the appearance of one non-integrable adelic line bundle. In Sects. 8.2 and 8.3, we introduce a numerical invariant, the asymptotic maximal slope, to measure the weak relative positivity of an adelic line bundle. In Sect. 8.4, we show that the asymptotic maximal slope does not decrease under the pull-back by a surjective projective morphism. In Sect. 8.5, we prove a relative version of Fujita’s approximation theorem for the asymptotic maximal slope of an adelic line bundle by the asymptotic minimal slope of relatively nef adelic line subbundles. In Sect. 8.6, we prove a strong lower bound of the arithmetic intersection product with the appearance of the asymptotic maximal slope of one adelic line bundle instead of its asymptotic minimal slope. In Sect. 8.7, we discuss asymptotic first minimum, which is similar, but in general not equal, to the asymptotic maximal slope. In Sect. 8.8, we compare the asymptotic maximal slope to the normalized height. In Sect. 8.9, we introduce the condition of strong Minkowskianness for adelic line bundles. Under this condition, the adelic line bundles behave similarly to the classic number field case. In Sect. 8.10, we study the successive minima of the normalized height function and discuss its connection with sectional invariants such as the asymptotic maximal and minimal slopes. In Sect. 8.11, we prove an equidistribution theorem for a generic sequence of integral closed subschemes.