Boundedness and volume of generalised pairs

In this paper we investigate boundedness and volume of generalised pairs and give applications to usual algebraic varieties and pairs, especially to a class of pairs that we call stable minimal models. We prove a result about descent of nef divisors to bounded families. This is the key to proving various other results in this paper and elsewhere. As an application, fixing the dimension and a DCC set controlling coefficients, we will show that the set of volumes of all projective generalised lc pairs $(X,B+M)$, under the given data, satisfies the descending chain condition (DCC). Futhermore, we will show that in the klt case, the set of such pairs with ample $K_X+B+M$ and fixed volume forms a bounded family. We will then apply the above to study projective lc pairs $(X,B)$ with abundant $K_X+B$ of arbitrary Kodaira dimension. In particular, we show that the set of Iitaka volumes of such pairs satisfies DCC under some natural boundedness assumptions on the fibres of the Iitaka fibration. We define strongly stable minimal models. Fixing appropriate invariants, we show that such models form a bounded family. This and other results of this paper are crucial ingredients of the construction of moduli spaces of stable minimal models in a sequel work.