An inductive approach to generalized abundance using nef reduction (Q2681087)

Let \((X,B)\) be a klt pair with \(K_X+B\) pseudoeffective. Suppose that \(L\) is a nef divisor on \(X\) such that \(D=K_X+B+L\) is also nef. The Generalized Abundance conjecture says that \(D\) is numerically equivalent to a semiample \(\mathbb{Q}\)-divisor. Assuming some conjectures in Algebraic Geometry (up to some dimension), this short article proves that this conjecture holds. Those conjectures are Termination of klt flips, Abundance conjecture, Semiampleness conjecture and Generalized Nonvanishing conjecture. The last one and the Generalized Abundance conjecture were introduced by \textit{V. Lazić} and \textit{T. Peternell} [Publ. Res. Inst. Math. Sci. 56, No. 2, 353--389 (2020; Zbl 1466.14019)]. The techniques used are typical of MMP.