Nonvanishing and Abundance for cones of movable divisors

Abstract

Let $\overline{\mathrm{Mov}}^k(X)$ be the closure of the cone $\mathrm{Mov}^k(X)$ generated by classes of effective divisors on a projective variety $X$ with stable base locus of codimension at least $k+1$. We propose a generalized version of the Log Nonvanishing Conjecture and of the Log Abundance Conjecture for a klt pair $(X,\Delta)$, that is: if $K_X+\Delta \in \overline{\mathrm{Mov}}^{k}(X)$, then $K_X+\Delta \in \mathrm{Mov}^{k}(X)$. Moreover, we prove that if the Log Minimal Model Program, the Log Nonvanishing, and the Log Abundance hold, then so does our conjecture.