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.

4 pages, comments welcome