Two Non-Vanishing results concerning the Anti-Canonical Bundle

Abstract

Let $(X, \Delta)$ be a klt pair with nef anti-log canonical bundle $-(K_X+\Delta)$. We prove that $\kappa(X, -(K_X+\Delta))\geq 0$ assuming that either $X$ is of low dimension or that the generalised (numerical) non-vanishing conjecture holds. To do so, we prove a more general equivariant non-vanishing result for anti-log canonical bundles.Comment: Comments are very welcome