Two Non-Vanishing results concerning the Anti-Canonical Bundle

Let $(X,\mathrm{\Delta })$ be a klt threefold pair with nef anti-log canonical bundle $-(K_X+\mathrm{\Delta })$. We show that $\kappa (X,-(K_X+\mathrm{\Delta }))\ge 0$. To do so, we prove a more general equivariant non-vanishing result for anti-log canonical bundles, which is valid in any dimension.