Bounded anticanonical metrics on klt pairs and torus quotients

Let $(W,D)$ be a projective ℚ-factorial klt pair with effective rational boundary. We prove that a positive Cartier multiple of $-(K_W+D)$ has a nonzero section if this divisor is nef, its pullback to a resolution admits a semipositive metric with locally bounded weights, and the resolution has nonzero structure-sheaf Euler characteristic. Consequently a smooth rationally connected projective variety with smoothly semipositive anticanonical bundle has a nonzero naturally invariant anticanonical plurisection for every algebraic torus action.