The positive Kodaira dimension conjecture for projective klt pairs

Let $(X,\Delta)$ be a projective kawamata log terminal pair. If $K_X+\Delta$ is pseudo-effective and, for some ample divisor $H$,

$$h^0(X,\lfloor m(K_X+\Delta)\rfloor+H)$$

is not a bounded function of $m$, then $\kappa(X,K_X+\Delta)\geq 1$. Positive Kodaira dimension conjecture. This strengthens the expected conclusion in the nonvanishing setting when the relevant spaces of sections are unbounded; together with nonvanishing and results of Kawamata, it is presented as implying abundance.