The nonvanishing conjecture for projective klt pairs

Let $(X,\Delta)$ be a projective kawamata log terminal pair. Nonvanishing conjecture. If $K_X+\Delta$ is pseudo-effective, then $\kappa(X,K_X+\Delta)\geq 0$. This is identified as the main obstruction to extending the paper's results beyond varieties of log general type and is a central nonvanishing problem in the minimal model program.