Extension theorems, Non-vanishing and the existence of good minimal models

We prove an extension theorem for effective plt pairs $(X,S+B)$ of non-negative Kodaira dimension $\kappa (K_X+S+B)\geq 0$. The main new ingredient is a refinement of the Ohsawa-Takegoshi $L^2$ extension theorem involving singular hermitian metrics. As a consequence of the extension theorem, we show that the Non-vanishing Conjecture implies the existence of good minimal models.