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We prove an extension theorem for effective purely log-terminal pairs ($X, S$+$B$) of non-negative Kodaira dimension $${\kappa (K_X+S+B)\ge 0}$$ . The main new ingredient is a refinement of the Ohsawa–Takegoshi$L$^{2}extension theorem involving singular Hermitian metrics.

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Let$D$be a convex domain with smooth boundary in complex space and let$f$be a continuous function on the boundary of$D$. Suppose that$f$holomorphically extends to the extremal discs tangent to a convex subdomain of$D$. We prove that f holomorphically extends to$D$. The result partially answers a conjecture by Globevnik and Stout of 1991.

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