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We give a simplified proof of an optimal version of the Ohsawa-Takegoshi$L^2$-extension theorem. We follow the variational proof by Berndtsson-Lempertand use the method in the paper of McNeal-Varolin. As an application, we givean optimal estimate for extensions from possibly non-reduced subvarieties.

A simplified proof of optimal $L^2$-extension theorem and extensions from non-reduced subvarieties