We give a proof of the Ohsawa–Takegoshi extension theorem with sharp estimates. The proof is based on ideas of Błocki to use variations of domains to simplify his proof of the Suita conjecture, and also uses positivity properties of direct image bundles.
"A proof of the Ohsawa–Takegoshi theorem with sharp estimates." J. Math. Soc. Japan 68 (4) 1461 - 1472, October, 2016. https://doi.org/10.2969/jmsj/06841461