The recent extension theorem of Demailly is an Ohsawa--Takegoshi type extension theorem which extends holomorphic sections on subvarieties defined by multiplier ideal sheaves to holomorphic sections on complete Kӓhler manifolds. I will present my result which improves the theorem of Demailly in the sense that estimates of the extensions are provided. This result, together with the techniques developed by Demailly, is believed to be the key to overcome the obstacles in generalising Demailly--Hacon--Pŭaun's extension theorem on plt pairs to that on dlt pairs, an important ingredient in solving the Abundance Conjecture. If time permits, I will explain how the estimates help to prove the dlt extension.

 

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