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Hosono obtained sharper estimates of the Ohsawa--Takegoshi $L^2$-extentiontheorem by allowing the constant depending on the weight function for a domainin $\mathbb{C}$. In this article, we show the higher dimensional case ofsharper estimates of the Ohsawa--Takegoshi $L^2$-extention theorem. To provethe higher dimensional case of them, we establish an analogue ofBerndtsson--Lempert type $L^2$-extension theorem by using the pluricomplexGreen functions with poles along subvarieties. As a special case, we considerthe sharper estimates in terms of the Azukawa pseudometric and show that thehigher dimensional case of sharper estimate provides the $L^2$-minimumextension for radial case.

Sharper estimates of Ohsawa--Takegoshi $L^2$-extension theorem in higher dimensional case