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