# Hermitian metric on conic Kaehler-Einstein setting

I have a technical question :

Consider the triple $(M,D,\omega)$ where $M$ is a Fano manifold, $D$ is a smooth divisor whose Poincare dual is $\lambda c_1(M)$ and $\omega$ is a conic Kaehler -Einstein metric on $M$ with cone angle $2\pi\beta$ along $D$. Let $S$ is a difining section of $D$ . In local holomorphic coordinates $z_1,z_2,...,z_n$ , write $S=f\big(\frac{\partial}{\partial z_1}\wedge...\wedge \frac{\partial}{\partial z_n}\big)$ and $H(.,.)$ be a Hermitian metric on $K_M^{-1}$ and $\omega=i\sum g_{\alpha\bar\beta}dz_\alpha\wedge d\bar z_\beta$

then $H$ can be represented by

$$\text{det}(g_{\alpha\bar\beta})^{\frac{1}{\mu}}\mid f \mid^{\frac{2(1-\beta)}{\mu}}$$

for some $\mu$

This fact has been appeared in most of the papers about Kahler-Einstein metrics. But I am looking for a reason!

See this paper for details

• What is the relation between $H$, $\omega$ and $S$? – YangMills Feb 4 '15 at 21:11
• This is certainly false if you don't impose any relation among $H$, $\omega$ and $S$, since as you wrote it now $H$ is an arbitrary Hermitian metric on $K_M^{-1}$. – YangMills Feb 5 '15 at 18:59
• You just took your notations from page 16 here arxiv.org/pdf/1211.4669.pdf. I believe everything is explained there. – YangMills Feb 5 '15 at 21:32
• In fact for defining section $S$ of divisor $D$ there exists unique $H$ which satisfies $\int_MH(S,S)\omega^n=1$. Tian left this part without any detatil. So my question make sense – user21574 Feb 5 '15 at 21:41