Lemma 3.29 . [0511] Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.
Source coverage notes Source fidelity gap: the original arXiv HTML omits an author TeX footnote attached to equation e:def-omega, including its reference to Remark r:error-function. The retained HTML is preserved as published; the original author TeX remains available. TeX correspondence is not complete. Complete original source context · Original author HTML
Lemma 3.29 .
We may choose the above holomorphic coordinates { w i } i = 1 n − 1 \{w_{i}\}_{i=1}^{n-1} p p H H w 1 = 0 w_{1}=0
(3.318) ω D = − 1 2 g i j ¯ d w i ∧ d w ¯ j , \omega_{D}=\frac{\sqrt{-1}}{2}g_{i\bar{j}}dw_{i}\wedge d\bar{w}_{j},
where
(3.319) { g i j ¯ ( 0 ) = δ i j , 1 ≤ i , j ≤ n − 1 , ∂ w 1 g i j ¯ ( 0 ) = 0 , 1 ≤ i , j ≤ n − 1 , ∂ w k g 1 1 ¯ ( 0 ) = ∂ w k g i j ¯ ( 0 ) = 0 , 2 ≤ i , j , k ≤ n − 1 . \displaystyle\begin{cases}g_{i\bar{j}}(0)=\delta_{ij},&1\leq i,j\leq n-1,\\
\partial_{w_{1}}g_{i\bar{j}}(0)=0,&1\leq i,j\leq n-1,\\
\partial_{w_{k}}g_{1\bar{1}}(0)=\partial_{w_{k}}g_{i\bar{j}}(0)=0,&2\leq i,j,k\leq n-1.\end{cases}