Lemma 10.2.11B12EA9BB1DC1

Proof. Choose holomorphic coordinates $z$ centered at $x$. Since the matrix of $\sigma $ at $x$ is positive definite Hermitian, after a linear change of coordinates $\sigma =\frac{\mathrm i}{2}\sum _{k,l}g_{k\bar l}\, \mathrm dz_ k\wedge \mathrm d\bar z_ l$ with $g_{k\bar l}(0)=\delta _{kl}$; moreover $\overline{g_{l\bar k}}=g_{k\bar l}$, since $\sigma $ is real. The $(2,1)$-part of $\mathrm d\sigma =0$ is $\partial \sigma =\frac{\mathrm i}{2}\sum _{k,l,m}\partial _ mg_{k\bar l}\, \mathrm dz_ m\wedge \mathrm dz_ k\wedge \mathrm d\bar z_ l=0$, which gives $\partial _ mg_{k\bar l}=\partial _ kg_{m\bar l}$.

By the holomorphic inverse function theorem (proposition 5.2.4(1)), the equations $z_ l=w_ l-\frac12\sum _{k,m}\partial _ mg_{k\bar l}(0)w_ kw_ m$ define holomorphic coordinates $w$ near $0$, since the Jacobian matrix of this map at $0$ is the identity. Substituting $\mathrm dz_ l=\mathrm dw_ l-\sum _{k,m}\partial _ mg_{k\bar l}(0)w_ m\, \mathrm dw_ k$, which uses the symmetry in $k$ and $m$, and $g_{k\bar l}(z)=\delta _{kl}+\sum _ m\partial _ mg_{k\bar l}(0)w_ m+\sum _ m\partial _{\bar m}g_{k\bar l}(0)\bar w_ m+O(|w|^2)$, we find that the coefficient of $\frac{\mathrm i}{2}\mathrm dw_ k\wedge \mathrm d\bar w_ l$ in $\sigma $ is $\delta _{kl}$ plus terms of order two in $w$: the terms $\sum _ m\partial _ mg_{k\bar l}(0)w_ m$ of $g_{k\bar l}$ cancel against the correction in $\mathrm dz_ k$, and their conjugates $\sum _ m\partial _{\bar m}g_{k\bar l}(0)\bar w_ m=\sum _ m\overline{\partial _ mg_{l\bar k}(0)}\, \bar w_ m$ cancel against the correction in $\mathrm d\bar z_ l$. Hence the first derivatives of the new coefficients vanish at $0$. $\square$