$$
\Vert f\Vert_{\varepsilon,p}=\Bigg(\sum_{s=1}^k
\big(\varepsilon_s|\xi_1|^{s-1}|\xi_s|\big)^{2p/(2s-1)}\Bigg)^{1/2p},
\leqno(3.5)
$$
under the assumption that $\xi_s\in(\xi_1)^\perp$ for $s\geq 2$
(the first term in the sum is equal to $\varepsilon_1|\xi_1|^{2p}$
and we will in fact take $\varepsilon_1=1$). The curvature of 
$\cO_{(\bC^r)_k}(1)$ fiberwise (i.e.\ on the rational variety~$\bR_{r,k}$)
is given by ${i\over 2\pi}\ddbar\log\Vert f\Vert_{\varepsilon,p}^2$, provided
holomorphic coordinates are used. We compute the curvature
form at a point \hbox{$\xi^0=(\xi_s^0)_{1\leq s\leq k}$} such that
$\xi^0_1\neq 0$. By applying a dilation $t\mapsto f(\lambda t)$,
$\lambda\in\bC^*$, we can assume that $|\xi^0_1|=1$. A nearby point
$\xi=(\xi_s)_{1\leq s\leq k}$ can be written as
$\xi_1=\xi_1^0+\zeta$ and $\xi_s$, $s\geq 2$, with
$\zeta,\xi_s\in (\xi^0_1)^\perp$. Notice that $((\xi^0_1)^\perp)^k$ is
a $k(r-1)$-dimensional complex subspace that defines an affine chart
of $\bR_{r,k}$ containing $\xi^0$. The difficulty
is that we do not necessarily have $\langle\xi_s,\xi_1\rangle=0$ any more,
however $\langle\xi_s,\xi_1\rangle=\langle\xi_s,\zeta\rangle=O(|\zeta|)$
for $s\geq 2$, and we have to correct this by applying an element
$\varphi\in\bG'_k$ close to identity. When computing $i\ddbar(...)$
at $\zeta=0$, all terms $O(\zeta^2,\smash{\ol\zeta}^{\kern 1pt 2},|\zeta|^3)$
can be neglected, and the calculations performed below will be made modulo
such terms. In particular, higher powers of
$\langle\xi_s,\zeta\rangle$ can be be neglected as they are of the form
$O(\smash{\ol\zeta}^{\kern 1pt 2})$. A suitable choice is
$$
\varphi(t)=t-\sum_{s=2}^k\langle\xi_s,\zeta\rangle t^s.
$$
Then
$$
f\circ\varphi(t)=x+\sum_{s=1}^kt^s\tilde\xi_s+O(t^{k+1})
$$
where $\tilde\xi_1=\xi_1$ and
$$
\tilde\xi_s=\xi_s- \langle\xi_s,\zeta\rangle\xi_1-\sum_{j=2}^{s-1}j
\langle\xi_{s-j+1},\zeta\rangle\xi_j~~\mod
O(\zeta^2,\smash{\ol\zeta}^{\kern 1pt 2},|\zeta|^3),\quad s\geq 2
$$
(the final summation is obtained by expanding the
terms $(t-\sum_{\ell\geq 2}\langle\xi_\ell,\zeta\rangle t^\ell)^j\xi_j$
for $2\leq j\leq s-1$ and $\ell=s-j+1$). Observe that
$|\xi_1|^2=1+ |\zeta|^2$ and
$\langle\xi_j,\xi_1\rangle=\langle\xi_j,\xi^0_1\rangle+
\langle\xi_j,\zeta\rangle=\langle\xi_j,\zeta\rangle$ for $j\geq 2$, hence
$$
\langle\tilde\xi_s,\xi_1\rangle=\langle \xi_s,\zeta\rangle
-\langle \xi_s,\zeta\rangle(1+|\zeta|^2)
-\sum_{j=2}^{s-1}j\langle\xi_{s-j+1},\zeta\rangle\langle \xi_j,\zeta\rangle
=O(\zeta^2,\smash{\ol\zeta}^{\kern 1pt 2},|\zeta|^3),
$$
so we do not need any more accurate correction. Moreover, by a straightforward
calculation
$$
|\tilde\xi_s|^2=
\bigg|\xi_s-\sum_{j=2}^{s-1}j\langle\xi_{s-j+1},\zeta\rangle\xi_j\bigg|^2
-|\langle\xi_s,\zeta\rangle|^2
~~\mod O(\zeta^2,\smash{\ol\zeta}^{\kern 1pt 2},|\zeta|^3).
$$
Therefore, for $\varepsilon_1=1$, we find
$$
\leqalignno{
\Vert f\Vert_{\varepsilon,p}
&=\Bigg(|\xi_1|^{2p}+\sum_{s=2}^k\big(\varepsilon_s|\xi_1|^{s-1}
|\tilde\xi_s|\big)^{2p/(2s-1)}\Bigg)^{1/2p}\cr
&=\Bigg((1+|\zeta|^2)^p+\sum_{s=2}^k\big(\varepsilon_s^2(1+|\zeta|^2)^{s-1}
|\tilde\xi_s|^2\big)^{p/(2s-1)}\Bigg)^{1/2p}&(3.6)\cr}
$$
We take profit of the constants $\varepsilon_s$ to perform a rescaling
and set $u_s=\varepsilon_s\xi_s$. Then our formula becomes
$$
\Vert f\Vert_{\varepsilon,p}^2=
\bigg((1+|\zeta|^2)^p+\sum_{s=2}^k\big((1+|\zeta|^2)^{s-1}
|\tilde u_s|^2\big)^{p/(2s-1)}\bigg)^{1/p}\leqno(3.7)
$$
where (modulo negligible terms) the $u_s$ and $\tilde u_s$ are defined by
$u_1=u^0_1+\zeta$, $|u^0_1|^2=1$,
$$
\eqalign{
\tilde u_s&=u_s- \langle u_s,\zeta\rangle u_1-\sum_{j=2}^{s-1}
{\varepsilon_s\over \varepsilon_j\varepsilon_{s-j+1}}
j\langle u_{s-j+1},\zeta\rangle u_j\cr
|\tilde u_s|^2&=
\bigg|u_s-\sum_{j=2}^{s-1}{\varepsilon_s\over \varepsilon_j\varepsilon_{s-j+1}}
j\langle u_{s-j+1},\zeta\rangle u_j\bigg|^2-|\langle u_s,\zeta\rangle|^2.\cr}
$$
By taking $\varepsilon_k\ll \varepsilon_{k-1}\ll\cdots\ll\varepsilon_2\ll 1$,
all terms $\varepsilon_s/(\varepsilon_j\varepsilon_{s-j+1})$ become
negligible (e.g.\ equal to $\varepsilon$ if we put
$\varepsilon_s=\varepsilon^{2s-3}$, $s\geq 2$).

