The order of vanishing of the determinant alone does not determine the normal distance. An upper bound for the trace bounds all eigenvalues from above; combined with a lower bound for the determinant, it bounds the smallest eigenvalue from below, and hence the normal length. We use the local length distance $d_\eta ^ U$ of theorem 5.39.1, the infimum of $\eta $-lengths of piecewise smooth paths in $U\setminus D$; the distance $d^ U_\eta (x,D)$ to $D$ is the infimum of the lengths of such paths starting at $x$ and tending to $D$.
Proposition 10.21.6.B9DD61C0FD5E Let $n\geq 2$, let $U\subset \mathbb C\times \mathbb C^{n-1}$ be a polydisk with coordinates $(z,w)$, and put $D=\{ z=0\} $. Let $\eta $ be a smooth semipositive $(1,1)$-form on $U$, positive off $D$, whose Euclidean trace is bounded and whose determinant is comparable to $|z|^{2\ell }$ for an integer $\ell \geq 1$. Then on a smaller concentric polydisk,
BC6A5A3DBD7A
10.21.7
\begin{equation} c|z|^{\ell +1}\leq d_\eta ^ U((z,w),D)\leq C|z| \end{equation}
for constants $c,C>0$. Both bounds are attained, up to constants, by smooth closed forms with the same determinant order. If the block of $\eta $ in the $w$-directions is uniformly positive, the distance is comparable to $|z|^{\ell +1}$.
Proof.
The least eigenvalue satisfies $\lambda _{\min }\geq \det \eta /(\operatorname {tr}\eta )^{n-1}\geq c|z|^{2\ell }$. Every path to $D$ therefore has length at least a constant times the integral of $|z|^\ell |\mathrm{d} z|$ along it, which gives the lower bound; a radial path and the trace bound give the upper bound. To see that both bounds occur, let $n=2$ first. For an integer $0\leq a\leq \ell $, take $\eta _ a=\mathrm{dd}^{\mathrm{c}}|G_ a|^2$ with $G_ a(z,w)=(z^{a+1}/(a+1),z^{\ell -a}w)$; for $n>2$, append Euclidean coordinates. The Jacobian determinant of $G_ a$ is $z^\ell $, but at $(z,0)$ the first component of $G_ a$ forces a distance of order $|z|^{a+1}$, attained along a radial path. The cases $a=\ell $ and $a=0$ give the two endpoints.
Finally, write the Hermitian coefficient matrix of $\eta $ in the normal and tangential directions as $\left(\begin{smallmatrix}h& b^*\\ b& B\end{smallmatrix}\right)$. When $B$ is uniformly positive, its Schur complement $h-b^*B^{-1}b=\det \eta /\det B$ is comparable to $|z|^{2\ell }$. Completing the square gives the lower bound of order $|z|^{\ell +1}$, and solving $\mathrm{d} w/\mathrm{d} s=-B^{-1}b\, \mathrm{d} z/\mathrm{d} s$ along a radial normal curve, on nested charts, gives a path of matching length.
$\square$
Pome
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