Pome

Corollary 11.19.2029DAFB20616

Corollary 11.19.2.029DAFB20616 For every $m>0$,

\[ \mathcal I(m\log q) =\mathcal I_ Z^{\max \{ \lfloor m\rfloor -r+1,0\} }. \]

In particular, if $k\geq 1$ is an integer and $m\geq r+k-1$, a holomorphic function that is locally square integrable against $q^{-m}$ times a smooth positive volume form vanishes to order at least $k$ along $Z$. If $m<r+k-1$, this conclusion fails for some local holomorphic functions.

Proof. In a local bundle frame, $\log q=\log |v|^2+O(1)$ near $Z$. Apply lemma 5.37.1 in codimension $r$. Its radial integrability condition for a homogeneous fiber term of degree $a$ is $a+r>m$. For $m<r+k-1$, a monomial of degree $k-1$ satisfies this condition and has insufficient vanishing order. Smooth positive volume forms do not change local integrability. $\square$

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