Lemma: Finite-jet dependence along a smooth missing locus

Suppose $Z$ is locally a smooth embedded submanifold of $X$. In a product coordinate neighborhood $(s,h)$ with $Z=\{h=0\}$, let $T_0$ have distributional order at most $M$ on a fixed compact coordinate neighborhood.

If $f\in C^{\infty }(X)$ satisfies $f{T}^{\circ }=0$ off $Z$ and

then $f{T}_0=0$ on the interior of that neighborhood. Thus, among annihilator functions, $f{T}_0$ depends locally only on the transverse jet of $f$ through order $M$. In particular, an annihilator flat along $Z$ produces no natural defect.

Proof. Let $\varphi$ be a test supported in a smaller compact coordinate neighborhood. The distribution $u=f{T}_0$ is supported on $Z$. Choose $\rho \in \mathcal{D}({\mathbb{R}}^{\dim h})$ equal to one near zero and write ${\rho }_{ϵ}(h)=\rho (h/ϵ)$. Since $u$ is supported on $Z$,

Taylor's theorem and (eq:vanishingjet) imply, uniformly on the fixed compact set,

The derivatives of ${\rho }_{ϵ}$ contribute the corresponding powers of ${ϵ}^{-1}$. The Leibniz rule therefore gives

for sufficiently small $ϵ$. The order-$M$ estimate for $T_0$ implies $⟨u,\varphi ⟩=0$ on letting $ϵ\downarrow 0$.

If two annihilator functions have the same transverse jet through order $M$, their difference satisfies (eq:vanishingjet), so their defects agree. Flatness implies vanishing of every finite transverse jet.