on the $\Lambda^*(Y)$ factors in the decomposition and $\star_{(Z, {\textbf{k}})}$, the $y$-dependent Riemannian structure induced on the fibers, on the $x^{k\operatorname{\mathbf{N}}}\Lambda^*(Z)$ factors. (So, for example, $\star_Z x^k dz_\alpha = x^{(f - 1) k} \star_Z dz_\alpha$.) Thus, we may also define unambiguously (still locally over the boundary) the Hodge-de Rham operators $\eth_Y$ and $\eth_Z$. Indeed, a homogeneous differential form $\alpha$, in the coordinates $(x, y, z)$ can be written $\alpha = (f(x, y, z) dy^I) \wedge
x^{kp}dz^A(\wedge dx) = \pm(f(x, y, z) x^{kp}dz^A) \wedge dy^I
(\wedge dx)$, where $I$ and $A$ are multi-indices and $|I|$, resp.$|A|$, is their order. One can act via $\eth_Y = d_Y + (-1)^{{b}(|I|-1)+1} \star_Y d_Y \star_Y$ on $f(x, y, z) dy^I$, where $\star_Y$ is the Hodge star operator of $(Y, h)$, and thus define $\eth_Y \alpha$, or act via $\eth_Z = d_Z + (-1)^{f (|A| -
  1) + 1} \star_Z d_Z \star_Z$ on $f(x, y, z) x^{kp}dz^A$ where $\star_Z$ is the Hodge star operator of $(Z, {\textbf{k}})$, and thus define $\eth_Z \alpha$. Note that $\eth_Z$ is thus a *family* of operators on the fiber $Z$ parametrized by the base point $y$. The Hodge-de Rham operator can be decomposed locally over the boundary in terms of the 