Let $z_V = (z_1, \dots, z_n)$ and $z_{\hat{V}} = (\hat{z}_1, \dots, \hat{z}_n)$ be the coordinate systems of $V$ and $\hat{V}$ on which (iv), (v) and (vi) are true. We assert that $\hat{z}_j = u_j.z_j$ on $V \cap \hat{V}$, where $u_j(z) \neq 0 \forall z \in V \cap \hat{V}$, $\forall 1 \le j \le n-1$.

In fact, if we fix $1 \le j \le n-1$, by (vi) we must have $\hat{z}_j = u.z_i$, where $u(z) \neq 0 \forall z \in V \cap \hat{V}$, for some $1 \le i \le n-1$. However, the fact that $\rho_\ell \neq \rho_j$ if $\ell \neq j$ implies that $i = j$. It follows that

$$ \theta_{\hat{V}}^{j} = \frac{d\hat{z}_{j}}{\hat{z}_{j}} - \frac{\rho_{j}}{\rho_{1}} \frac{d\hat{z}_{1}}{\hat{z}_{1}} = \frac{dz_{j}}{z_{j}} - \frac{\rho_{j}}{\rho_{1}} \frac{dz_{1}}{z_{1}} + dv = \theta_{V}^{j} + dv, \quad 2 \le j \le n-1, $$

where $v = \log(u_j) - \frac{\rho_j}{\rho_1} \log(u_1)$ is holomorphic. Now, (iv) and (v) imply that

$$ \begin{align*} i_{X_V} \theta_{\hat{V}}^j &= i_{X_V} \theta_V^j = i_{Y_V} \theta_{\hat{V}}^j = i_{Y_V} \theta_V^j = 0 \\ &\implies X_V(v) = Y_V(v) = 0 \\ &\implies v \text{ is a first integral of both vector fields } X_V \text{ and } Y_V \end{align*} $$

Since $X_V = \frac{\partial}{\partial z_n}$ we get $v = v(z_1, \dots, z_{n-1})$. Since the eigenvalues of $Y_V$ are in the Poincaré domain $v(z_1, \dots, z_{n-1})$ must be a constant and $dv = 0$. Hence, $\theta_{\hat{V}}^j = \theta_V^j$ on $V \cap \hat{V}$, as asserted. Therefore there are closed logarithmic 1-forms $\theta_2, \dots, \theta_{n-1}$, defined on $W = \cup_V V$, such that $\mathcal{F}|_W$ is defined by $\theta_2 \wedge \dots \wedge \theta_{n-1}$, as asserted. Let us prove that the forms $\theta_j$ extend to a neighborhood of any point in $F$.

Given $p \in F$ let $\omega$ be a germ of (n-2)-form defining the germ of $\mathcal{F}$ at $p$. Let $(U, z = (x = z_1, \dots, z_n))$ be a coordinate system around $p$ as in (III), so that $z(p) = 0$ and $S \cap U = (z_1 = \dots = z_{n-1} = 0)$. The rotational $X$ of $\omega$ has eigenvalues $\lambda_1, \dots, \lambda_n$ and there exists $a \neq 0$ such that $\mathrm{Re}(a.\lambda_n) < 0$ and $\mathrm{Re}(a.\lambda_j) > 0$, $\forall 1 \le j \le n-1$. Since $p=0$ is an isolated singularity of $X$ there exists another germ of vector field $Z$ such that $\omega = i_Z i_X \nu$, $\nu = dz_1 \wedge \dots \wedge dz_n$. The vector fields $X$ and $Z$ generate the germ of $\mathcal{F}$ at 0.

**LEMMA 4.8.** — There are germs at $p$ of vector fields $\tilde{X}$ and $\tilde{Z}$ that generate the germ of $\mathcal{F}$ at $p$ and a holomorphic coordinate system $(U_1, w = (w_1, \dots, w_n))$ around $p$, with the following properties:

(a) $w(p) = 0$ and $S \cap U_1 = (w_1 = \dots = w_{n-1} = 0)$.

(b) $\tilde{Z}(w) = \sum_{j=1}^{n-1} \rho_j w_j \frac{\partial}{\partial w_j}$. In particular, $\tilde{Z}$ is the normal type of $\mathcal{F}$ along $S \setminus F$.

(c) $\tilde{X} = \sum_{j=1}^n \lambda_j w_j (1 + \phi_j(w_n)) \frac{\partial}{\partial w_j}$, where $\phi_j(0) = 0, \forall 1 \le j \le n-1$.

*Proof.* — Let $W_u$ be the hyperplane of $T_p\mathbb{P}^n$ generated by the eigenspaces of $DX(p)$ associated to the eigenvalues $\lambda_1, \dots, \lambda_{n-1}$ and $W_s$ be the eigenspace associated to $\lambda_n$. Recall that we have assumed $W_s = T_pS$, which implies that $W_u$ is transverse to $S$ at $p$. The conditions $\mathrm{Re}(a.\lambda_n) < 0$ and $\mathrm{Re}(a.\lambda_j) > 0$
Let $z_V = (z_1, \dots, z_n)$ and $z_{\hat{V}} = (\hat{z}_1, \dots, \hat{z}_n)$ be the coordinate systems of $V$ and $\hat{V}$ on which (iv), (v) and (vi) are true. We assert that $\hat{z}_j = u_j.z_j$ on $V \cap \hat{V}$, where $u_j(z) \neq 0 \forall z \in V \cap \hat{V}$, $\forall 1 \le j \le n-1$.

In fact, if we fix $1 \le j \le n-1$, by (vi) we must have $\hat{z}_j = u.z_i$, where $u(z) \neq 0 \forall z \in V \cap \hat{V}$, for some $1 \le i \le n-1$. However, the fact that $\rho_\ell \neq \rho_j$ if $\ell \neq j$ implies that $i = j$. It follows that

$$ \theta_{\hat{V}}^{j} = \frac{d\hat{z}_{j}}{\hat{z}_{j}} - \frac{\rho_{j}}{\rho_{1}} \frac{d\hat{z}_{1}}{\hat{z}_{1}} = \frac{dz_{j}}{z_{j}} - \frac{\rho_{j}}{\rho_{1}} \frac{dz_{1}}{z_{1}} + dv = \theta_{V}^{j} + dv, \quad 2 \le j \le n-1, $$

where $v = \log(u_j) - \frac{\rho_j}{\rho_1} \log(u_1)$ is holomorphic. Now, (iv) and (v) imply that

$$ \begin{align*} i_{X_V} \theta_{\hat{V}}^j &= i_{X_V} \theta_V^j = i_{Y_V} \theta_{\hat{V}}^j = i_{Y_V} \theta_V^j = 0 \\ &\implies X_V(v) = Y_V(v) = 0 \\ &\implies v \text{ is a first integral of both vector fields } X_V \text{ and } Y_V \end{align*} $$

Since $X_V = \frac{\partial}{\partial z_n}$ we get $v = v(z_1, \dots, z_{n-1})$. Since the eigenvalues of $Y_V$ are in the Poincaré domain $v(z_1, \dots, z_{n-1})$ must be a constant and $dv = 0$. Hence, $\theta_{\hat{V}}^j = \theta_V^j$ on $V \cap \hat{V}$, as asserted. Therefore there are closed logarithmic 1-forms $\theta_2, \dots, \theta_{n-1}$, defined on $W = \cup_V V$, such that $\mathcal{F}|_W$ is defined by $\theta_2 \wedge \dots \wedge \theta_{n-1}$, as asserted. Let us prove that the forms $\theta_j$ extend to a neighborhood of any point in $F$.

Given $p \in F$ let $\omega$ be a germ of (n-2)-form defining the germ of $\mathcal{F}$ at $p$. Let $(U, z = (x = z_1, \dots, z_n))$ be a coordinate system around $p$ as in (III), so that $z(p) = 0$ and $S \cap U = (z_1 = \dots = z_{n-1} = 0)$. The rotational $X$ of $\omega$ has eigenvalues $\lambda_1, \dots, \lambda_n$ and there exists $a \neq 0$ such that $\mathrm{Re}(a.\lambda_n) < 0$ and $\mathrm{Re}(a.\lambda_j) > 0$, $\forall 1 \le j \le n-1$. Since $p=0$ is an isolated singularity of $X$ there exists another germ of vector field $Z$ such that $\omega = i_Z i_X \nu$, $\nu = dz_1 \wedge \dots \wedge dz_n$. The vector fields $X$ and $Z$ generate the germ of $\mathcal{F}$ at 0.

**LEMMA 4.8.** — There are germs at $p$ of vector fields $\tilde{X}$ and $\tilde{Z}$ that generate the germ of $\mathcal{F}$ at $p$ and a holomorphic coordinate system $(U_1, w = (w_1, \dots, w_n))$ around $p$, with the following properties:

(a) $w(p) = 0$ and $S \cap U_1 = (w_1 = \dots = w_{n-1} = 0)$.

(b) $\tilde{Z}(w) = \sum_{j=1}^{n-1} \rho_j w_j \frac{\partial}{\partial w_j}$. In particular, $\tilde{Z}$ is the normal type of $\mathcal{F}$ along $S \setminus F$.

(c) $\tilde{X} = \sum_{j=1}^n \lambda_j w_j (1 + \phi_j(w_n)) \frac{\partial}{\partial w_j}$, where $\phi_j(0) = 0, \forall 1 \le j \le n-1$.

*Proof.* — Let $W_u$ be the hyperplane of $T_p\mathbb{P}^n$ generated by the eigenspaces of $DX(p)$ associated to the eigenvalues $\lambda_1, \dots, \lambda_{n-1}$ and $W_s$ be the eigenspace associated to $\lambda_n$. Recall that we have assumed $W_s = T_pS$, which implies that $W_u$ is transverse to $S$ at $p$. The conditions $\mathrm{Re}(a.\lambda_n) < 0$ and $\mathrm{Re}(a.\lambda_j) > 0$
implies that the vector field a.X has an unique invariant smooth hypersurface Σ_u tangent to W_u, which meets S transversely at p. This is a consequence of the existence of invariant manifolds for hyperbolic singularities of vector fields (see [15]). The hypersurface Σ_u is the unstable manifold of the vector field a.X. We assert that Σ_u is also Z-invariant. For simplicity, we will assume a = 1.

In the proof we will use the relation:

$$[Z, X] = hX \qquad (4.4)$$

where $h \in O_n$ and $h(0) = 0$. Let us assume (4.4) and prove that Σ is Z-invariant.

Take representatives of Z, X and h defined in some small ball B around 0. Let Z_t and X_t be the local flows of Z and X, respectively. Since Σ_u is the unstable manifold of X the real flow X_t of X satisfies $\lim_{t \to -\infty} X_t(z) = 0 \forall z \in \Sigma \cap B$. Integrating (4.4) we get

$$Z_t^*(X) = \phi_t.X, \quad \text{where } \phi_t(z) = \exp\left(\int_0^t h(Z_s(z))ds\right).$$

The above relation implies that $Z_t$ sends orbits of X on orbits of X. Given $z \in \Sigma \cap B$ denote $O(z) = \{X_t(z) | t \le 0\}$. Since $\lim_{t \to -\infty} X_t(z) = 0$ we get $\overline{O(z)} = O(z) \cup \{0\}$. Let $\tilde{O}(z)$ be the germ of $\overline{O(z)}$ at 0. Note that $\tilde{O}(z) \subset \Sigma$ and that $Z_t(\tilde{O}(z))$ is a germ of curve through 0 such that $Z_t(\tilde{O}(z)) \setminus \{0\}$ is an orbit of X. This of course implies that $Z_t(\tilde{O}(z)) \subset \Sigma$. Hence, Σ is Z-invariant.

*Proof of (4.4).* — Since $d\omega = i_X\nu$ and $\omega = i_Zi_X\nu$ we have

$$\begin{align*} L_Z\omega &= i_Z(d\omega) + d(i_Z\omega) = \omega \\ &\implies L_Z(d\omega) = d\omega \\ &\implies i_X\nu = L_Z(i_X\nu) = i_{[Z,X]}\nu + i_X L_Z\nu = i_{[Z,X]}\nu + \nabla(Z)i_X\nu, \end{align*}$$

where $\nabla(Z) = \sum_j \frac{\partial Z_j}{\partial z_j}$ is the divergence of Z. From this relation we get $[Z, X] = hX$, where $h = 1 - \nabla(Z)$. We assert that $h(0) = 0$.

In fact, let $X_1 = DX(0)$ and $Z_1 = DZ(0)$. Relation (4.4) implies that

$$[Z_1, X_1] = h(0)X_1.$$

The above relation implies that if $h(0) \ne 0$ then $X_1$ is nilpotent, so that $\lambda_1 = \dots = \lambda_n = 0$, a contradiction (see [17]). In particular, we have proved that $X_1$ and $Z_1$ commute. □

Let us continue the proof of Lemma 4.8. After a holomorphic change of variables, we can assume that $\Sigma_u \subset (z_n = 0)$. Since $(z_n = 0)$ is invariant for both vector fields, in the new coordinate system we can write the $n^{th}$
implies that the vector field a.X has an unique invariant smooth hypersurface Σ_u tangent to W_u, which meets S transversely at p. This is a consequence of the existence of invariant manifolds for hyperbolic singularities of vector fields (see [15]). The hypersurface Σ_u is the unstable manifold of the vector field a.X. We assert that Σ_u is also Z-invariant. For simplicity, we will assume a = 1.

In the proof we will use the relation:

$$[Z, X] = hX \qquad (4.4)$$

where $h \in O_n$ and $h(0) = 0$. Let us assume (4.4) and prove that Σ is Z-invariant.

Take representatives of Z, X and h defined in some small ball B around 0. Let $Z_t$ and $X_t$ be the local flows of Z and X, respectively. Since Σ_u is the unstable manifold of X the real flow $X_t$ of X satisfies $\lim_{t \to -\infty} X_t(z) = 0 \forall z \in \Sigma \cap B$. Integrating (4.4) we get

$$Z_t^*(X) = \phi_t.X, \quad \text{where } \phi_t(z) = \exp\left(\int_0^t h(Z_s(z))ds\right).$$

The above relation implies that $Z_t$ sends orbits of X on orbits of X. Given $z \in \Sigma \cap B$ denote $O(z) = \{X_t(z) | t \le 0\}$. Since $\lim_{t \to -\infty} X_t(z) = 0$ we get $\overline{O(z)} = O(z) \cup \{0\}$. Let $\tilde{O}(z)$ be the germ of $\overline{O(z)}$ at 0. Note that $\tilde{O}(z) \subset \Sigma$ and that $Z_t(\tilde{O}(z))$ is a germ of curve through 0 such that $Z_t(\tilde{O}(z)) \setminus \{0\}$ is an orbit of X. This of course implies that $Z_t(\tilde{O}(z)) \subset \Sigma$. Hence, Σ is Z-invariant.

*Proof of (4.4).* — Since $d\omega = i_X\nu$ and $\omega = i_Zi_X\nu$ we have

$$\begin{align*}
L_Z\omega &= i_Z(d\omega) + d(i_Z\omega) = \omega \\
&\implies L_Z(d\omega) = d\omega \\
&\implies i_X\nu = L_Z(i_X\nu) = i_{[Z,X]}\nu + i_X L_Z\nu = i_{[Z,X]}\nu + \nabla(Z)i_X\nu,
\end{align*}$$

where $\nabla(Z) = \sum_j \frac{\partial Z_j}{\partial z_j}$ is the divergence of Z. From this relation we get $[Z, X] = hX$, where $h = 1 - \nabla(Z)$. We assert that $h(0) = 0$.

In fact, let $X_1 = DX(0)$ and $Z_1 = DZ(0)$. Relation (4.4) implies that

$$[Z_1, X_1] = h(0)X_1.$$

The above relation implies that if $h(0) \ne 0$ then $X_1$ is nilpotent, so that $\lambda_1 = \dots = \lambda_n = 0$, a contradiction (see [17]). In particular, we have proved that $X_1$ and $Z_1$ commute. □

Let us continue the proof of Lemma 4.8. After a holomorphic change of variables, we can assume that $\Sigma_u \subset (z_n = 0)$. Since $(z_n = 0)$ is invariant for both vector fields, in the new coordinate system we can write the $n^{th}$
component of X and Z as $\lambda_n z_n(1+h_1(z))$ and $z_n f(z)$, respectively, where $h_1(0) = 0$. If we set $\Psi := -\frac{f(z)}{\lambda_n(1+h_1(z))}$ then the $n^{th}$ component of $\tilde{Z} := Z + \Psi X$ vanishes. Moreover, $\omega = i_{\tilde{Z}}i_X\nu = i_Zi_X\nu$ and $[\tilde{Z}, X] = gX$, where $g = h - X(\Psi)$ and $g(0) = 0$. We assert that there are coordinates $(W, w = (w_1, \dots, w_{n-1}, w_n))$ around $p$ such that

$$ (i) \quad w(p) = 0, \Sigma_u \cap W = (w_n = 0) \text{ and } S \cap W = (w_1 = \dots = w_{n-1} = 0). $$

$$ (ii) \quad \tilde{Z} = \phi(w_n). \sum_{j=1}^{n-1} \rho_j w_j \frac{\partial}{\partial w_j}, \text{ where } \phi(0) \neq 0. $$

In fact, since the $n^{th}$ component of $\tilde{Z}$ vanishes, the hyperplanes $\Sigma_c := (z_n = c)$ are $\tilde{Z}$-invariant. On the other hand, if $c \neq 0$ then $\Sigma_c$ is transverse to $S = (z_1 = \dots = z_{n-1} = 0)$ and so $\tilde{Z}|_{\Sigma_c}$ represents the normal type of $\mathcal{F}$ in the section $\Sigma_c$. Therefore, the eigenvalues of $D\tilde{Z}(0, c)$ are proportional to $\rho_1, \dots, \rho_{n-1}$. In other words there exists a $\phi \in \mathcal{O}_1$ such that the eigenvalues of $D\tilde{Z}(0, c)|_{\Sigma_c}$ are $\phi(c).\rho_1, \dots, \phi(c).\rho_{n-1}$. Considering $\tilde{Z}$ as a 1-parameter family of germs of vector fields at $0 \in \mathbb{C}^{n-1}$ and applying Theorem 4.7 to this family we get (ii) of the assertion. Now, we assert that there exists $\Phi \in \mathcal{O}_n$ such that if we set $\tilde{X} := e^{\Phi}.X$ then

$$ [\tilde{Z}, \tilde{X}] = 0. \qquad (4.5) $$

In fact, if $\Phi \in \mathcal{O}_n$ then

$$ [\tilde{Z}, \tilde{X}] = [\tilde{Z}, e^{\Phi}.X] = e^{\Phi}.\tilde{Z}(\Phi).X + e^{\Phi}.[\tilde{Z}, X] = e^{\Phi}(\tilde{Z}(\Phi) + g)X. $$

Therefore, we have to prove that $\tilde{Z}(\Phi) = -g$ has a solution $\Phi \in \mathcal{O}_n$. Recall that $\tilde{Z} = \phi(w_n).L$, where $L = \sum_{j=1}^{n-1} \rho_j w_j \frac{\partial}{\partial w_j}$. Set $w = (x, w_n)$, $x = (w_1, \dots, w_{n-1})$. We can write

$$ -g(x, w_n) = \sum_{\sigma} b_{\sigma}(w_n) x^{\sigma} $$

where $\sigma = (\sigma_1, \dots, \sigma_{n-1}) \in \mathbb{Z}_{\ge 0}^{n-1}$, $b_{\sigma} \in \mathcal{O}_1$ and $x^{\sigma} = w_1^{\sigma_1} \dots w_{n-1}^{\sigma_{n-1}}$.

Let $\sigma_0 = (0, \dots, 0)$. We will prove below that $b_{\sigma_0} \equiv 0$. Assuming this fact, the equation $\tilde{Z}(\Phi) = -g$ has a formal solution $\Phi = \sum_{\sigma} c_{\sigma}(w_n)x^{\sigma}$ where

$$ c_{\sigma}(w_n) = \frac{b_{\sigma}(w_n)}{\phi(w_n) \langle \rho, \sigma \rangle}, \quad \langle \rho, \sigma \rangle = \sum_{j=1}^{n-1} \rho_j \sigma_j. $$

Since $\rho_1, \dots, \rho_{n-1}$ are in the Poincaré domain we have

$$ \inf \left\{ \left| \langle \rho, \sigma \rangle \right| \mid \sigma \in \mathbb{Z}_{\ge 0}^{n-1}, \sigma \neq (0, \dots, 0) \right\} \ge C, $$

where $C > 0$. This implies that the formal series converges; $\Phi \in \mathcal{O}_n$.
component of X and Z as $\lambda_n z_n(1+h_1(z))$ and $z_n f(z)$, respectively, where $h_1(0) = 0$. If we set $\Psi := -\frac{f(z)}{\lambda_n(1+h_1(z))}$ then the $n^{th}$ component of $\tilde{Z} := Z + \Psi X$ vanishes. Moreover, $\omega = i_{\tilde{Z}}i_X\nu = i_Zi_X\nu$ and $[\tilde{Z}, X] = gX$, where $g = h - X(\Psi)$ and $g(0) = 0$. We assert that there are coordinates $(W, w = (w_1, \dots, w_{n-1}, w_n))$ around $p$ such that

$$ (i) \quad w(p) = 0, \Sigma_u \cap W = (w_n = 0) \text{ and } S \cap W = (w_1 = \dots = w_{n-1} = 0). $$

$$ (ii) \quad \tilde{Z} = \phi(w_n). \sum_{j=1}^{n-1} \rho_j w_j \frac{\partial}{\partial w_j}, \text{ where } \phi(0) \neq 0. $$

In fact, since the $n^{th}$ component of $\tilde{Z}$ vanishes, the hyperplanes $\Sigma_c := (z_n = c)$ are $\tilde{Z}$-invariant. On the other hand, if $c \neq 0$ then $\Sigma_c$ is transverse to $S = (z_1 = \dots = z_{n-1} = 0)$ and so $\tilde{Z}|_{\Sigma_c}$ represents the normal type of $\mathcal{F}$ in the section $\Sigma_c$. Therefore, the eigenvalues of $D\tilde{Z}(0, c)$ are proportional to $\rho_1, \dots, \rho_{n-1}$. In other words there exists a $\phi \in \mathcal{O}_1$ such that the eigenvalues of $D\tilde{Z}(0, c)|_{\Sigma_c}$ are $\phi(c).\rho_1, \dots, \phi(c).\rho_{n-1}$. Considering $\tilde{Z}$ as a 1-parameter family of germs of vector fields at $0 \in \mathbb{C}^{n-1}$ and applying Theorem 4.7 to this family we get (ii) of the assertion. Now, we assert that there exists $\Phi \in \mathcal{O}_n$ such that if we set $\tilde{X} := e^{\Phi}.X$ then

$$ [\tilde{Z}, \tilde{X}] = 0. \tag{4.5} $$

In fact, if $\Phi \in \mathcal{O}_n$ then

$$ [\tilde{Z}, \tilde{X}] = [\tilde{Z}, e^{\Phi}.X] = e^{\Phi}.\tilde{Z}(\Phi).X + e^{\Phi}.[\tilde{Z}, X] = e^{\Phi}(\tilde{Z}(\Phi) + g)X. $$

Therefore, we have to prove that $\tilde{Z}(\Phi) = -g$ has a solution $\Phi \in \mathcal{O}_n$. Recall that $\tilde{Z} = \phi(w_n).L$, where $L = \sum_{j=1}^{n-1} \rho_j w_j \frac{\partial}{\partial w_j}$. Set $w = (x, w_n)$, $x = (w_1, \dots, w_{n-1})$. We can write

$$ -g(x, w_n) = \sum_{\sigma} b_{\sigma}(w_n) x^{\sigma} $$

where $\sigma = (\sigma_1, \dots, \sigma_{n-1}) \in \mathbb{Z}_{\ge 0}^{n-1}$, $b_{\sigma} \in \mathcal{O}_1$ and $x^{\sigma} = w_1^{\sigma_1} \dots w_{n-1}^{\sigma_{n-1}}$.

Let $\sigma_0 = (0, \dots, 0)$. We will prove below that $b_{\sigma_0} \equiv 0$. Assuming this fact, the equation $\tilde{Z}(\Phi) = -g$ has a formal solution $\Phi = \sum_{\sigma} c_{\sigma}(w_n)x^{\sigma}$ where

$$ c_{\sigma}(w_n) = \frac{b_{\sigma}(w_n)}{\phi(w_n) \langle \rho, \sigma \rangle}, \quad \langle \rho, \sigma \rangle = \sum_{j=1}^{n-1} \rho_j \sigma_j. $$

Since $\rho_1, \dots, \rho_{n-1}$ are in the Poincaré domain we have

$$ \inf \left\{ \left| \langle \rho, \sigma \rangle \right| \mid \sigma \in \mathbb{Z}_{\ge 0}^{n-1}, \sigma \neq (0, \dots, 0) \right\} \ge C, $$

where $C > 0$. This implies that the formal series converges; $\Phi \in \mathcal{O}_n$.
Proof that $b_{\sigma_0}(w_n) \equiv 0$, or equivalently $g(0, w_n) \equiv 0$. — First of all, the $n^{th}$ component of $[\tilde{Z}, X]$ is $\tilde{Z}(\tilde{X}_n)$, where $\tilde{X}_n = \lambda_n w_n(1 + h_1(x, w_n))$ is the $n^{th}$ component of $X$. Hence, $[\tilde{Z}, X] = gX$ implies that

$$ 
\begin{aligned}
g \cdot \lambda_n w_n(1+h_1) &= \tilde{Z} (\lambda_n w_n(1+h_1)) = \lambda_n w_n \tilde{Z}(h_1) = \lambda_n w_n \phi(w_n) L(h_1) \\
&\implies g(x, w_n) = \frac{\phi(w_n) \sum_{j=1}^{n-1} \rho_j w_j \frac{\partial h_1}{\partial w_j}}{1+h_1} \\
&\implies g(0, w_n) \equiv 0 \quad \Box
\end{aligned}
$$

From $\tilde{Z} = \phi(w_n) \cdot L$ we get that (4.5) implies:

(1) $D\tilde{X}(0)$ commutes with $L$. In particular, $D\tilde{X}(0)$ is diagonal in the coordinate system $w$ and we can write $\tilde{X} = \sum_{j=1}^n X^j(w) \frac{\partial}{\partial w_j}$, where $X^j(w) = \lambda_j w_j + \text{h.o.t.}, 1 \le j \le n$.

(2) $L(X^n) = 0$. Since the first integrals of $L$ are functions of $w_n$ we get $X^n = \lambda_n w_n(1 + \psi_n(w_n)), \psi_n \in \mathcal{O}_1$.

(3) $\phi(w_n) \cdot L(X^j) = \tilde{X}(\phi(w_n) \rho_j w_j), 1 \le j \le n-1 \iff L(X^j) - \rho_j X^j = \frac{\phi'(w_n)}{\phi(w_n)}(1 + \psi_n(w_n))\rho_n \rho_j w_j w_n, 1 \le j \le n$. (4.6)

Note that (4.6) implies that $w_j|X^j$ if $j < n$ and so $X^j = \lambda_j w_j u$, where $u(0) = 1$. But, in this case we get that $\frac{1}{w_j}(L(X^j) - \rho_j X_j) \in \mathcal{O}_1^*$ which can happens only if $\phi' \equiv 0$ and $L(X^j) = \rho_j X_j$. We can assume that $\phi \equiv 1$. Finally, the solutions of $L(X^j) = \rho_j X^j$ with linear part $\lambda_j w_j$ are of the form $X^j(w) = \lambda_j w_j(1 + \psi_j(w_n)), \psi_j(0) = 0$. This finishes the proof of Lemma 4.8. $\square$

Let us finish the proof that the forms $\theta_j, 2 \le j \le n-1$, extend to a neighborhood of $p \in F$. Define closed logarithmic 1-forms $\tilde{\theta}_j, 2 \le j \le n-1$, by

$$ \tilde{\theta}_j = \frac{dw_j}{w_j} - \frac{\rho_j}{\rho_1} \frac{dw_1}{w_1} - \zeta_j(w_n) \frac{dw_n}{w_n}, $$

where

$$ \zeta_j(w_n) = \frac{\lambda_j(1 + \phi_j(w_n)) - \frac{\rho_j}{\rho_1}\lambda_1(1 + \phi_1(w_n))}{\lambda_n(1 + \phi_n(w_n))}. $$

Note that $\zeta_j(0) \ne 0$, because $\rho_1\lambda_j - \rho_j\lambda_1 \ne 0$. In particular, the pole divisor of $\tilde{\theta}_j$ contains $w_n$ with multiplicity one.

The reader can check directly that $i_{\tilde{Z}}\tilde{\theta}_j = i_{\tilde{X}}\tilde{\theta}_j = 0, \forall 2 \le j \le n-1$, so that $\tilde{\eta} := \tilde{\theta}_2 \wedge \dots \wedge \tilde{\theta}_{n-1}$ defines the germ of $\mathcal{F}$ at $p$. Taking representatives,
Proof that $b_{\sigma_0}(w_n) \equiv 0$, or equivalently $g(0, w_n) \equiv 0$. — First of all, the $n^{th}$ component of $[\tilde{Z}, X]$ is $\tilde{Z}(\tilde{X}_n)$, where $\tilde{X}_n = \lambda_n w_n(1 + h_1(x, w_n))$ is the $n^{th}$ component of $X$. Hence, $[\tilde{Z}, X] = gX$ implies that

$$ 
\begin{aligned}
g \cdot \lambda_n w_n (1 + h_1) &= \tilde{Z} (\lambda_n w_n (1 + h_1)) = \lambda_n w_n \tilde{Z} (h_1) = \lambda_n w_n \phi(w_n) L(h_1) \\
&\implies g(x, w_n) = \frac{\phi(w_n) \sum_{j=1}^{n-1} \rho_j w_j \frac{\partial h_1}{\partial w_j}}{1 + h_1} \\
&\implies g(0, w_n) \equiv 0 \quad \Box
\end{aligned}
$$

From $\tilde{Z} = \phi(w_n) \cdot L$ we get that (4.5) implies:

(1) $D\tilde{X}(0)$ commutes with $L$. In particular, $D\tilde{X}(0)$ is diagonal in the coordinate system $w$ and we can write $\tilde{X} = \sum_{j=1}^{n} X^j(w) \frac{\partial}{\partial w_j}$, where $X^j(w) = \lambda_j w_j + \text{h.o.t.}, 1 \le j \le n$.

(2) $L(X^n) = 0$. Since the first integrals of $L$ are functions of $w_n$ we get $X^n = \lambda_n w_n(1 + \psi_n(w_n))$, $\psi_n \in \mathcal{O}_1$.

(3) $\phi(w_n) \cdot L(X^j) = \tilde{X}(\phi(w_n) \rho_j w_j)$, $1 \le j \le n-1 \iff L(X^j) - \rho_j X^j = \frac{\phi'(w_n)}{\phi(w_n)}(1 + \psi_n(w_n))\rho_n \rho_j w_j w_n, 1 \le j \le n$. (4.6)

Note that (4.6) implies that $w_j|X^j$ if $j < n$ and so $X^j = \lambda_j w_j u$, where $u(0) = 1$. But, in this case we get that $\frac{1}{w_j}(L(X^j) - \rho_j X_j) \in \mathcal{O}_1^*$ which can happens only if $\phi' \equiv 0$ and $L(X^j) = \rho_j X_j$. We can assume that $\phi \equiv 1$. Finally, the solutions of $L(X^j) = \rho_j X^j$ with linear part $\lambda_j w_j$ are of the form $X^j(w) = \lambda_j w_j(1 + \psi_j(w_n))$, $\psi_j(0) = 0$. This finishes the proof of Lemma 4.8. $\square$

Let us finish the proof that the forms $\theta_j, 2 \le j \le n-1$, extend to a neighborhood of $p \in F$. Define closed logarithmic 1-forms $\tilde{\theta}_j, 2 \le j \le n-1$, by

$$ \tilde{\theta}_j = \frac{dw_j}{w_j} - \frac{\rho_j}{\rho_1} \frac{dw_1}{w_1} - \zeta_j(w_n) \frac{dw_n}{w_n}, $$

where

$$ \zeta_j(w_n) = \frac{\lambda_j(1 + \phi_j(w_n)) - \frac{\rho_j}{\rho_1}\lambda_1(1 + \phi_1(w_n))}{\lambda_n(1 + \phi_n(w_n))}. $$

Note that $\zeta_j(0) \ne 0$, because $\rho_1\lambda_j - \rho_j\lambda_1 \ne 0$. In particular, the pole divisor of $\tilde{\theta}_j$ contains $w_n$ with multiplicity one.

The reader can check directly that $i_{\tilde{Z}}\tilde{\theta}_j = i_{\tilde{X}}\tilde{\theta}_j = 0, \forall 2 \le j \le n-1$, so that $\tilde{\eta} := \tilde{\theta}_2 \wedge \cdots \wedge \tilde{\theta}_{n-1}$ defines the germ of $\mathcal{F}$ at $p$. Taking representatives,
we can assume that the $\tilde{\theta}_{j's}$ are defined in some polydisc $\widetilde{W}$ containing $p$ and with $F \cap \widetilde{W} = \{p\}$. We assert that $\tilde{\theta}_j = \theta_j$ on $\widetilde{W} \cap W$, $2 \le j \le n-1$.

In fact, fix a point $q \in S \cap \widetilde{W} \cap W$. We have seen that there are coordinates $(V, z = (z_1, \dots, z_{n-1}, z_n))$ around $q$ such that $z(q) = 0$, $S \cap V = (z_1 = \dots = z_{n-1} = 0)$, $\mathcal{F}|_V$ is generated by the vector fields $X = \frac{\partial}{\partial z_n}$ and $Y = \sum_{j=1}^{n-1} \rho_j z_j \frac{\partial}{\partial z_j}$ and $\theta_j|_V = \frac{dz_j}{z_j} - \frac{\rho_j}{\rho_1} \frac{dz_1}{z_1}$, $2 \le j \le n-1$. Note that $w_n|_V \in \mathcal{O}^*(V)$ and that $w_j|_V = v_j.z_j$, where $v_j \in \mathcal{O}^*(V)$, $1 \le j \le n-1$. This implies that $\tilde{\theta}_j|_V = \theta_j|_V + df_j$, where $f_j$ is a primitive of the closed holomorphic form

$$ \frac{dv_j}{v_j} - \frac{\rho_j}{\rho_1} \frac{dv_1}{v_1} - \zeta_j(w_n) \frac{dw_n}{w_n}|_V. $$

Finally, the fact that $i_X\theta_j = i_Y\theta_j = i_X\tilde{\theta}_j = i_Y\tilde{\theta}_j = 0$ implies that $X(f_j) = Y(f_j) = 0$ and so $f_j$ is a constant and $\tilde{\theta}_j|_V = \theta_j|_V$, $2 \le j \le n-1$.

We have proved that there are closed logarithmic 1-forms $\theta_2, \dots, \theta_{n-1}$ defined in a neighborhood $U$ of $S$ such that $\eta := \theta_2 \wedge \dots \wedge \theta_{n-1}$ defines $\mathcal{F}|_U$. By the extension theorem in [22] the form $\theta_j$ can be extended to closed meromorphic 1-forms on $\mathbb{P}^n$, $2 \le j \le n-1$, denoted by the same symbol. The pole divisor of $\theta_j$ must be reduced because the pole divisor of the restriction $\theta_j|_U$ is reduced. Therefore $\theta_j$ is logarithmic, $2 \le j \le n-1$, and $\eta$ is totally decomposable into logarithmic forms. In particular, there exist $g_1, \dots, g_r$ such that $\eta \in \mathcal{L}_{td}^{n-1}(g_1, \dots, g_r)$. This finishes the proof of Theorem 4.3. $\square$

## 4.4. Proof of Theorem 4.1

Let $M \subset \mathbb{P}^n$ be a $m$-dimensional smooth algebraic submanifold, where $m < n$, and $\mathcal{F}$ be a codimension $p$ holomorphic foliation on $\mathbb{P}^n$, where $p+1 \le m$. Assume that:

(a) The set of tangencies of $\mathcal{F}$ with $M$ has codimension $\ge 2$ on $M$.

(b) $\mathcal{F}|_M$ can be defined by a meromorphic closed $p$-form $\eta$.

We want to prove that $\eta$ admits a closed meromorphic extension $\tilde{\eta}$ defining $\mathcal{F}$ on $\mathbb{P}^n$. In fact, this proof is similar to the proof of the extension theorem of [4] (see also [16, Proposition 3.1.1]). The idea is to prove that $\eta$ admits a closed extension $\hat{\eta}$, defined in a neighborhood $U$ of $M$, such that $\mathcal{F}|_U$ is represented by $\hat{\eta}$. After that, by [2] and [22], the form $\hat{\eta}$ admits a meromorphic extension $\tilde{\eta}$ to $\mathbb{P}^n$. Since $U$ is an open non-empty subset of $\mathbb{P}^n$, it is clear that $\tilde{\eta}$ is closed and defines $\mathcal{F}$ on $\mathbb{P}^n$.
we can assume that the $\tilde{\theta}_{j's}$ are defined in some polydisc $\widetilde{W}$ containing $p$ and with $F \cap \widetilde{W} = \{p\}$. We assert that $\tilde{\theta}_j = \theta_j$ on $\widetilde{W} \cap W$, $2 \le j \le n-1$.

In fact, fix a point $q \in S \cap \widetilde{W} \cap W$. We have seen that there are coordinates $(V, z = (z_1, \dots, z_{n-1}, z_n))$ around $q$ such that $z(q) = 0$, $S \cap V = (z_1 = \dots = z_{n-1} = 0)$, $\mathcal{F}|_V$ is generated by the vector fields $X = \frac{\partial}{\partial z_n}$ and $Y = \sum_{j=1}^{n-1} \rho_j z_j \frac{\partial}{\partial z_j}$ and $\theta_j|_V = \frac{dz_j}{z_j} - \frac{\rho_j}{\rho_1} \frac{dz_1}{z_1}$, $2 \le j \le n-1$. Note that $w_n|_V \in \mathcal{O}^*(V)$ and that $w_j|_V = v_j.z_j$, where $v_j \in \mathcal{O}^*(V)$, $1 \le j \le n-1$. This implies that $\tilde{\theta}_j|_V = \theta_j|_V + df_j$, where $f_j$ is a primitive of the closed holomorphic form

$$ \frac{dv_j}{v_j} - \frac{\rho_j}{\rho_1} \frac{dv_1}{v_1} - \zeta_j(w_n) \frac{dw_n}{w_n}|_V. $$

Finally, the fact that $i_X\theta_j = i_Y\theta_j = i_X\tilde{\theta}_j = i_Y\tilde{\theta}_j = 0$ implies that $X(f_j) = Y(f_j) = 0$ and so $f_j$ is a constant and $\tilde{\theta}_j|_V = \theta_j|_V$, $2 \le j \le n-1$.

We have proved that there are closed logarithmic 1-forms $\theta_2, \dots, \theta_{n-1}$ defined in a neighborhood $U$ of $S$ such that $\eta := \theta_2 \wedge \dots \wedge \theta_{n-1}$ defines $\mathcal{F}|_U$. By the extension theorem in [22] the form $\theta_j$ can be extended to closed meromorphic 1-forms on $\mathbb{P}^n$, $2 \le j \le n-1$, denoted by the same symbol. The pole divisor of $\theta_j$ must be reduced because the pole divisor of the restriction $\theta_j|_U$ is reduced. Therefore $\theta_j$ is logarithmic, $2 \le j \le n-1$, and $\eta$ is totally decomposable into logarithmic forms. In particular, there exist $g_1, \dots, g_r$ such that $\eta \in \mathcal{L}_{td}^{n-1}(g_1, \dots, g_r)$. This finishes the proof of Theorem 4.3. $\square$

## 4.4. Proof of Theorem 4.1

Let $M \subset \mathbb{P}^n$ be a $m$-dimensional smooth algebraic submanifold, where $m < n$, and $\mathcal{F}$ be a codimension $p$ holomorphic foliation on $\mathbb{P}^n$, where $p+1 \le m$. Assume that:

(a) The set of tangencies of $\mathcal{F}$ with $M$ has codimension $\ge 2$ on $M$.

(b) $\mathcal{F}|_M$ can be defined by a meromorphic closed $p$-form $\eta$.

We want to prove that $\eta$ admits a closed meromorphic extension $\tilde{\eta}$ defining $\mathcal{F}$ on $\mathbb{P}^n$. In fact, this proof is similar to the proof of the extension theorem of [4] (see also [16, Proposition 3.1.1]). The idea is to prove that $\eta$ admits a closed extension $\tilde{\eta}$, defined in a neighborhood $U$ of $M$, such that $\mathcal{F}|_U$ is represented by $\tilde{\eta}$. After that, by [2] and [22], the form $\tilde{\eta}$ admits a meromorphic extension $\tilde{\eta}$ to $\mathbb{P}^n$. Since $U$ is an open non-empty subset of $\mathbb{P}^n$, it is clear that $\tilde{\eta}$ is closed and defines $\mathcal{F}$ on $\mathbb{P}^n$.
Let $X = \text{Sing}(\mathcal{F}|_M)$. Note that $X = \text{Tang}(\mathcal{F}, M) \cup (\text{Sing}(\mathcal{F}) \cap M)$, where $\text{Tang}(\mathcal{F}, M)$ denotes the set of tangencies of $\mathcal{F}$ and $M$. By (a) we have $\text{cod}_M(X) \ge 2$. We begin by extending $\eta$ to a neighborhood of $M \setminus X$.

**Extension to a neighborhood of $M \setminus X$.** — By definition, the foliation $\mathcal{F}$ is transverse to $M$ at the points of $M \setminus X$. In particular, given $q \in M \setminus X$ there exists a local coordinate system around $q$, $z = (z_1, \dots, z_n) : W \to \mathbb{C}^n$, with $z(W)$ a polydisc of $\mathbb{C}^n$, $z(q) = 0 \in \mathbb{C}^n$, and such that

(i) $M \cap W = (z_{m+1} = \dots = z_n = 0)$.

(ii) The leaves of $\mathcal{F}|_W$ are the levels $z_1 = ct_1, \dots, z_p = ct_p$.

In particular, $\mathcal{F}|_W$ is defined by the form $\Omega_W = dz_1 \wedge \dots \wedge dz_p$. Since $\mathcal{F}|_{W \cap M}$ is also defined by $\eta|_{W \cap M}$ we must have $\eta|_{W \cap M} = f \cdot \Omega_W|_{W \cap M}$, where $f = f(z_1, \dots, z_m)$ is meromorphic on $W \cap M$. Since $\eta$ and $\Omega_W$ are closed we get $\text{d}f \wedge \Omega_W = 0$, which is equivalent to

$$ \frac{\partial f}{\partial z_j} = 0, \quad \forall p+1 \le j \le m $$

⇒ $f(z_1, \dots, z_m) = f(z_1, \dots, z_p) : f$ depends only of $z_1, \dots, z_p$.

In particular, $\eta|_{W \cap M}$ admits a unique closed meromorphic extension to $W$ defining $\mathcal{F}|_W: \hat{\eta}_W = f(z_1, \dots, z_p)dz_1 \wedge \dots \wedge dz_p$. This proves that $\eta|_{M \setminus X}$ admits an unique closed meromorphic extension $\hat{\eta}$ to a neighborhood $V$ of $M \setminus X$ representing $\mathcal{F}|_V$.

**Extension of $\hat{\eta}$ to a neighborhood of $M$.** — Since $\text{cod}_M(X) \ge 2$, given $q \in X$ we can find a Hartog's domain $H \subset V$ such that $q \in \hat{H}$, the holomorphic closure of $H$ (for the details see [16]). Therefore, $\hat{\eta}$ admits a meromorphic extension to a neighborhood of $q$, by Levi's extension theorem [24]. In particular, $\eta$ can be extended to a closed meromorphic p-form $\tilde{\eta}$ defining $\mathcal{F}$ on $\mathbb{P}^n$ by [2] and [22].

Let us assume now that $\eta$ is logarithmic and let $(\tilde{\eta})_\infty = \tilde{S}_1^{k_1} \dots \tilde{S}_r^{k_r}$ be the decomposition of the pole divisor of $\tilde{\eta}$ into irreducible components. The pole divisor of $\eta$ will be then $(\eta)_\infty = (\tilde{\eta})_\infty \cap M$, which is reduced because $\eta$ is logarithmic. Hence, $k_1 = \dots = k_r = 1$ and $\tilde{\eta}$ is logarithmic. $\square$

### 4.5. End of the proof of Theorem 1.17

Recall that we want to prove that, if $k \ge 3$, $n \ge 5$ and $r \ge n-k+2 := p+2$ then $\mathcal{L}_{td}(d_1, \dots, d_r; k, n)$ is an irreducible component of $\text{Fol}(D, k, n)$, where $D = \sum_j d_j - n + k - 1$. Fix $f_1, \dots, f_r$ homogeneous polynomials on $\mathbb{C}^{n+1}$ with the following properties:
Let $X = \text{Sing}(\mathcal{F}|_M)$. Note that $X = \text{Tang}(\mathcal{F}, M) \cup (\text{Sing}(\mathcal{F}) \cap M)$, where $\text{Tang}(\mathcal{F}, M)$ denotes the set of tangencies of $\mathcal{F}$ and $M$. By (a) we have $\text{cod}_M(X) \ge 2$. We begin by extending $\eta$ to a neighborhood of $M \setminus X$.

**Extension to a neighborhood of $M \setminus X$.** — By definition, the foliation $\mathcal{F}$ is transverse to $M$ at the points of $M \setminus X$. In particular, given $q \in M \setminus X$ there exists a local coordinate system around $q$, $z = (z_1, \dots, z_n) : W \to \mathbb{C}^n$, with $z(W)$ a polydisc of $\mathbb{C}^n$, $z(q) = 0 \in \mathbb{C}^n$, and such that

(i) $M \cap W = (z_{m+1} = \cdots = z_n = 0)$.

(ii) The leaves of $\mathcal{F}|_W$ are the levels $z_1 = ct_1, \dots, z_p = ct_p$.

In particular, $\mathcal{F}|_W$ is defined by the form $\Omega_W = \text{d}z_1 \wedge \cdots \wedge \text{d}z_p$. Since $\mathcal{F}|_{W \cap M}$ is also defined by $\eta|_{W \cap M}$ we must have $\eta|_{W \cap M} = f \cdot \Omega_W|_{W \cap M}$, where $f = f(z_1, \dots, z_m)$ is meromorphic on $W \cap M$. Since $\eta$ and $\Omega_W$ are closed we get $\text{d}f \wedge \Omega_W = 0$, which is equivalent to

$$ \frac{\partial f}{\partial z_j} = 0, \quad \forall p+1 \le j \le m \\ \implies f(z_1, \dots, z_m) = f(z_1, \dots, z_p) : f \text{ depends only of } z_1, \dots, z_p. $$

In particular, $\eta|_{W \cap M}$ admits a unique closed meromorphic extension to $W$ defining $\mathcal{F}|_W: \hat{\eta}_W = f(z_1, \dots, z_p)\text{d}z_1 \wedge \cdots \wedge \text{d}z_p$. This proves that $\eta|_{M \setminus X}$ admits a unique closed meromorphic extension $\hat{\eta}$ to a neighborhood $V$ of $M \setminus X$ representing $\mathcal{F}|_V$.

**Extension of $\hat{\eta}$ to a neighborhood of $M$.** — Since $\text{cod}_M(X) \ge 2$, given $q \in X$ we can find a Hartog's domain $H \subset V$ such that $q \in \hat{H}$, the holomorphic closure of $H$ (for the details see [16]). Therefore, $\hat{\eta}$ admits a meromorphic extension to a neighborhood of $q$, by Levi's extension theorem [24]. In particular, $\eta$ can be extended to a closed meromorphic p-form $\tilde{\eta}$ defining $\mathcal{F}$ on $\mathbb{P}^n$ by [2] and [22].

Let us assume now that $\eta$ is logarithmic and let $(\tilde{\eta})_\infty = \tilde{S}_1^{k_1} \dots \tilde{S}_r^{k_r}$ be the decomposition of the pole divisor of $\tilde{\eta}$ into irreducible components. The pole divisor of $\eta$ will be then $(\eta)_\infty = (\tilde{\eta})_\infty \cap M$, which is reduced because $\eta$ is logarithmic. Hence, $k_1 = \cdots = k_r = 1$ and $\tilde{\eta}$ is logarithmic. $\square$

### 4.5. End of the proof of Theorem 1.17

Recall that we want to prove that, if $k \ge 3$, $n \ge 5$ and $r \ge n-k+2 := p+2$ then $\mathcal{L}_{td}(d_1, \dots, d_r; k, n)$ is an irreducible component of $\text{Fol}(D, k, n)$, where $D = \sum_j d_j - n + k - 1$. Fix $f_1, \dots, f_r$ homogeneous polynomials on $\mathbb{C}^{n+1}$ with the following properties:
(i) $\deg(f_j) = d_j, 1 \le j \le r.$

(ii) the hypersurface $(f_1 \dots f_r = 0)$ has strictly ordinary singularities outside $0 \in \mathbb{C}^{n+1}$.

Set $m = n - k + 2$ and let $\mathbb{P}^m \simeq \Sigma \subset \mathbb{P}^n$ be a m-plane such that:

(iii) If $\mathbb{C}^{m+1} \simeq E = \Pi^{-1}(\Sigma) \cup \{0\} \subset \mathbb{C}^{n+1}$ and $\tilde{f}_j = f_j|_E, 1 \le j \le r,$
then $(\tilde{f}_1 \dots \tilde{f}_r = 0)$ has strictly ordinary singularities outside $0 \in E$.

Such m-plane $E$ exists by transversality theory. In fact, it is sufficient to choose $E$ in such a way that for any sequence $I = (i_1 < \dots < i_s) \in S_s^r$, where $1 \le s \le n-1$, then the algebraic smooth set $\Pi(f_{i_1} = \dots = f_{i_s} = 0) \subset \mathbb{P}^n$ meets transversely $\Sigma = \Pi(E)$ (see Definition 1.2). We leave the details for the reader.

Since $m-2=n-k=p$, then for any $\mathcal{F} \in \mathcal{L}_{td}^p(f_1, \dots, f_r)$ we have
$\mathcal{F}|_{\Sigma} \in \mathcal{L}_{td}^{m-2}(\tilde{f}_1, \dots, \tilde{f}_r)$, so that $\mathcal{F}|_{\Sigma}$ is a two dimensional foliation. Given
a 1-form $\theta = \sum_j \lambda_j \frac{df_j}{f_j} \in \mathcal{L}_{\mathcal{F}}^1(f_1, \dots, f_r)$, we set $\tilde{\theta} = \sum_{j=1}^r \lambda_j \frac{d\tilde{f}_j}{\tilde{f}_j}$.

Choose $\mathcal{F}_o \in \mathcal{L}_{td}^p(f_1, \dots, f_r)$ defined by a logarithmic form $\eta_o = \theta_2 \wedge \dots \wedge \theta_{p+1}$, where $\theta_2, \dots, \theta_{p+1}$ are as in Lemma 4.4. We assume also $\mathcal{F}_o|_{\Sigma}$ satisfies Remark 4.5: if $\mathcal{F}_o|_{\Sigma} \in \mathcal{L}_{td}(d'_1, \dots, d'_s; 2, m)$ then $s=r$ and $d'_j = d_j$, $1 \le j \le r$.

Let $(\mathcal{F}_t)_{t \in (\mathbb{C}, 0)}$ be a germ of holomorphic 1-parameter family of foliations in $\text{Fol}(D, k, n)$ such that $\mathcal{F}_t|_{(t=0)} = \mathcal{F}_o$. Consider the germ of 1-parameter family of two dimensional foliations $\tilde{\mathcal{F}}_t := \mathcal{F}_t|_{\Sigma}$, $t \in (\mathbb{C}, 0)$. By the proof in Section 4.2 we get $\tilde{\mathcal{F}}_t \in \mathcal{L}_{td}(d_1, \dots, d_r; 2, m)$, $\forall t \in (\mathbb{C}, 0)$, so that $\tilde{\mathcal{F}}_t$ can be defined in homogeneous coordinates by a $m-2=n-k$ logarithmic form $\tilde{\eta}_t \in \mathcal{L}^{m-2}(\tilde{f}_{1t}, \dots, \tilde{f}_{rt})$, where $\tilde{f}_{jt}|_{t=0} = \tilde{f}_j, 1 \le j \le r$. By Theorem 4.1 the foliation $\mathcal{F}_t \in \text{Fol}(D, k, n)$ is logarithmic, $\forall t \in (\mathbb{C}, 0)$, so that $\mathcal{F}_t \in \mathcal{L}(d_1(t), \dots, d_{st}(t); k, n)$. We assert that $s_t = r$ and $d_j(t) = d_j, 1 \le j \le r$.

In fact, since $\mathcal{F}_t \in \mathcal{L}(d_1(t), \dots, d_{st}(t); k, n)$ we get $\tilde{\mathcal{F}}_t \in \mathcal{L}(d_1(t), \dots, d_{st}(t); 2, m)$. Therefore, as in the proof of the two dimensional case, we have $s_t = r$ and $d_j(t) = d_j, 1 \le j \le r, \forall t \in (\mathbb{C}, 0)$. Finally, by Corollary 3.6 we get $\mathcal{F}_t \in \mathcal{L}_{td}(d_1, \dots, d_r; k, n), \forall t \in (\mathbb{C}, 0)$. This finishes the proof of Theorem 1.17. $\square$

**5. Linear pull-back foliations**

The purpose of this section is to prove a result of “trivialisation” of holomorphic foliations on projective spaces:
(i) $\deg(f_j) = d_j, 1 \le j \le r.$

(ii) the hypersurface $(f_1 \dots f_r = 0)$ has strictly ordinary singularities outside $0 \in \mathbb{C}^{n+1}$.

Set $m = n - k + 2$ and let $\mathbb{P}^m \simeq \Sigma \subset \mathbb{P}^n$ be a m-plane such that:

(iii) If $\mathbb{C}^{m+1} \simeq E = \Pi^{-1}(\Sigma) \cup \{0\} \subset \mathbb{C}^{n+1}$ and $\tilde{f}_j = f_j|_E, 1 \le j \le r,$
then $(\tilde{f}_1 \dots \tilde{f}_r = 0)$ has strictly ordinary singularities outside $0 \in E$.

Such m-plane $E$ exists by transversality theory. In fact, it is sufficient to choose $E$ in such a way that for any sequence $I = (i_1 < \dots < i_s) \in S_s^r$, where $1 \le s \le n-1$, then the algebraic smooth set $\Pi(f_{i_1} = \dots = f_{i_s} = 0) \subset \mathbb{P}^n$ meets transversely $\Sigma = \Pi(E)$ (see Definition 1.2). We leave the details for the reader.

Since $m-2=n-k=p$, then for any $\mathcal{F} \in \mathcal{L}_{td}^p(f_1, \dots, f_r)$ we have
$\mathcal{F}|_{\Sigma} \in \mathcal{L}_{td}^{m-2}(\tilde{f}_1, \dots, \tilde{f}_r)$, so that $\mathcal{F}|_{\Sigma}$ is a two dimensional foliation. Given
a 1-form $\theta = \sum_j \lambda_j \frac{df_j}{f_j} \in \mathcal{L}_{\mathcal{F}}^1(f_1, \dots, f_r)$, we set $\tilde{\theta} = \sum_{j=1}^r \lambda_j \frac{d\tilde{f}_j}{\tilde{f}_j}$.

Choose $\mathcal{F}_o \in \mathcal{L}_{td}^p(f_1, \dots, f_r)$ defined by a logarithmic form $\eta_o = \theta_2 \wedge \dots \wedge \theta_{p+1}$, where $\theta_2, \dots, \theta_{p+1}$ are as in Lemma 4.4. We assume also $\mathcal{F}_o|_{\Sigma}$ satisfies Remark 4.5: if $\mathcal{F}_o|_{\Sigma} \in \mathcal{L}_{td}(d'_1, \dots, d'_s; 2, m)$ then $s=r$ and $d'_j = d_j$, $1 \le j \le r$.

Let $(\mathcal{F}_t)_{t \in (\mathbb{C}, 0)}$ be a germ of holomorphic 1-parameter family of foliations in $\text{Fol}(D, k, n)$ such that $\mathcal{F}_t|_{(t=0)} = \mathcal{F}_o$. Consider the germ of 1-parameter family of two dimensional foliations $\tilde{\mathcal{F}}_t := \mathcal{F}_t|_{\Sigma}$, $t \in (\mathbb{C}, 0)$. By the proof in Section 4.2 we get $\tilde{\mathcal{F}}_t \in \mathcal{L}_{td}(d_1, \dots, d_r; 2, m)$, $\forall t \in (\mathbb{C}, 0)$, so that $\tilde{\mathcal{F}}_t$ can be defined in homogeneous coordinates by a $m-2=n-k$ logarithmic form $\tilde{\eta}_t \in \mathcal{L}^{m-2}(\tilde{f}_{1t}, \dots, \tilde{f}_{rt})$, where $\tilde{f}_{jt}|_{t=0} = \tilde{f}_j, 1 \le j \le r$. By Theorem 4.1 the foliation $\mathcal{F}_t \in \text{Fol}(D, k, n)$ is logarithmic, $\forall t \in (\mathbb{C}, 0)$, so that $\mathcal{F}_t \in \mathcal{L}(d_1(t), \dots, d_{st}(t); k, n)$. We assert that $s_t = r$ and $d_j(t) = d_j, 1 \le j \le r$.

In fact, since $\mathcal{F}_t \in \mathcal{L}(d_1(t), \dots, d_{st}(t); k, n)$ we get $\tilde{\mathcal{F}}_t \in \mathcal{L}(d_1(t), \dots, d_{st}(t); 2, m)$. Therefore, as in the proof of the two dimensional case, we have $s_t = r$ and $d_j(t) = d_j, 1 \le j \le r, \forall t \in (\mathbb{C}, 0)$. Finally, by Corollary 3.6 we get $\mathcal{F}_t \in \mathcal{L}_{td}(d_1, \dots, d_r; k, n), \forall t \in (\mathbb{C}, 0)$. This finishes the proof of Theorem 1.17. $\square$

**5. Linear pull-back foliations**

The purpose of this section is to prove a result of “trivialisation” of holomorphic foliations on projective spaces:
**THEOREM 5.1.** — Let $\mathcal{G}$ be a codimension $p$ foliation on $\mathbb{P}^n$, where $n \ge p+2$. Assume that there is a $p+1$ plane $\Sigma \simeq \mathbb{P}^{p+1}$ such that the foliation $\mathcal{F} := \mathcal{G}|_{\Sigma}$ has singular set $\text{Sing}(\mathcal{F})$ of codimension $\ge 3$. Then $\mathcal{G}$ is the pull-back of $\mathcal{F}$ by some linear projection $T: \mathbb{P}^n \to \Sigma$. In particular, there exists an affine coordinate system $(z, w) \in \mathbb{C}^{p+1} \times \mathbb{C}^{n-p-1} = \mathbb{C}^n \subset \mathbb{P}^n$ such that $\mathcal{G}$ is represented in these coordinates by a $p$-form depending only of $z$ and $dz$:

$$ \eta = \sum_{j=1}^{p+1} P_j(z) dz_1 \wedge \cdots \wedge \widehat{dz_j} \wedge \cdots \wedge dz_{p+1} = i_X dz_1 \wedge \cdots \wedge dz_n, $$ 
where $X = \sum_{j=1}^{p+1} (-1)^{j-1} P_j(z) \frac{\partial}{\partial z_j}.$

Theorem 5.2, that will be used in the proof of Theorem 5.1, is a local version of it. Let $Z_o \neq 0$ be a germ at $(\mathbb{C}^{p+1}, 0)$ of holomorphic vector field, where $p+1 \ge 3$. The germ of foliation defined by $Z_o$ is also defined by the germ of $p$-form $\eta_o = i_{Z_o} \nu$, where $\nu = dz_1 \wedge \cdots \wedge dz_{p+1}$. If $Z_o = \sum_{j=1}^{p+1} f_j(z) \frac{\partial}{\partial z_j}$ then

$$ \eta_o = \sum_{j=1}^{p+1} (-1)^{j-1} f_j(z) dz_1 \wedge \cdots \wedge \widehat{dz_j} \wedge \cdots \wedge dz_{p+1}. $$

We will assume that there is a germ of integrable holomorphic $p$-form $\eta$ at $0 \in \mathbb{C}^n$, where $\mathbb{C}^n = \mathbb{C}^{p+1} \times \mathbb{C}^{n-p-1}$, $n > p+1$, such that $\eta_o = i^*\eta$, where $i$ is the inclusion $\mathbb{C}^{p+1} \mapsto \mathbb{C}^{p+1} \times \mathbb{C}^{n-p-1}$.

**THEOREM 5.2.** — In the above situation, assume that cod($\text{Sing}(Z_o)$) $\ge 3$. Then there exists a local coordinate system $(z, w) \in (\mathbb{C}^{p+1} \times \mathbb{C}^{n-p-1}, (0,0))$ and an unity $\phi \in \mathcal{O}_n^*$ such that

$$ \eta = \phi \sum_{j=1}^{p+1} (-1)^{j-1} f_j(z) dz_1 \wedge \cdots \wedge \widehat{dz_j} \wedge \cdots \wedge dz_{p+1} = \phi i_{Z_o} dz_1 \wedge \cdots \wedge dz_{p+1}. $$

In particular, the foliation generated by $\eta$ is equivalent to the product of the singular one dimensional foliation generated by $Z_o$ by the non-singular foliation of dimension $n-p-1$ with leaves $z = \text{constant}$.

We begin with the proof of Theorem 5.2.

## 5.1. Proof of Theorem 5.2

Let $\eta_o = i_{Z_o} \nu$ be the germ of $p$-form on $(\mathbb{C}^{p+1}, 0)$ which can be extended to a germ of integrable $p$-form $\eta$ on $(\mathbb{C}^n = \mathbb{C}^{p+1} \times \mathbb{C}^{n-p-1}, (0,0))$. As in the hypothesis of Theorem 5.2 we will assume that cod($\text{Sing}(Z_o)$) $\ge 3$.
**THEOREM 5.1.** — Let $\mathcal{G}$ be a codimension $p$ foliation on $\mathbb{P}^n$, where $n \ge p+2$. Assume that there is a $p+1$ plane $\Sigma \simeq \mathbb{P}^{p+1}$ such that the foliation $\mathcal{F} := \mathcal{G}|_{\Sigma}$ has singular set $\text{Sing}(\mathcal{F})$ of codimension $\ge 3$. Then $\mathcal{G}$ is the pull-back of $\mathcal{F}$ by some linear projection $T: \mathbb{P}^n \to \Sigma$. In particular, there exists an affine coordinate system $(z, w) \in \mathbb{C}^{p+1} \times \mathbb{C}^{n-p-1} = \mathbb{C}^n \subset \mathbb{P}^n$ such that $\mathcal{G}$ is represented in these coordinates by a $p$-form depending only of $z$ and $dz$:

$$ \eta = \sum_{j=1}^{p+1} P_j(z) dz_1 \wedge \cdots \wedge \widehat{dz_j} \wedge \cdots \wedge dz_{p+1} = i_X dz_1 \wedge \cdots \wedge dz_n, $$ 
$$ \text{where } X = \sum_{j=1}^{p+1} (-1)^{j-1} P_j(z) \frac{\partial}{\partial z_j}. $$

Theorem 5.2, that will be used in the proof of Theorem 5.1, is a local version of it. Let $Z_o \neq 0$ be a germ at $(\mathbb{C}^{p+1}, 0)$ of holomorphic vector field, where $p+1 \ge 3$. The germ of foliation defined by $Z_o$ is also defined by the germ of $p$-form $\eta_o = i_{Z_o} \nu$, where $\nu = dz_1 \wedge \cdots \wedge dz_{p+1}$. If $Z_o = \sum_{j=1}^{p+1} f_j(z) \frac{\partial}{\partial z_j}$ then

$$ \eta_o = \sum_{j=1}^{p+1} (-1)^{j-1} f_j(z) dz_1 \wedge \cdots \wedge \widehat{dz_j} \wedge \cdots \wedge dz_{p+1}. $$

We will assume that there is a germ of integrable holomorphic $p$-form $\eta$ at $0 \in \mathbb{C}^n$, where $\mathbb{C}^n = \mathbb{C}^{p+1} \times \mathbb{C}^{n-p-1}$, $n > p+1$, such that $\eta_o = i^*\eta$, where $i$ is the inclusion $\mathbb{C}^{p+1} \mapsto \mathbb{C}^{p+1} \times \mathbb{C}^{n-p-1}$.

**THEOREM 5.2.** — In the above situation, assume that cod($\text{Sing}(Z_o)$) $\ge 3$. Then there exists a local coordinate system $(z, w) \in (\mathbb{C}^{p+1} \times \mathbb{C}^{n-p-1}, (0,0))$ and an unity $\phi \in \mathcal{O}_n^*$ such that

$$ \eta = \phi \sum_{j=1}^{p+1} (-1)^{j-1} f_j(z) dz_1 \wedge \cdots \wedge \widehat{dz_j} \wedge \cdots \wedge dz_{p+1} = \phi i_{Z_o} dz_1 \wedge \cdots \wedge dz_{p+1}. $$

In particular, the foliation generated by $\eta$ is equivalent to the product of the singular one dimensional foliation generated by $Z_o$ by the non-singular foliation of dimension $n-p-1$ with leaves $z = \text{constant}$.

We begin with the proof of Theorem 5.2.

## 5.1. Proof of Theorem 5.2

Let $\eta_o = i_{Z_o} \nu$ be the germ of $p$-form on $(\mathbb{C}^{p+1}, 0)$ which can be extended to a germ of integrable $p$-form $\eta$ on $(\mathbb{C}^n = \mathbb{C}^{p+1} \times \mathbb{C}^{n-p-1}, (0,0))$. As in the hypothesis of Theorem 5.2 we will assume that cod($\text{Sing}(Z_o)$) $\ge 3$.
The points in $\mathbb{C}^n = \mathbb{C}^{p+1} \times \mathbb{C}^{n-p-1}$ will be denoted $(z, y)$, where $z \in \mathbb{C}^{p+1}$ and $y \in \mathbb{C}^{n-p-1}$. We will consider representatives of $Z_o$, $\eta_o$ and $\eta$, denoted by the same letters, the first two defined in a neighborhood $V \subset \mathbb{C}^{p+1}$ of a closed polydisc $\bar{U}$ and the last defined in a neighborhood of $\bar{U} \times \{0\}$ in $\mathbb{C}^n$, so that

$$ \eta_o = \eta|_{V \times \{0\}} = i_{Z_o} \mathrm{d}z_1 \wedge \cdots \wedge \mathrm{d}z_{p+1}. $$

We will assume $\mathrm{cod}_V(\mathrm{Sing}(Z_o)) \ge 3$. Define a holomorphic vector field $Z$ in a neighborhood of $\bar{U} \times \{0\}$ by $Z(z, y_o) = Z_{y_o}(z) = \sum_{j=1}^{p+1} g_j(z, y_o) \frac{\partial}{\partial z_j}$, where

$$ \eta|_{(y=y_o)} = i_{Z_{y_o}} \mathrm{d}z_1 \wedge \cdots \wedge \mathrm{d}z_{p+1}. $$

Since $\eta$ is locally totally decomposable. outside its singular set, we have $i_Z \eta = 0$.

Note that $Z(z, 0) = Z_o$. Therefore, the hypothesis implies that there is a neighborhood $W$ of $\bar{U} \times \{0\}$ in $\mathbb{C}^n$ such that $\mathrm{cod}(\mathrm{Sing}(Z)) \ge 3$ and $W \cap (y=0) = V \times \{0\}$.

We assert that there are holomorphic vector fields $X_1, \dots, X_{n-p-1}$ defined in a smaller neighborhood of $\bar{U} \times \{0\}$, such that $i_{X_j} \eta = 0$ and

$$ X_j(x, y) = \frac{\partial}{\partial y_j} + \sum_{i=1}^{p+1} h_{ji}(z, y) \frac{\partial}{\partial z_i}, \quad \forall 1 \le j \le n-p-1. \quad (5.1) $$

First of all, we note that the above assertion is true in a neighborhood of any point $(z_o, 0) \in (V \times \{0\}) \setminus \mathrm{Sing}(Z_o)$. This is true because for $(z, y)$ in a neighborhood $U_\alpha$ of $(z_o, 0)$ some component of $Z(z, y)$ does not vanishes, say $g_{p+1}(z, y) \ne 0$, so that

$$ \frac{(-1)^p}{g_{p+1}} \eta|_{U_\alpha} = \mathrm{d}z_1 \wedge \cdots \wedge \mathrm{d}z_p + \tilde{\Theta}, $$

where $\tilde{\Theta} \wedge \mathrm{d}z_{p+1} \wedge \mathrm{d}y_1 \wedge \cdots \wedge \mathrm{d}y_{n-p-1} \equiv 0$. As the reader can check this implies the existence of holomorphic vector fields $X_{j\alpha}$ on $U_\alpha$ as in (5.1), $1 \le j \le n-p-1$. It follows that there are:

• a polydisc $Q = Q_1 \times Q_2 \subset \mathbb{C}^{p+1} \times \mathbb{C}^{n-p-1}$, with $Q_1 \supset \bar{U}$ and $0 \in Q_2$.

• a covering $\mathcal{U} = (U_\alpha)_{\alpha \in A}$ of $Q \setminus \mathrm{Sing}(\eta)$ by polydiscs,

• n-p-1 collections of holomorphic vector fields $(X_{j\alpha})_{\alpha \in A}$, $1 \le j \le n-p-1$, $X_{j\alpha} \in \mathcal{X}(U_\alpha)$,

such that

(i) $\mathrm{cod}_Q(\mathrm{Sing}(Z|_Q)) \ge 3$.

(ii) $X_{j\alpha} = \frac{\partial}{\partial y_j} + \sum_{i=1}^{p+1} g_{i\alpha}(z, y) \frac{\partial}{\partial z_i}$.

(iii) $i_{X_{j\alpha}} \eta = 0, \forall 1 \le j \le n-p-1, \forall \alpha \in A$.
The points in $\mathbb{C}^n = \mathbb{C}^{p+1} \times \mathbb{C}^{n-p-1}$ will be denoted $(z, y)$, where $z \in \mathbb{C}^{p+1}$ and $y \in \mathbb{C}^{n-p-1}$. We will consider representatives of $Z_o$, $\eta_o$ and $\eta$, denoted by the same letters, the first two defined in a neighborhood $V \subset \mathbb{C}^{p+1}$ of a closed polydisc $\bar{U}$ and the last defined in a neighborhood of $\bar{U} \times \{0\}$ in $\mathbb{C}^n$, so that

$$ \eta_o = \eta|_{V \times \{0\}} = i_{Z_o} \mathrm{d}z_1 \wedge \cdots \wedge \mathrm{d}z_{p+1}. $$

We will assume $\mathrm{cod}_V(\mathrm{Sing}(Z_o)) \ge 3$. Define a holomorphic vector field $Z$ in a neighborhood of $\bar{U} \times \{0\}$ by $Z(z, y_o) = Z_{y_o}(z) = \sum_{j=1}^{p+1} g_j(z, y_o) \frac{\partial}{\partial z_j}$, where

$$ \eta|_{(y=y_o)} = i_{Z_{y_o}} \mathrm{d}z_1 \wedge \cdots \wedge \mathrm{d}z_{p+1}. $$

Since $\eta$ is locally totally decomposable. outside its singular set, we have $i_Z \eta = 0$.

Note that $Z(z, 0) = Z_o$. Therefore, the hypothesis implies that there is a neighborhood $W$ of $\bar{U} \times \{0\}$ in $\mathbb{C}^n$ such that $\mathrm{cod}(\mathrm{Sing}(Z)) \ge 3$ and $W \cap (y=0) = V \times \{0\}$.

We assert that there are holomorphic vector fields $X_1, \dots, X_{n-p-1}$ defined in a smaller neighborhood of $\bar{U} \times \{0\}$, such that $i_{X_j} \eta = 0$ and

$$ X_j(x, y) = \frac{\partial}{\partial y_j} + \sum_{i=1}^{p+1} h_{ji}(z, y) \frac{\partial}{\partial z_i}, \quad \forall 1 \le j \le n-p-1. \quad (5.1) $$

First of all, we note that the above assertion is true in a neighborhood of any point $(z_o, 0) \in (V \times \{0\}) \setminus \mathrm{Sing}(Z_o)$. This is true because for $(z, y)$ in a neighborhood $U_\alpha$ of $(z_o, 0)$ some component of $Z(z, y)$ does not vanishes, say $g_{p+1}(z, y) \ne 0$, so that

$$ \frac{(-1)^p}{g_{p+1}} \eta|_{U_\alpha} = \mathrm{d}z_1 \wedge \cdots \wedge \mathrm{d}z_p + \tilde{\Theta}, $$

where $\tilde{\Theta} \wedge \mathrm{d}z_{p+1} \wedge \mathrm{d}y_1 \wedge \cdots \wedge \mathrm{d}y_{n-p-1} \equiv 0$. As the reader can check this implies the existence of holomorphic vector fields $X_{j\alpha}$ on $U_\alpha$ as in (5.1), $1 \le j \le n-p-1$. It follows that there are:

* a polydisc $Q = Q_1 \times Q_2 \subset \mathbb{C}^{p+1} \times \mathbb{C}^{n-p-1}$, with $Q_1 \supset \bar{U}$ and $0 \in Q_2$.

* a covering $\mathcal{U} = (U_\alpha)_{\alpha \in A}$ of $Q \setminus \mathrm{Sing}(\eta)$ by polydiscs,

* n-p-1 collections of holomorphic vector fields $(X_{j\alpha})_{\alpha \in A}$, $1 \le j \le n-p-1$, $X_{j\alpha} \in \mathcal{X}(U_\alpha)$,

such that

(i) $\mathrm{cod}_Q(\mathrm{Sing}(Z|_Q)) \ge 3$.

(ii) $X_{j\alpha} = \frac{\partial}{\partial y_j} + \sum_{i=1}^{p+1} g_{i\alpha}(z, y) \frac{\partial}{\partial z_i}$.

(iii) $i_{X_{j\alpha}} \eta = 0, \forall 1 \le j \le n-p-1, \forall \alpha \in A$.
(iv) for all $q = (z, y) \in U_\alpha$ then $\ker(\eta(q)) = \langle Z(q), X_{1\alpha}(q), \dots, X_{n-p-1\alpha}(q) \rangle_{\mathbb{C}}$.

If $1 \le j \le n-p-1$ and $U_\alpha \cap U_\beta \ne \emptyset$ then

$$X_{j\alpha} - X_{j\beta} = \sum_{i=1}^{p+1} (g_{j\alpha} - g_{j\beta}) \frac{\partial}{\partial z_i} = h_{\alpha\beta}^j Z,$$

where $h_{\alpha\beta}^j \in \mathcal{O}(U_\alpha \cap U_\beta)$. The collection $(h_{\alpha\beta}^j)_{U_\alpha \ne \emptyset}$ is an additive cocycle. Since $\mathrm{cod}(\mathrm{Sing}(Z)) \ge 3$ by Cartan's theorem (cf. [5] and [12]) the cocycle is trivial; $h_{\alpha\beta}^j = h_\alpha^j - h_\beta^j, h_\alpha^j \in \mathcal{O}(U_\alpha)$. Hence, there exists a holomorphic vector field $X_j$ on $Q \setminus \mathrm{Sing}(Z)$ as in (5.1) such that $i_{X_j}\eta = 0$; $X_j|_{U_\alpha} = X_{j\alpha} - h_\alpha^j Z$. By Hartog's theorem $X_j$ can be extended to a holomorphic vector field on $Q$, denoted by the same letter. In particular, we have

$$\ker(\eta)|_Q = \langle Z, X_1, \dots, X_{n-p-1} \rangle_{\mathcal{O}(Q)} \qquad (5.2)$$

Finally, (5.2) and [6, Theorem 11] imply the theorem:

(i) There exists a smaller polydisc $\bar{U} \times \{0\} \subset Q' \subset Q$ and holomorphic vector fields $Z', Y_1, \dots, Y_{n-p-1} \in \langle Z, X_1, \dots, X_{n-p-1} \rangle_{\mathcal{O}(Q')}$ such that $[Y_i, Y_j] = 0, [Z', Y_j] = 0, \forall 1 \le j \le n-p-1$, and

$$\langle Z', Y_1, \dots, Y_{n-p-1} \rangle_{\mathcal{O}(Q')} = \langle Z|_{Q'}, X_1|_{Q'}, \dots, X_{n-p-1}|_{Q'} \rangle_{\mathcal{O}(Q')}.$$

(ii) There are coordinates $(z, w) = (z, w_1, \dots, w_{n-p-1})$ in $Q'$ such that $Y_j = \frac{\partial}{\partial w_j}, \forall 1 \le j \le n-p-1$.

This finishes the proof of Theorem 5.2. $\square$

A simple consequence of Theorem 5.2 is the following:

**COROLLARY 5.3.** — $\mathrm{Sing}(\eta)$ is biholomorphic to $\mathrm{Sing}(Z_o) \times (\mathbb{C}^{n-p-1}, 0)$.

## 5.2. Proof of Theorem 5.1

In this section we consider a holomorphic codimension $p$ foliation $\mathcal{G}$ on $\mathbb{P}^n, 2 \le p \le n-2$. We assume that there is a $p+1$ plane $\mathbb{P}^{p+1} = \Sigma_o \subset \mathbb{P}^n$ such that $\mathrm{cod}_\Sigma(\mathcal{G}|_{\Sigma_o}) \ge 3$. We want to prove that there is a linear projection $T: \mathbb{P}^n \to \Sigma_o$ such that $\mathcal{G} = T^*(\mathcal{G}|_{\Sigma_o})$. We will prove Theorem 5.1 in the case $n = p+2$, or equivalently, when the foliation is two-dimensional. The general case will be reduced to this case using [18, Section 3.4].

The foliation $\mathcal{G}|_{\Sigma_o}$ is one dimensional and so it can be defined by a finite covering $(Q_\alpha)_{\alpha \in A}$ of $\Sigma_o$ by polydiscs of $\Sigma_o$, a collection $(X_\alpha)_{\alpha \in A}$ of holomorphic vector fields $X_\alpha \in \mathcal{X}(Q_\alpha)$, and a multiplicative cocycle $(g_{\alpha\beta})_{Q_\alpha \cap Q_\beta \ne \emptyset}$
(iv) for all $q = (z, y) \in U_\alpha$ then $\ker(\eta(q)) = \langle Z(q), X_{1\alpha}(q), \dots, X_{n-p-1\alpha}(q) \rangle_{\mathbb{C}}$.

If $1 \le j \le n-p-1$ and $U_\alpha \cap U_\beta \ne \emptyset$ then

$$X_{j\alpha} - X_{j\beta} = \sum_{i=1}^{p+1} (g_{j\alpha} - g_{j\beta}) \frac{\partial}{\partial z_i} = h_{\alpha\beta}^j Z,$$

where $h_{\alpha\beta}^j \in \mathcal{O}(U_\alpha \cap U_\beta)$. The collection $(h_{\alpha\beta}^j)_{U_\alpha \ne \emptyset}$ is an additive cocycle. Since $\mathrm{cod}(\mathrm{Sing}(Z)) \ge 3$ by Cartan's theorem (cf. [5] and [12]) the cocycle is trivial; $h_{\alpha\beta}^j = h_\alpha^j - h_\beta^j, h_\alpha^j \in \mathcal{O}(U_\alpha)$. Hence, there exists a holomorphic vector field $X_j$ on $Q \setminus \mathrm{Sing}(Z)$ as in (5.1) such that $i_{X_j}\eta = 0$; $X_j|_{U_\alpha} = X_{j\alpha} - h_\alpha^j Z$. By Hartog's theorem $X_j$ can be extended to a holomorphic vector field on $Q$, denoted by the same letter. In particular, we have

$$\ker(\eta)|_Q = \langle Z, X_1, \dots, X_{n-p-1} \rangle_{\mathcal{O}(Q)}. \qquad (5.2)$$

Finally, (5.2) and [6, Theorem 11] imply the theorem:

(i) There exists a smaller polydisc $\bar{U} \times \{0\} \subset Q' \subset Q$ and holomorphic vector fields $Z', Y_1, \dots, Y_{n-p-1} \in \langle Z, X_1, \dots, X_{n-p-1} \rangle_{\mathcal{O}(Q')}$ such that $[Y_i, Y_j] = 0, [Z', Y_j] = 0, \forall 1 \le j \le n-p-1$, and

$$\langle Z', Y_1, \dots, Y_{n-p-1} \rangle_{\mathcal{O}(Q')} = \langle Z|_{Q'}, X_1|_{Q'}, \dots, X_{n-p-1}|_{Q'} \rangle_{\mathcal{O}(Q')}.$$

(ii) There are coordinates $(z, w) = (z, w_1, \dots, w_{n-p-1})$ in $Q'$ such that $Y_j = \frac{\partial}{\partial w_j}, \forall 1 \le j \le n-p-1$.

This finishes the proof of Theorem 5.2. $\square$

A simple consequence of Theorem 5.2 is the following:

**COROLLARY 5.3.** — $\mathrm{Sing}(\eta)$ is biholomorphic to $\mathrm{Sing}(Z_o) \times (\mathbb{C}^{n-p-1}, 0)$.

## 5.2. Proof of Theorem 5.1

In this section we consider a holomorphic codimension $p$ foliation $\mathcal{G}$ on $\mathbb{P}^n, 2 \le p \le n-2$. We assume that there is a $p+1$ plane $\mathbb{P}^{p+1} = \Sigma_o \subset \mathbb{P}^n$ such that $\mathrm{cod}_\Sigma(\mathcal{G}|_{\Sigma_o}) \ge 3$. We want to prove that there is a linear projection $T: \mathbb{P}^n \to \Sigma_o$ such that $\mathcal{G} = T^*(\mathcal{G}|_{\Sigma_o})$. We will prove Theorem 5.1 in the case $n = p+2$, or equivalently, when the foliation is two-dimensional. The general case will be reduced to this case using [18, Section 3.4].

The foliation $\mathcal{G}|_{\Sigma_o}$ is one dimensional and so it can be defined by a finite covering $(Q_\alpha)_{\alpha \in A}$ of $\Sigma_o$ by polydiscs of $\Sigma_o$, a collection $(X_\alpha)_{\alpha \in A}$ of holomorphic vector fields $X_\alpha \in \mathcal{X}(Q_\alpha)$, and a multiplicative cocycle $(g_{\alpha\beta})_{Q_\alpha \cap Q_\beta \ne \emptyset}$
such that $X_\alpha = g_{\alpha\beta} \cdot X_\beta$ on $Q_\alpha \cap Q_\beta \neq \emptyset$. A consequence of Theorem 5.2 is the following:

**COROLLARY 5.4.** — There is a finite covering of $\Sigma_o$ by polydiscs of $\mathbb{P}^{p+2}$, say $(U_\alpha)_\alpha$, and two collections of holomorphic vector fields $(Z_\alpha)_{\alpha \in A}$ and $(Y_\alpha)_{\alpha \in A}$, $Z_\alpha, Y_\alpha \in \mathcal{X}(U_\alpha)$, with the following properties:

(a) $U_\alpha \cap \Sigma_o = Q_\alpha$ and $Z_\alpha$ is an extension of $X_\alpha$ to $U_\alpha$. In particular, $Z_\alpha$ is tangent to $\Sigma_o$ along $Q_\alpha$.

(b) $\mathrm{Sing}(Y_\alpha) = \emptyset$ and $Y_\alpha$ is transverse to $\Sigma_o$ along $Q_\alpha$.

(c) If $z \notin \mathrm{Sing}(\mathcal{G}) \cap U_\alpha$ then $T_z\mathcal{G} = \langle Z_\alpha(z), Y_\alpha(z) \rangle_{\mathbb{C}}$.

(d) If $z \in \mathrm{Sing}(\mathcal{G}) \cap U_\alpha$ then $Z_\alpha(z) = 0$. Moreover, the orbit of $Y_\alpha$ through $z$ is contained in $\mathrm{Sing}(\mathcal{G})$.

The proof is a straightforward consequence of Theorem 5.2 and is left to the reader.

Our goal now is to prove the following:

**LEMMA 5.5.** — Under the hypothesis of Theorem 5.1 assume that $n = p+2$. Then there is a one-dimensional foliation $\mathcal{H}$ of degree zero transverse to $\Sigma_o$ whose leaves are $\mathcal{G}$-invariant.

*Proof.* — The foliation $\mathcal{H}$ will be constructed in homogeneous coordinates. Let $\Pi: \mathbb{C}^{p+3} \setminus \{0\} \to \mathbb{P}^{p+2}$ be the canonical projection and $\tilde{\mathcal{G}} = \Pi^*(\mathcal{G})$. Consider homogeneous coordinates $z = (z_0, \dots, z_{p+2}) \in \mathbb{C}^{p+3}$ such that $\Pi^{-1}(\Sigma_o) \cup \{0\} = (z_o = 0) := \tilde{\Sigma}_o$. In these homogeneous coordinates the foliation $\tilde{\mathcal{G}}$ is defined by an integrable homogeneous p-form $\eta$ such that $i_R\eta = 0$, where $R$ denotes the radial vector field on $\mathbb{C}^{p+3}$. The foliation $\mathcal{H}$ will be defined in homogeneous coordinates by $R$ and a constant vector field $v$ such that $i_v\eta = 0$.

The idea is to construct a formal series of vector fields of the form
$V = \frac{\partial}{\partial z_0} + \sum_{j \ge 0} z_0^j V_j$, where $V_j = \sum_{i=1}^{p+2} f_{ji}(z_1, \dots, z_{p+2}) \frac{\partial}{\partial z_i}$, the $f_{ji}$'s are holomorphic in some polydisc $Q$ of $\mathbb{C}^{p+2}$ containing the origin of $\mathbb{C}^{p+2}$ and such that $i_V \eta = 0$. If $v := V(0) = \frac{\partial}{\partial z_0} + V_0(0) \ne 0$ then $i_v \eta = 0$ because the coefficients of $\eta$ are homogeneous of the same degree. The constant vector field $v$ and $R$ induce a foliation $\mathcal{H}$ of degree zero on $\mathbb{P}^{p+2}$ tangent to $\mathcal{G}$.

Let us construct the series $V$. The covering $(U_\alpha)_{\alpha \in A}$, given by Corollary 5.4, induces the covering $(\tilde{U}_\alpha = \Pi^{-1}(U_\alpha))_{\alpha \in A}$ of $\tilde{\Sigma}_o \setminus \{0\}$. Without loss of generality, we can suppose that for any $\alpha \in A$ then $U_\alpha$ is contained in some affine chart $(z_{j(\alpha)} \ne 0)$, where $j(\alpha) \ne 0$.
such that $X_\alpha = g_{\alpha\beta} \cdot X_\beta$ on $Q_\alpha \cap Q_\beta \neq \emptyset$. A consequence of Theorem 5.2 is the following:

**COROLLARY 5.4.** — There is a finite covering of $\Sigma_o$ by polydiscs of $\mathbb{P}^{p+2}$, say $(U_\alpha)_\alpha$, and two collections of holomorphic vector fields $(Z_\alpha)_{\alpha \in A}$ and $(Y_\alpha)_{\alpha \in A}$, $Z_\alpha, Y_\alpha \in \mathcal{X}(U_\alpha)$, with the following properties:

(a) $U_\alpha \cap \Sigma_o = Q_\alpha$ and $Z_\alpha$ is an extension of $X_\alpha$ to $U_\alpha$. In particular, $Z_\alpha$ is tangent to $\Sigma_o$ along $Q_\alpha$.

(b) $\mathrm{Sing}(Y_\alpha) = \emptyset$ and $Y_\alpha$ is transverse to $\Sigma_o$ along $Q_\alpha$.

(c) If $z \notin \mathrm{Sing}(\mathcal{G}) \cap U_\alpha$ then $T_z\mathcal{G} = \langle Z_\alpha(z), Y_\alpha(z) \rangle_{\mathbb{C}}$.

(d) If $z \in \mathrm{Sing}(\mathcal{G}) \cap U_\alpha$ then $Z_\alpha(z) = 0$. Moreover, the orbit of $Y_\alpha$ through $z$ is contained in $\mathrm{Sing}(\mathcal{G})$.

The proof is a straightforward consequence of Theorem 5.2 and is left to the reader.

Our goal now is to prove the following:

**LEMMA 5.5.** — Under the hypothesis of Theorem 5.1 assume that $n = p+2$. Then there is a one-dimensional foliation $\mathcal{H}$ of degree zero transverse to $\Sigma_o$ whose leaves are $\mathcal{G}$-invariant.

*Proof.* — The foliation $\mathcal{H}$ will be constructed in homogeneous coordinates. Let $\Pi: \mathbb{C}^{p+3} \setminus \{0\} \to \mathbb{P}^{p+2}$ be the canonical projection and $\tilde{\mathcal{G}} = \Pi^*(\mathcal{G})$. Consider homogeneous coordinates $z = (z_0, \dots, z_{p+2}) \in \mathbb{C}^{p+3}$ such that $\Pi^{-1}(\Sigma_o) \cup \{0\} = (z_o = 0) := \tilde{\Sigma}_o$. In these homogeneous coordinates the foliation $\tilde{\mathcal{G}}$ is defined by an integrable homogeneous p-form $\eta$ such that $i_R\eta = 0$, where $R$ denotes the radial vector field on $\mathbb{C}^{p+3}$. The foliation $\mathcal{H}$ will be defined in homogeneous coordinates by $R$ and a constant vector field $v$ such that $i_v\eta = 0$.

The idea is to construct a formal series of vector fields of the form
$V = \frac{\partial}{\partial z_0} + \sum_{j \ge 0} z_0^j V_j$, where $V_j = \sum_{i=1}^{p+2} f_{ji}(z_1, \dots, z_{p+2}) \frac{\partial}{\partial z_i}$, the $f_{ji}$'s are holomorphic in some polydisc $Q$ of $\mathbb{C}^{p+2}$ containing the origin of $\mathbb{C}^{p+2}$ and such that $i_V \eta = 0$. If $v := V(0) = \frac{\partial}{\partial z_0} + V_0(0) \ne 0$ then $i_v \eta = 0$ because the coefficients of $\eta$ are homogeneous of the same degree. The constant vector field $v$ and $R$ induce a foliation $\mathcal{H}$ of degree zero on $\mathbb{P}^{p+2}$ tangent to $\mathcal{G}$.

Let us construct the series $V$. The covering $(U_\alpha)_{\alpha \in A}$, given by Corollary 5.4, induces the covering $(\tilde{U}_\alpha = \Pi^{-1}(U_\alpha))_{\alpha \in A}$ of $\tilde{\Sigma}_o \setminus \{0\}$. Without loss of generality, we can suppose that for any $\alpha \in A$ then $U_\alpha$ is contained in some affine chart $(z_{j(\alpha)} \ne 0)$, where $j(\alpha) \ne 0$.
CLAIM 5.6. — There are collections of holomorphic vector fields $(\tilde{Z}_\alpha)_{\alpha \in A}$ and $(\tilde{Y}_\alpha)_{\alpha \in A}$, with $\tilde{Z}_\alpha, \tilde{Y}_\alpha \in \mathcal{X}(\tilde{U}_\alpha)$ $\forall \alpha \in A$, with the following properties:

(i) $D\Pi(z).\tilde{Z}_\alpha(z) = Z_\alpha \circ \Pi(z)$ and $D\Pi(z).\tilde{Y}_\alpha(z) = Y_\alpha \circ \Pi(z)$, $\forall z \in \tilde{U}_\alpha$. In particular, $\tilde{Y}_\alpha$ and $\tilde{Z}_\alpha$ are tangent to $\tilde{G}|_{\tilde{U}_\alpha}: i_{\tilde{Y}_\alpha} \eta = 0$ and $i_{\tilde{Z}_\alpha} \eta = 0$, $\forall \alpha$.

(ii) $\tilde{Y}_\alpha, \tilde{Z}_\alpha$ and $R$ generate $\tilde{G}$ in the sense that:

• if $z \in \tilde{U}_\alpha \setminus \text{Sing}(\tilde{G})$ then $T_z\tilde{G} = \langle \tilde{Y}_\alpha(z), \tilde{Z}_\alpha(z), R(z) \rangle_\mathcal{C}$.

• $z \in \tilde{U}_\alpha \cap \text{Sing}(\tilde{G}) \iff \tilde{Y}_\alpha(z) \wedge \tilde{Z}_\alpha(z) \wedge R(z) = 0$.

(iii) $\tilde{Z}_\alpha$ is tangent to $\tilde{\Sigma}_o$ along $\tilde{\Sigma}_o \cap \tilde{U}_\alpha$, $\forall \alpha \in A$. This means that

$$ \tilde{Z}_{\alpha}(0, z_1, \dots, z_{p+2}) \in \left\langle \frac{\partial}{\partial z_1}, \dots, \frac{\partial}{\partial z_{p+2}} \right\rangle_{\mathcal{O}}. $$

(iv) $\tilde{Y}_\alpha = g_\alpha(z) \frac{\partial}{\partial z_0} + V_\alpha$, where $V_\alpha \in \left\langle \frac{\partial}{\partial z_1}, \dots, \frac{\partial}{\partial z_{p+2}} \right\rangle_\mathcal{O}$ and $g_\alpha \in \mathcal{O}^*(\tilde{U}_\alpha)$.

In particular, $\text{Sing}(\tilde{Y}_\alpha) = \emptyset$ and $\tilde{Y}_\alpha$ is transverse to $\tilde{\Sigma}_o$ along $\tilde{\Sigma}_o \cap \tilde{U}_\alpha$, $\forall \alpha \in A$.

Proof. — Let us construct $\tilde{Y}_\alpha$ and $\tilde{Z}_\alpha$, $\alpha \in A$. Let $j \neq 0$ be such that $U_\alpha \subset (z_j \neq 0)$. Let us assume that $U_\alpha \subset (z_n = 1)$, for instance, and that $Y_\alpha$ and $Z_\alpha$ are vector fields tangent to the affine plane $(z_n = 1)$: $Y_\alpha = \sum_{i<n} g_i^\alpha(z_0, \dots, z_{n-1}) \frac{\partial}{\partial z_i}$ and $Z_\alpha = \sum_{i<n} h_i^\alpha(z_0, \dots, z_{n-1}) \frac{\partial}{\partial z_i}$, where $g_i^\alpha, h_i^\alpha \in \mathcal{O}(U_\alpha)$, $\forall \alpha$. Since $Y_\alpha$ is transverse to $\Sigma_o$ we have $g_0^\alpha \in \mathcal{O}^*(\tilde{U}_\alpha)$, $\forall \alpha$. The vector fields $\tilde{Y}_\alpha$ and $\tilde{Z}_\alpha$ are then constructed by extending $Y_\alpha$ and $Z_\alpha$ “radially”: we set $\tilde{Y}_\alpha := \sum_{i<n} \tilde{g}_i^\alpha(z) \frac{\partial}{\partial z_i}$ and $\tilde{Z}_\alpha := \sum_{i<n} \tilde{h}_i^\alpha(z) \frac{\partial}{\partial z_i}$, where $\tilde{g}_i^\alpha(z) = z_0 g_i^\alpha(z_0/z_n, \dots, z_{n-1}/z_n)$ and $\tilde{h}_i^\alpha(z) = z_0 h_i^\alpha(z_0/z_n, \dots, z_{n-1}/z_n)$. We leave the proof of (i), (ii), (iii) and (iv) for the reader. $\square$

We now define a multiplicative cocycle of $3 \times 3$ matrices $(A_{\alpha\beta})_{\tilde{U}_\alpha \cap \tilde{U}_\beta \neq \emptyset}$. Since $\text{cod}(\text{Sing}(\mathcal{G}|_{\tilde{U}_\alpha})) \geq 3$, we get $\text{cod}(\text{Sing}(\mathcal{G}|_{\tilde{U}_\alpha})) \geq 3$, which implies

$$ \text{cod} \left( \{ z \in \tilde{U}_{\alpha} | \tilde{Y}_{\alpha}(z) \wedge \tilde{Z}_{\alpha}(z) \wedge R(z) = 0 \} \right) \geq 3. $$

From this and (ii) we get that, if $\tilde{U}_\alpha \cap \tilde{U}_\beta \neq \emptyset$ then we can write

$$ \left\{ \begin{array}{l} \tilde{Y}_{\alpha}(z) = a_{\alpha\beta}(z)\tilde{Y}_{\beta}(z) + b_{\alpha\beta}(z)\tilde{Z}_{\beta}(z) + c_{\alpha\beta}(z)R(z) \\ \tilde{Z}_{\alpha}(z) = d_{\alpha\beta}(z)\tilde{Y}_{\alpha\beta}(z) + e_{\alpha\beta}(z)\tilde{Z}_{\beta}(z) + f_{\alpha\beta}(z)R(z), \quad \forall z \in \tilde{U}_{\alpha} \cap \tilde{U}_{\beta}, \end{array} \right. $$

where $a_{\alpha\beta}, \dots, f_{\alpha\beta} \in \mathcal{O}(\tilde{U}_\alpha \cap \tilde{U}_\beta)$. The matrix

$$ A_{\alpha\beta} := \begin{pmatrix} a_{\alpha\beta} & b_{\alpha\beta} & c_{\alpha\beta} \\ d_{\alpha\beta} & e_{\alpha\beta} & f_{\alpha\beta} \\ 0 & 0 & 1 \end{pmatrix} $$
CLAIM 5.6. — There are collections of holomorphic vector fields $(\tilde{Z}_\alpha)_{\alpha \in A}$ and $(\tilde{Y}_\alpha)_{\alpha \in A}$, with $\tilde{Z}_\alpha, \tilde{Y}_\alpha \in \mathcal{X}(\tilde{U}_\alpha)$ $\forall \alpha \in A$, with the following properties:

(i) $D\Pi(z).\tilde{Z}_\alpha(z) = Z_\alpha \circ \Pi(z)$ and $D\Pi(z).\tilde{Y}_\alpha(z) = Y_\alpha \circ \Pi(z)$, $\forall z \in \tilde{U}_\alpha$. In particular, $\tilde{Y}_\alpha$ and $\tilde{Z}_\alpha$ are tangent to $\tilde{G}|_{\tilde{U}_\alpha}: i_{\tilde{Y}_\alpha}\eta = 0$ and $i_{\tilde{Z}_\alpha}\eta = 0$, $\forall \alpha$.

(ii) $\tilde{Y}_\alpha, \tilde{Z}_\alpha$ and $R$ generate $\tilde{G}$ in the sense that:

• if $z \in \tilde{U}_\alpha \setminus \text{Sing}(\tilde{G})$ then $T_z\tilde{G} = \langle \tilde{Y}_\alpha(z), \tilde{Z}_\alpha(z), R(z) \rangle_\mathcal{C}$.

• $z \in \tilde{U}_\alpha \cap \text{Sing}(\tilde{G}) \iff \tilde{Y}_\alpha(z) \wedge \tilde{Z}_\alpha(z) \wedge R(z) = 0$.

(iii) $\tilde{Z}_\alpha$ is tangent to $\tilde{\Sigma}_o$ along $\tilde{\Sigma}_o \cap \tilde{U}_\alpha$, $\forall \alpha \in A$. This means that

$$ \tilde{Z}_{\alpha}(0, z_1, \dots, z_{p+2}) \in \left\langle \frac{\partial}{\partial z_1}, \dots, \frac{\partial}{\partial z_{p+2}} \right\rangle_{\mathcal{O}}. $$

(iv) $\tilde{Y}_\alpha = g_\alpha(z) \frac{\partial}{\partial z_0} + V_\alpha$, where $V_\alpha \in \left\langle \frac{\partial}{\partial z_1}, \dots, \frac{\partial}{\partial z_{p+2}} \right\rangle_\mathcal{O}$ and $g_\alpha \in \mathcal{O}^*(\tilde{U}_\alpha)$.

In particular, $\text{Sing}(\tilde{Y}_\alpha) = \emptyset$ and $\tilde{Y}_\alpha$ is transverse to $\tilde{\Sigma}_o$ along $\tilde{\Sigma}_o \cap \tilde{U}_\alpha$, $\forall \alpha \in A$.

Proof. — Let us construct $\tilde{Y}_\alpha$ and $\tilde{Z}_\alpha$, $\alpha \in A$. Let $j \neq 0$ be such that $U_\alpha \subset (z_j \neq 0)$. Let us assume that $U_\alpha \subset (z_n = 1)$, for instance, and that $Y_\alpha$ and $Z_\alpha$ are vector fields tangent to the affine plane $(z_n = 1)$: $Y_\alpha = \sum_{i<n} g_i^\alpha(z_0, \dots, z_{n-1}) \frac{\partial}{\partial z_i}$ and $Z_\alpha = \sum_{i<n} h_i^\alpha(z_0, \dots, z_{n-1}) \frac{\partial}{\partial z_i}$, where $g_i^\alpha, h_i^\alpha \in \mathcal{O}(U_\alpha)$, $\forall \alpha$. Since $Y_\alpha$ is transverse to $\Sigma_o$ we have $g_0^\alpha \in \mathcal{O}^*(\tilde{U}_\alpha)$, $\forall \alpha$. The vector fields $\tilde{Y}_\alpha$ and $\tilde{Z}_\alpha$ are then constructed by extending $Y_\alpha$ and $Z_\alpha$ “radially”: we set $\tilde{Y}_\alpha := \sum_{i<n} \tilde{g}_i^\alpha(z) \frac{\partial}{\partial z_i}$ and $\tilde{Z}_\alpha := \sum_{i<n} \tilde{h}_i^\alpha(z) \frac{\partial}{\partial z_i}$, where $\tilde{g}_i^\alpha(z) = z_0 g_i^\alpha(z_0/z_n, \dots, z_{n-1}/z_n)$ and $\tilde{h}_i^\alpha(z) = z_0 h_i^\alpha(z_0/z_n, \dots, z_{n-1}/z_n)$. We leave the proof of (i), (ii), (iii) and (iv) for the reader. $\square$

We now define a multiplicative cocycle of $3 \times 3$ matrices $(A_{\alpha\beta})_{\tilde{U}_\alpha \cap \tilde{U}_\beta \neq \emptyset}$. Since $\text{cod}(\text{Sing}(\mathcal{G}|_{\tilde{U}_\alpha})) \geq 3$, we get $\text{cod}(\text{Sing}(\mathcal{G}|_{\tilde{U}_\alpha})) \geq 3$, which implies

$$ \text{cod} \left( \{z \in \tilde{U}_\alpha | \tilde{Y}_\alpha(z) \wedge \tilde{Z}_\alpha(z) \wedge R(z) = 0\} \right) \geq 3. $$

From this and (ii) we get that, if $\tilde{U}_\alpha \cap \tilde{U}_\beta \neq \emptyset$ then we can write

$$ \left\{ \begin{array}{l} \tilde{Y}_{\alpha}(z) = a_{\alpha\beta}(z)\tilde{Y}_{\beta}(z) + b_{\alpha\beta}(z)\tilde{Z}_{\beta}(z) + c_{\alpha\beta}(z)R(z) \\ \tilde{Z}_{\alpha}(z) = d_{\alpha\beta}(z)\tilde{Y}_{\alpha\beta}(z) + e_{\alpha\beta}(z)\tilde{Z}_{\beta}(z) + f_{\alpha\beta}(z)R(z), \quad \forall z \in \tilde{U}_{\alpha} \cap \tilde{U}_{\beta}, \end{array} \right. $$

where $a_{\alpha\beta}, \dots, f_{\alpha\beta} \in \mathcal{O}(\tilde{U}_\alpha \cap \tilde{U}_\beta)$. The matrix

$$ A_{\alpha\beta} := \begin{pmatrix} a_{\alpha\beta} & b_{\alpha\beta} & c_{\alpha\beta} \\ d_{\alpha\beta} & e_{\alpha\beta} & f_{\alpha\beta} \\ 0 & 0 & 1 \end{pmatrix} $$
defines the transition

$$ \begin{pmatrix} \tilde{Y}_{\alpha} \\ \tilde{Z}_{\alpha} \\ R \end{pmatrix} = A_{\alpha\beta} \cdot \begin{pmatrix} \tilde{Y}_{\beta} \\ \tilde{Z}_{\beta} \\ R \end{pmatrix}. \qquad (5.3) $$

Of course, $A_{\alpha\beta} = A_{\beta\alpha}^{-1}$ and if $\tilde{U}_{\alpha} \cap \tilde{U}_{\beta} \cap \tilde{U}_{\gamma} \neq \emptyset$ then $A_{\alpha\beta}A_{\beta\gamma}A_{\gamma\alpha} = I$ on $\tilde{U}_{\alpha} \cap \tilde{U}_{\beta} \cap \tilde{U}_{\gamma}$. Since $\tilde{U}_{\alpha} \cap \tilde{U}_{\beta}$ is a neighborhood of $Q_{\alpha} \subset (z_0 = 0)$ in $\mathbb{C}^{p+3}$ we can write $A_{\alpha\beta}$ as a power series in $z_0$:

$$ A_{\alpha\beta} = \sum_{j \ge 0} z_0^j A_{\alpha\beta}^j, $$

where $A_{\alpha\beta}^j$ is a matrix with coefficients in $\mathcal{O}(Q_{\alpha} \cap Q_{\beta})$, $Q_{\alpha} = \tilde{U}_{\alpha} \cap (z_0 = 0)$. Now, the proof of Lemma 5.5 can be reduced to the following:

**LEMMA 5.7.** — The matrix cocycle $(A_{\alpha\beta})_{\tilde{U}_{\alpha} \cap \tilde{U}_{\beta} \neq \emptyset}$ is formally trivial: there exist a collection $(A_{\alpha})_{\alpha \in A}$ of formal power series

$$ A_{\alpha} = \sum_{j \ge 0} z_0^j A_{\alpha}^j, $$

where

(a) $A_{\alpha}^j$ is a matrix with coefficients in $\mathcal{O}(Q_{\alpha})$, $Q_{\alpha} = \tilde{U}_{\alpha} \cap (z_0 = 0)$, $\forall \alpha, \forall j \ge 0$.

(b) $A_{\alpha}$ is invertible as a matrix formal series and its third line is $(0, 0, 1)$, $\forall \alpha$.

(c) if $Q_{\alpha} \cap Q_{\beta} \neq \emptyset$ then $A_{\alpha\beta} = A_{\alpha}^{-1} \cdot A_{\beta}$.

(d) $A_{\alpha}^0$ is triangular superior $\forall \alpha \in A$.

The proof of Lemma 5.7 will be done at the end of the section. Let us see how it implies Lemma 5.5. From (5.3) we have

$$ \begin{pmatrix} \tilde{Y}_{\alpha} \\ \tilde{Z}_{\alpha} \\ R \end{pmatrix} = A_{\alpha}^{-1} \cdot A_{\beta} \cdot \begin{pmatrix} \tilde{Y}_{\beta} \\ \tilde{Z}_{\beta} \\ R \end{pmatrix} \implies A_{\alpha} \cdot \begin{pmatrix} \tilde{Y}_{\alpha} \\ \tilde{Z}_{\alpha} \\ R \end{pmatrix} = A_{\beta} \cdot \begin{pmatrix} \tilde{Y}_{\beta} \\ \tilde{Z}_{\beta} \\ R \end{pmatrix}. $$

Since the third line of $A_{\alpha}$ and $A_{\beta}$ is $(0, 0, 1)$, it follows that there are formal series of vector fields $Y = \sum_{j \ge 0} z_0^j Y_j$ and $Z = \sum_{j \ge 0} z_0^j Z_j$ such that

$$ \left. \begin{pmatrix} Y \\ Z \\ R \end{pmatrix} \right|_{Q_{\alpha} \times (\mathbb{C}, 0)} = A_{\alpha} \cdot \begin{pmatrix} \tilde{Y}_{\alpha} \\ \tilde{Z}_{\alpha} \\ R \end{pmatrix}, \forall \alpha. $$

Note that $i_Y \eta = 0$. Since the coefficients of $\eta$ are homogeneous of the same degree, we obtain $i_{\nu} \eta = 0$, where $\nu = Y(0) = Y_0(0)$. Therefore, it
defines the transition

$$ \begin{pmatrix} \tilde{Y}_{\alpha} \\ \tilde{Z}_{\alpha} \\ R \end{pmatrix} = A_{\alpha\beta} \cdot \begin{pmatrix} \tilde{Y}_{\beta} \\ \tilde{Z}_{\beta} \\ R \end{pmatrix}. \qquad (5.3) $$

Of course, $A_{\alpha\beta} = A_{\beta\alpha}^{-1}$ and if $\tilde{U}_{\alpha} \cap \tilde{U}_{\beta} \cap \tilde{U}_{\gamma} \neq \emptyset$ then $A_{\alpha\beta}A_{\beta\gamma}A_{\gamma\alpha} = I$ on $\tilde{U}_{\alpha} \cap \tilde{U}_{\beta} \cap \tilde{U}_{\gamma}$. Since $\tilde{U}_{\alpha} \cap \tilde{U}_{\beta}$ is a neighborhood of $Q_{\alpha} \subset (z_0 = 0)$ in $\mathbb{C}^{p+3}$ we can write $A_{\alpha\beta}$ as a power series in $z_0$:

$$ A_{\alpha\beta} = \sum_{j \ge 0} z_0^j A_{\alpha\beta}^j, $$

where $A_{\alpha\beta}^j$ is a matrix with coefficients in $\mathcal{O}(Q_{\alpha} \cap Q_{\beta})$, $Q_{\alpha} = \tilde{U}_{\alpha} \cap (z_0 = 0)$. Now, the proof of Lemma 5.5 can be reduced to the following:

**LEMMA 5.7.** — The matrix cocycle $(A_{\alpha\beta})_{\tilde{U}_{\alpha} \cap \tilde{U}_{\beta} \neq \emptyset}$ is formally trivial: there exist a collection $(A_{\alpha})_{\alpha \in A}$ of formal power series

$$ A_{\alpha} = \sum_{j \ge 0} z_0^j A_{\alpha}^j, $$

where

(a) $A_{\alpha}^j$ is a matrix with coefficients in $\mathcal{O}(Q_{\alpha})$, $Q_{\alpha} = \tilde{U}_{\alpha} \cap (z_0 = 0)$, $\forall \alpha, \forall j \ge 0$.

(b) $A_{\alpha}$ is invertible as a matrix formal series and its third line is $(0, 0, 1)$, $\forall \alpha$.

(c) if $Q_{\alpha} \cap Q_{\beta} \neq \emptyset$ then $A_{\alpha\beta} = A_{\alpha}^{-1} \cdot A_{\beta}$.

(d) $A_{\alpha}^0$ is triangular superior $\forall \alpha \in A$.

The proof of Lemma 5.7 will be done at the end of the section. Let us see how it implies Lemma 5.5. From (5.3) we have

$$ \begin{pmatrix} \tilde{Y}_{\alpha} \\ \tilde{Z}_{\alpha} \\ R \end{pmatrix} = A_{\alpha}^{-1} \cdot A_{\beta} \cdot \begin{pmatrix} \tilde{Y}_{\beta} \\ \tilde{Z}_{\beta} \\ R \end{pmatrix} \implies A_{\alpha} \cdot \begin{pmatrix} \tilde{Y}_{\alpha} \\ \tilde{Z}_{\alpha} \\ R \end{pmatrix} = A_{\beta} \cdot \begin{pmatrix} \tilde{Y}_{\beta} \\ \tilde{Z}_{\beta} \\ R \end{pmatrix}. $$

Since the third line of $A_{\alpha}$ and $A_{\beta}$ is $(0, 0, 1)$, it follows that there are formal series of vector fields $Y = \sum_{j \ge 0} z_0^j Y_j$ and $Z = \sum_{j \ge 0} z_0^j Z_j$ such that

$$ \left. \begin{pmatrix} Y \\ Z \\ R \end{pmatrix} \right|_{Q_{\alpha} \times (\mathbb{C}, 0)} = A_{\alpha} \cdot \begin{pmatrix} \tilde{Y}_{\alpha} \\ \tilde{Z}_{\alpha} \\ R \end{pmatrix}, \forall \alpha. $$

Note that $i_Y \eta = 0$. Since the coefficients of $\eta$ are homogeneous of the same degree, we obtain $i_{\nu} \eta = 0$, where $\nu = Y(0) = Y_0(0)$. Therefore, it
is sufficient to see that $Y(0) = \sum_{j=0}^{n} a_j \frac{\partial}{\partial z_j}$, where $a_0 \neq 0$. This is a consequence of Lemma 5.7(d) and the fact that the $\frac{\partial}{\partial z_0}$ component of $\tilde{Y}_\alpha$ does not vanishes at $(z_0 = 0)$, as the reader can check. This finishes the proof of Theorem 5.1. $\square$

### 5.2.1. Proof of Lemma 5.7

The restriction of $A_{\alpha\beta}$ to $\tilde{\Sigma}_o \cap \tilde{U}_\alpha \cap \tilde{U}_\beta$ is triangular:

$$A_{\alpha\beta}|_{\tilde{\Sigma}_o \cap \tilde{U}_\alpha \cap \tilde{U}_\beta} = \begin{pmatrix} a_{\alpha\beta} & b_{\alpha\beta} & c_{\alpha\beta} \\ 0 & e_{\alpha\beta} & f_{\alpha\beta} \\ 0 & 0 & 1 \end{pmatrix}. \qquad (5.4)$$

The cocycle defined by (5.4) is trivial, when restricted to a domain of $\tilde{\Sigma}_o$ where we can apply Cartan's theorem [5]. Fix two polydiscs $Q_1, \bar{Q}_2 \subset \tilde{\Sigma}_o$, where $Q_1 = \{(z_1, \dots, z_{p+2}) \mid |z_i| < 1\}$ and $\bar{Q}_2 = \{(z_1, \dots, z_{p+2}) \mid |z_i| \le 1/2\}$, for instance. The open set $H := Q_1 \setminus \bar{Q}_2$ is a Hartog's domain in $\tilde{\Sigma}_o$, so that any $f \in \mathcal{O}(H)$ extends to a holomorphic function $\tilde{f} \in \mathcal{O}(Q_1)$ (cf. [24]). By Cartan's theorem [5] we have $H^1(H, \mathcal{O}) = 0$, because $n \ge 3$. Since $H^2(H, \mathbb{Z}) = 0$ we have also $H^1(H, \mathcal{O}^*) = 0$. Consider the Leray covering $(W_\alpha)_{\alpha \in A}$ of $H$ given by $W_\alpha = \tilde{U}_\alpha \cap \tilde{\Sigma}_o$. The restriction $A_{\alpha\beta}|_{W_\alpha \cap W_\beta}$ in (5.4) will be denoted by $B_{\alpha\beta}$. Since $B_{\alpha\beta}$ is triangular, the entries $a_{\alpha\beta}$ and $e_{\alpha\beta}$ define multiplicative cocycles $(a_{\alpha\beta})_{W_\alpha \cap W_\beta \neq \emptyset}$ and $(e_{\alpha\beta})_{W_\alpha \cap W_\beta \neq \emptyset}$, which are trivial: there are collections $(a_\alpha)_{\alpha \in A}$ and $(e_\alpha)_{\alpha \in A}$, $a_\alpha, e_\alpha \in \mathcal{O}^*(W_\alpha)$ such that $a_{\alpha\beta} = a_\alpha^{-1} \cdot a_\beta$ and $e_{\alpha\beta} = e_\alpha^{-1} \cdot e_\beta$ on $W_\alpha \cap W_\beta \neq \emptyset$. Hence, the cocycle $(B_{\alpha\beta})_{W_\alpha \cap W_\beta \neq \emptyset}$ is equivalent to a cocycle $(C_{\alpha\beta})_{W_\alpha \cap W_\beta \neq \emptyset}$, where

$$C_{\alpha\beta} = \begin{pmatrix} 1 & g_{\alpha\beta} & h_{\alpha\beta} \\ 0 & 1 & k_{\alpha\beta} \\ 0 & 0 & 1 \end{pmatrix}.$$

By writing explicitly that $(C_{\alpha\beta})_{W_\alpha \cap W_\beta \neq \emptyset}$ is a multiplicative cocycle, we get that $(g_{\alpha\beta})_{W_\alpha \cap W_\beta \neq \emptyset}$ and $(k_{\alpha\beta})_{W_\alpha \cap W_\beta \neq \emptyset}$ are additive cocycles. In particular, there are collections $(g_\alpha)_\alpha$ and $(k_\alpha)_\alpha$ with $g_\alpha, k_\alpha \in \mathcal{O}(W_\alpha)$ such that $g_{\alpha\beta} = g_\beta - g_\alpha$ and $k_{\alpha\beta} = k_\beta - k_\alpha$ on $W_\alpha \cap W_\beta \neq \emptyset$. If we set

$$M_\alpha = \begin{pmatrix} 1 & -g_\alpha & 0 \\ 0 & 1 & -k_\alpha \\ 0 & 0 & 1 \end{pmatrix}$$

then

$$D_{\alpha\beta} := M_{\alpha}^{-1} C_{\alpha\beta} M_{\beta} = \begin{pmatrix} 1 & 0 & \ell_{\alpha\beta} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$
is sufficient to see that $Y(0) = \sum_{j=0}^{n} a_j \frac{\partial}{\partial z_j}$, where $a_0 \neq 0$. This is a consequence of Lemma 5.7(d) and the fact that the $\frac{\partial}{\partial z_0}$ component of $\tilde{Y}_\alpha$ does not vanishes at $(z_0 = 0)$, as the reader can check. This finishes the proof of Theorem 5.1. $\square$

### 5.2.1. Proof of Lemma 5.7

The restriction of $A_{\alpha\beta}$ to $\tilde{\Sigma}_o \cap \tilde{U}_\alpha \cap \tilde{U}_\beta$ is triangular:

$$A_{\alpha\beta}|_{\tilde{\Sigma}_o \cap \tilde{U}_\alpha \cap \tilde{U}_\beta} = \begin{pmatrix} a_{\alpha\beta} & b_{\alpha\beta} & c_{\alpha\beta} \\ 0 & e_{\alpha\beta} & f_{\alpha\beta} \\ 0 & 0 & 1 \end{pmatrix}. \qquad (5.4)$$

The cocycle defined by (5.4) is trivial, when restricted to a domain of $\tilde{\Sigma}_o$ where we can apply Cartan's theorem [5]. Fix two polydiscs $Q_1, \bar{Q}_2 \subset \tilde{\Sigma}_o$, where $Q_1 = \{(z_1, \dots, z_{p+2}) \mid |z_i| < 1\}$ and $\bar{Q}_2 = \{(z_1, \dots, z_{p+2}) \mid |z_i| \le 1/2\}$, for instance. The open set $H := Q_1 \setminus \bar{Q}_2$ is a Hartog's domain in $\tilde{\Sigma}_o$, so that any $f \in \mathcal{O}(H)$ extends to a holomorphic function $\tilde{f} \in \mathcal{O}(Q_1)$ (cf. [24]). By Cartan's theorem [5] we have $H^1(H, \mathcal{O}) = 0$, because $n \ge 3$. Since $H^2(H, \mathbb{Z}) = 0$ we have also $H^1(H, \mathcal{O}^*) = 0$. Consider the Leray covering $(W_\alpha)_{\alpha \in A}$ of $H$ given by $W_\alpha = \tilde{U}_\alpha \cap \tilde{\Sigma}_o$. The restriction $A_{\alpha\beta}|_{W_\alpha \cap W_\beta}$ in (5.4) will be denoted by $B_{\alpha\beta}$. Since $B_{\alpha\beta}$ is triangular, the entries $a_{\alpha\beta}$ and $e_{\alpha\beta}$ define multiplicative cocycles $(a_{\alpha\beta})_{W_\alpha \cap W_\beta \neq \emptyset}$ and $(e_{\alpha\beta})_{W_\alpha \cap W_\beta \neq \emptyset}$, which are trivial: there are collections $(a_\alpha)_{\alpha \in A}$ and $(e_\alpha)_{\alpha \in A}$, $a_\alpha, e_\alpha \in \mathcal{O}^*(W_\alpha)$ such that $a_{\alpha\beta} = a_\alpha^{-1} \cdot a_\beta$ and $e_{\alpha\beta} = e_\alpha^{-1} \cdot e_\beta$ on $W_\alpha \cap W_\beta \neq \emptyset$. Hence, the cocycle $(B_{\alpha\beta})_{W_\alpha \cap W_\beta \neq \emptyset}$ is equivalent to a cocycle $(C_{\alpha\beta})_{W_\alpha \cap W_\beta \neq \emptyset}$, where

$$C_{\alpha\beta} = \begin{pmatrix} 1 & g_{\alpha\beta} & h_{\alpha\beta} \\ 0 & 1 & k_{\alpha\beta} \\ 0 & 0 & 1 \end{pmatrix}.$$

By writing explicitly that $(C_{\alpha\beta})_{W_\alpha \cap W_\beta \neq \emptyset}$ is a multiplicative cocycle, we get that $(g_{\alpha\beta})_{W_\alpha \cap W_\beta \neq \emptyset}$ and $(k_{\alpha\beta})_{W_\alpha \cap W_\beta \neq \emptyset}$ are additive cocycles. In particular, there are collections $(g_\alpha)_\alpha$ and $(k_\alpha)_\alpha$ with $g_\alpha, k_\alpha \in \mathcal{O}(W_\alpha)$ such that $g_{\alpha\beta} = g_\beta - g_\alpha$ and $k_{\alpha\beta} = k_\beta - k_\alpha$ on $W_\alpha \cap W_\beta \neq \emptyset$. If we set

$$M_\alpha = \begin{pmatrix} 1 & -g_\alpha & 0 \\ 0 & 1 & -k_\alpha \\ 0 & 0 & 1 \end{pmatrix}$$

then

$$D_{\alpha\beta} := M_{\alpha}^{-1} C_{\alpha\beta} M_{\beta} = \begin{pmatrix} 1 & 0 & \ell_{\alpha\beta} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$
Using that $(D_{\alpha\beta})_{V_\alpha \cap V_\beta \neq \emptyset}$ is a multiplicative cocycle we obtain that $(\ell_{\alpha\beta})_{W_\alpha \cap W_\beta \neq \emptyset}$ is an aditive cocycle and $\ell_{\alpha\beta} = \ell_\beta - \ell_\alpha$ on $W_\alpha \cap W_\beta \neq \emptyset$. Finally, $L_{\overline{\alpha}}^{-1} D_{\alpha\beta} L_\beta = I$, where

$$L_{\alpha} = \begin{pmatrix} 1 & 0 & -\ell_{\alpha} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$

as the reader can check. From this, we obtain that there is a collection of invertible triangular superior matrices $(B_\alpha)_\alpha$ such that $B_{\alpha\beta} = B_\alpha^{-1}B_\beta$ on $W_\alpha \cap W_\beta \neq \emptyset$. Let

$$B_{\alpha} = \begin{pmatrix} r_{\alpha} & s_{\alpha} & t_{\alpha} \\ 0 & u_{\alpha} & v_{\alpha} \\ 0 & 0 & 1 \end{pmatrix}.$$

If $W_\alpha \cap W_\beta \neq \emptyset$ then $r_\alpha \tilde{Y}_\alpha + s_\alpha \tilde{Z}_\alpha + t_\alpha R = r_\beta \tilde{Y}_\beta + s_\beta \tilde{Z}_\beta + t_\beta R$ and $u_\alpha Z_\alpha + v_\alpha R = u_\beta Z_\beta + v_\beta R$ on $W_\alpha \cap W_\beta$. This defines two holomorphic vector fields $V_0$ and $Z_0$ on $H$ by

$$V_0|_{W_\alpha} = r_\alpha \tilde{Y}_\alpha + s_\alpha \tilde{Z}_\alpha + t_\alpha R \quad \text{and} \quad Z_0|_{W_\alpha} = u_\alpha \tilde{Z}_\alpha + v_\alpha R.$$

Since $H$ is a Hartog's domain with holomorphic closure is the polydisc $Q_1$, $V_0$ and $Z_0$ can be extended to $Q_1$. We denote these extensions by the same symbols. Moreover, we have $i_{V_0} \eta = i_{Z_0} \eta = 0$. We assert that $V_0(0) \neq 0$.

In fact, write $V_0(z) = \sum_{j=0}^{p+2} g_j(z) \frac{\partial}{\partial z_j}$, $z \in Q_1$. If $V_0(0) = 0$ then $g_0(0) = 0$ and so the analytic set $C := \{z \in Q_1 | g_0(z) = 0\}$ must intersect the boundary $\partial Q_1$ of $Q_1$. If $z_0 \in C \cap \partial Q_1$ then there is $\alpha \in A$ such that $z_0 \in W_\alpha$. However, since $\tilde{Z}_\alpha$ and $R$ are tangent to $\tilde{\Sigma}_o$, we get $g_0(z_0) = r_\alpha(z_0) \cdot g_\alpha(z_0) \neq 0$ (see (iv)), because $g_\alpha \in \mathcal{O}^*(\tilde{U}_\alpha)$ and the matrix $B_\alpha$ is invertible.

Now, let us prove that there is a formal vector field $V = V_0 + \sum_{j \geq 1} z_0^j V_j$ such that $i_V \eta = 0$. To do that we recall that $A_{\alpha\beta}|_{W_\alpha \cap W_\beta} = B_{\alpha\beta}$ and $B_{\alpha\beta} = B_\alpha^{-1}B_\beta$. Consider a collection of invertible matrices $(\tilde{B}_\alpha)_{\alpha \in A}$, where $\tilde{B}_\alpha$ is an extension of $B_\alpha$ to $\tilde{U}_\alpha$. Consider also the cocycle of matrices $\tilde{A}_{\alpha\beta} := \tilde{B}_\alpha \cdot A_{\alpha\beta} \cdot \tilde{B}_\beta^{-1}$. This cocycle is equivalent to $A_{\alpha\beta}$ and $\tilde{A}_{\alpha\beta}|_{W_\alpha \cap W_\beta} = I$, $\forall W_\alpha \cap W_\beta \neq \emptyset$. Since $W_\alpha \cap W_\beta = (z_0 = 0) \cap \tilde{U}_\alpha \cap \tilde{U}_\beta$ we can write

$$\tilde{A}_{\alpha\beta} = I + \sum_{j \geq 1} z_0^j A_{\alpha\beta}^j,$$

where the entries of $A_{\alpha\beta}^j$ are holomorphic in $W_\alpha \cap W_\beta$. We claim that there are collections of power series of matrices of the form

$$A_{\alpha} = I + \sum_{j \geq 1} z_{0}^{j} A_{\alpha}^{j}, \quad \alpha \in A, \qquad (5.5)$$
Using that $(D_{\alpha\beta})_{V_\alpha \cap V_\beta \neq \emptyset}$ is a multiplicative cocycle we obtain that $(\ell_{\alpha\beta})_{W_\alpha \cap W_\beta \neq \emptyset}$ is an aditive cocycle and $\ell_{\alpha\beta} = \ell_\beta - \ell_\alpha$ on $W_\alpha \cap W_\beta \neq \emptyset$. Finally, $L_{\overline{\alpha}}^{-1} D_{\alpha\beta} L_\beta = I$, where

$$L_{\alpha} = \begin{pmatrix} 1 & 0 & -\ell_{\alpha} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$

as the reader can check. From this, we obtain that there is a collection of invertible triangular superior matrices $(B_\alpha)_\alpha$ such that $B_{\alpha\beta} = B_\alpha^{-1}B_\beta$ on $W_\alpha \cap W_\beta \neq \emptyset$. Let

$$B_{\alpha} = \begin{pmatrix} r_{\alpha} & s_{\alpha} & t_{\alpha} \\ 0 & u_{\alpha} & v_{\alpha} \\ 0 & 0 & 1 \end{pmatrix}.$$

If $W_\alpha \cap W_\beta \neq \emptyset$ then $r_\alpha \tilde{Y}_\alpha + s_\alpha \tilde{Z}_\alpha + t_\alpha R = r_\beta \tilde{Y}_\beta + s_\beta \tilde{Z}_\beta + t_\beta R$ and $u_\alpha Z_\alpha + v_\alpha R = u_\beta Z_\beta + v_\beta R$ on $W_\alpha \cap W_\beta$. This defines two holomorphic vector fields $V_0$ and $Z_0$ on $H$ by

$$V_0|_{W_\alpha} = r_\alpha \tilde{Y}_\alpha + s_\alpha \tilde{Z}_\alpha + t_\alpha R \quad \text{and} \quad Z_0|_{W_\alpha} = u_\alpha \tilde{Z}_\alpha + v_\alpha R.$$

Since $H$ is a Hartog's domain with holomorphic closure is the polydisc $Q_1$, $V_0$ and $Z_0$ can be extended to $Q_1$. We denote these extensions by the same symbols. Moreover, we have $i_{V_0} \eta = i_{Z_0} \eta = 0$. We assert that $V_0(0) \neq 0$.

In fact, write $V_0(z) = \sum_{j=0}^{p+2} g_j(z) \frac{\partial}{\partial z_j}$, $z \in Q_1$. If $V_0(0) = 0$ then $g_0(0) = 0$ and so the analytic set $C := \{z \in Q_1 | g_0(z) = 0\}$ must intersect the boundary $\partial Q_1$ of $Q_1$. If $z_0 \in C \cap \partial Q_1$ then there is $\alpha \in A$ such that $z_0 \in W_\alpha$. However, since $\tilde{Z}_\alpha$ and $R$ are tangent to $\tilde{\Sigma}_o$, we get $g_0(z_0) = r_\alpha(z_0) \cdot g_\alpha(z_0) \neq 0$ (see (iv)), because $g_\alpha \in \mathcal{O}^*(\tilde{U}_\alpha)$ and the matrix $B_\alpha$ is invertible.

Now, let us prove that there is a formal vector field $V = V_0 + \sum_{j \geq 1} z_0^j V_j$ such that $i_V \eta = 0$. To do that we recall that $A_{\alpha\beta}|_{W_\alpha \cap W_\beta} = B_{\alpha\beta}$ and $B_{\alpha\beta} = B_\alpha^{-1}B_\beta$. Consider a collection of invertible matrices $(\tilde{B}_\alpha)_{\alpha \in A}$, where $\tilde{B}_\alpha$ is an extension of $B_\alpha$ to $\tilde{U}_\alpha$. Consider also the cocycle of matrices $\tilde{A}_{\alpha\beta} := \tilde{B}_\alpha \cdot A_{\alpha\beta} \cdot \tilde{B}_\beta^{-1}$. This cocycle is equivalent to $A_{\alpha\beta}$ and $\tilde{A}_{\alpha\beta}|_{W_\alpha \cap W_\beta} = I$, $\forall W_\alpha \cap W_\beta \neq \emptyset$. Since $W_\alpha \cap W_\beta = (z_0 = 0) \cap \tilde{U}_\alpha \cap \tilde{U}_\beta$ we can write

$$\tilde{A}_{\alpha\beta} = I + \sum_{j \geq 1} z_0^j A_{\alpha\beta}^j,$$

where the entries of $A_{\alpha\beta}^j$ are holomorphic in $W_\alpha \cap W_\beta$. We claim that there are collections of power series of matrices of the form

$$A_{\alpha} = I + \sum_{j \geq 1} z_0^j A_{\alpha}^j, \quad \alpha \in A, \qquad (5.5)$$
such that the entries of $A_{\alpha}^j$ are holomorphic in $W_{\alpha}$ and $\tilde{A}_{\alpha\beta} = A_{\alpha}^{-1}A_{\beta}$. Given a power series in $z_0$, say $S = \sum_{j \ge 0} z_0^j S_j$, we will use the notation $J^k(S)$ for the truncation $\sum_{0 \le j \le k} z_0^j S_j$. The matrices $A_{\alpha}^j$ will be constructed by induction on $j \ge 0$ in such a way that

$$J^k \left( \left( I + \sum_{1 \le j \le k} z_0^j A_{\alpha}^j \right)^{-1} \cdot \tilde{A}_{\alpha\beta} \cdot \left( I + \sum_{1 \le j \le k} z_0^j A_{\beta}^j \right) \right) = I. \quad (I_k)$$

Note that $(I_0)$ is true and assume that we can construct collections $(A_{\alpha}^j)_{0 \le j \le l-1}$, $\alpha \in A$, such that $(I_k)$ is true for all $0 \le k \le l-1$. Set $\tilde{A}_{\alpha\beta}^l = I + \sum_{j=1}^l z_0^j A_{\alpha\beta}^j$ and $C_{\alpha}^{l-1} = I + \sum_{j=1}^{l-1} z_0^j A_{\alpha}^j$. Since $(I_{l-1})$ is true, we get

$$J^l [(C_{\alpha}^{l-1})^{-1} \cdot \tilde{A}_{\alpha\beta}^l \cdot C_{\beta}^{l-1}] = I + z_0^l A_{\alpha\beta}^l.$$

Writing explicitly that the above expression is a multiplicative cocycle of matrices we get that $(A_{\alpha\beta}^l)_{W_{\alpha} \cap W_{\beta} \neq \emptyset}$ is an additive cocycle. Since $H^1(H, \mathcal{O}) = 0$ we get collections $(A_{\alpha}^l)_{\alpha \in A}$ such that if $C_{\alpha}^l = I + \sum_{j=1}^l z_0^j A_{\alpha}^j$ then $(I_l)$ is true. In particular, the collection of formal series $C_{\alpha} = I + \sum_{j \ge 1} z_0^j A_{\alpha}^j$, $\alpha \in A$, satisfies $C_{\alpha}^{-1} \tilde{A}_{\alpha\beta} C_{\beta} = I$, so that

$$\tilde{A}_{\alpha\beta} = C_{\alpha} \cdot C_{\beta}^{-1} \implies A_{\alpha\beta} = \tilde{B}_{\alpha}^{-1} \cdot C_{\alpha} \cdot C_{\beta}^{-1} \cdot \tilde{B}_{\beta} = (\tilde{B}_{\alpha}^{-1} \cdot C_{\alpha}) \cdot (\tilde{B}_{\beta}^{-1} \cdot C_{\beta})^{-1}.$$

This proves that the cocycle $(A_{\alpha\beta})_{\alpha\beta}$ is formally trivial and finishes the proof of the existence of the constant vector field $v$ such that $i_v\eta = 0$.

# 6. Appendix. Proof of Theorem 2.10 (by Alcides Lins Neto)

Let $(f_1...f_s = 0)$ be a germ at $0 \in \mathbb{C}^n$ of hypersurface with a strictly ordinary singularity at $0$, where $s = n-k$ and $2 \le k \le n-1$. Set $X_s = (f_1 = \dots = f_s = 0)$ and $X_s^* = X_s \setminus \{0\}$. Note that $\dim_{\mathbb{C}}(X_s) = k$. Our aim is to prove the following:

**THEOREM 2.10.** — In the above situation there are representatives of $X_s$ and $X_s^*$ in a polydisc $Q \subset \mathbb{C}^n$, denoted by the same letters, such that:

(a) If $0 \le q \le k-2$ then any form $\tilde{\alpha} \in \Omega^q(X_s^*)$ can be extended to a form $\alpha \in \Omega^q(Q)$.

(b) If $q \ge 1$, $l \ge 0$ and $1 \le q+l \le k-2$ then $H^q(X_s^*, \Omega^l) = 0$.

**Example 6.1.** — We would like to observe that the statement of Theorem 2.10 is not true for $k$ and $k-1$ forms. For instance, let $f \in \mathbb{C}[z_0, z_1, ..., z_n]$, $n \ge 3$, be a homogeneous polynomial of degree $\ge n+1$ and $X = (f=0) \subset$
such that the entries of $A_{\alpha}^j$ are holomorphic in $W_{\alpha}$ and $\tilde{A}_{\alpha\beta} = A_{\alpha}^{-1}A_{\beta}$. Given a power series in $z_0$, say $S = \sum_{j \ge 0} z_0^j S_j$, we will use the notation $J^k(S)$ for the truncation $\sum_{0 \le j \le k} z_0^j S_j$. The matrices $A_{\alpha}^j$ will be constructed by induction on $j \ge 0$ in such a way that

$$J^k \left( \left( I + \sum_{1 \le j \le k} z_0^j A_{\alpha}^j \right)^{-1} \cdot \tilde{A}_{\alpha\beta} \cdot \left( I + \sum_{1 \le j \le k} z_0^j A_{\beta}^j \right) \right) = I. \quad (I_k)$$

Note that $(I_0)$ is true and assume that we can construct collections $(A_{\alpha}^j)_{0 \le j \le l-1}$, $\alpha \in A$, such that $(I_k)$ is true for all $0 \le k \le l-1$. Set $\tilde{A}_{\alpha\beta}^l = I + \sum_{j=1}^l z_0^j A_{\alpha\beta}^j$ and $C_{\alpha}^{l-1} = I + \sum_{j=1}^{l-1} z_0^j A_{\alpha}^j$. Since $(I_{l-1})$ is true, we get

$$J^l [(C_{\alpha}^{l-1})^{-1} \cdot \tilde{A}_{\alpha\beta}^l \cdot C_{\beta}^{l-1}] = I + z_0^l A_{\alpha\beta}^l.$$

Writing explicitly that the above expression is a multiplicative cocycle of matrices we get that $(A_{\alpha\beta}^l)_{W_{\alpha} \cap W_{\beta} \neq \emptyset}$ is an additive cocycle. Since $H^1(H, \mathcal{O}) = 0$ we get collections $(A_{\alpha}^l)_{\alpha \in A}$ such that if $C_{\alpha}^l = I + \sum_{j=1}^l z_0^j A_{\alpha}^j$ then $(I_l)$ is true. In particular, the collection of formal series $C_{\alpha} = I + \sum_{j \ge 1} z_0^j A_{\alpha}^j$, $\alpha \in A$, satisfies $C_{\alpha}^{-1} \tilde{A}_{\alpha\beta} C_{\beta} = I$, so that

$$\tilde{A}_{\alpha\beta} = C_{\alpha} \cdot C_{\beta}^{-1} \implies A_{\alpha\beta} = \tilde{B}_{\alpha}^{-1} \cdot C_{\alpha} \cdot C_{\beta}^{-1} \cdot \tilde{B}_{\beta} = (\tilde{B}_{\alpha}^{-1} \cdot C_{\alpha}) \cdot (\tilde{B}_{\beta}^{-1} \cdot C_{\beta})^{-1}.$$

This proves that the cocycle $(A_{\alpha\beta})_{\alpha\beta}$ is formally trivial and finishes the proof of the existence of the constant vector field $v$ such that $i_v\eta = 0$.

## 6. Appendix. Proof of Theorem 2.10 (by Alcides Lins Neto)

Let $(f_1...f_s = 0)$ be a germ at $0 \in \mathbb{C}^n$ of hypersurface with a strictly ordinary singularity at 0, where $s = n-k$ and $2 \le k \le n-1$. Set $X_s = (f_1 = \dots = f_s = 0)$ and $X_s^* = X_s \setminus \{0\}$. Note that $\dim_{\mathbb{C}}(X_s) = k$. Our aim is to prove the following:

**THEOREM 2.10.** — In the above situation there are representatives of $X_s$ and $X_s^*$ in a polydisc $Q \subset \mathbb{C}^n$, denoted by the same letters, such that:

(a) If $0 \le q \le k-2$ then any form $\tilde{\alpha} \in \Omega^q(X_s^*)$ can be extended to a form $\alpha \in \Omega^q(Q)$.

(b) If $q \ge 1$, $l \ge 0$ and $1 \le q+l \le k-2$ then $H^q(X_s^*, \Omega^l) = 0$.

**Example 6.1.** — We would like to observe that the statement of Theorem 2.10 is not true for $k$ and $k-1$ forms. For instance, let $f \in \mathbb{C}[z_0, z_1, ..., z_n]$, $n \ge 3$, be a homogeneous polynomial of degree $\ge n+1$ and $X = (f=0) \subset \mathbb{C}^n$.
$\mathbb{C}^{n+1}$, so that $k - 1 = n - 1$. Assume that $Z := \Pi(X^*) \subset \mathbb{P}^n$ is smooth. It is known that there exists a non-vanishing holomorphic (n-1)-form on $Z$, say $\alpha$. The (n-1)-form $\Pi^*(\alpha)$ is holomorphic on $X^*$ and has no holomorphic extension to any neighborhood of $0 \in \mathbb{C}^{n+1}$.

In the situation of the hypothesis of Theorem 2.10, if we fix representatives of $f_1, \dots, f_s$ in a polydisc $Q$ (denoted by the same letters), $0 \in Q \subset \mathbb{C}^n$, we use the notation $X_0 = Q$ and $X_t = \{z \in Q \mid f_1(z) = \dots = f_t(z) = 0\}$, $1 \le t \le s$. We set also $X_t^* = X_t \setminus \{0\}$.

The “strictly ordinary singularity” assumption implies that for any $1 \le t \le s$ then $X_t^* := (f_1 = \dots = f_t = 0) \setminus \{0\}$ has an isolated singularity at 0:

$$ \mathrm{d}f_1(z) \wedge \dots \wedge \mathrm{d}f_t(z) \ne 0, \forall z \in X_t^*. \quad (6.1) $$

From the above remark we get the following:

**LEMMA 6.2.** — There are representatives of $f_1, \dots, f_s$ in a polydisc $Q$ such that:

(a) 0 is the unique singularity of $X_t$ in $Q$, $\forall 1 \le t \le s$. In particular, $X_t^*$ is smooth of codimension $t$, $\forall 1 \le t \le s$.

(b) For all $0 \le t \le s-1$ the function $f_{t+1}|_{X_t^*}$ is a submersion. In particular, $\mathrm{d}f_{t+1}(z) \ne 0$ for all $z \in X_t^*$.

With Lemma 6.2 the proof of Theorem 2.10 is reduced to the following:

**CLAIM 6.3.** — Let $Q \subset \mathbb{C}^n$ be a polydisc with $0 \in Q$. Let $X \subset Q$ be a connected complete intersection with a singularity $0 \in X$, defined by $X = (f_1 = \dots = f_{n-k} = 0)$. Assume $2 \le k \le n-1$ and:

(1) $f_j$ has an isolated singularity at $0 \in Q$, $\forall 1 \le j \le n-k$.

(2) $\forall I = (i_1, \dots, i_t)$, where $i_j \ne i_k$ if $j \ne k$, and $\forall z \in (f_{i_1} = \dots = f_{i_t} = 0) \setminus \{0\}$ then $\mathrm{d}f_{i_1}(z) \wedge \dots \wedge \mathrm{d}f_{i_t}(z) \ne 0$. In particular:

(a) $(f_{i_1} = \dots = f_{i_t} = 0) \setminus \{0\}$ is smooth of codimension $t$.

(b) $\dim_{\mathbb{C}}(X) = k$.

If $0 \le \ell \le k-2$ then any $\ell$-form $\omega_\ell \in \Omega^\ell(X \setminus \{0\})$ admits an extension $\tilde{\omega}_\ell \in \Omega^\ell(Q)$.

In order to prove Claim 6.3 we will consider the situation below:

Let $Y$ be a connected complex manifold of dimension $n \ge 3$ and $Z \subset Y$ be a codimension one complex codimension one submanifold defined by $f=0$, where $f \in \mathcal{O}(Y)$ and 0 is a regular value of $f$. In particular, $Z$ is a smooth submanifold of $Y$. For simplicity, we will use the notations $\Omega^\ell$ for the sheaf of holomorphic $\ell$-forms on $Y$ and $Z$. Of course $\Omega^0 = \mathcal{O}$.
$\mathbb{C}^{n+1}$, so that $k-1=n-1$. Assume that $Z := \Pi(X^*) \subset \mathbb{P}^n$ is smooth. It is known that there exists a non-vanishing holomorphic (n-1)-form on $Z$, say $\alpha$. The (n-1)-form $\Pi^*(\alpha)$ is holomorphic on $X^*$ and has no holomorphic extension to any neighborhood of $0 \in \mathbb{C}^{n+1}$.

In the situation of the hypothesis of Theorem 2.10, if we fix representatives of $f_1, \dots, f_s$ in a polydisc $Q$ (denoted by the same letters), $0 \in Q \subset \mathbb{C}^n$, we use the notation $X_0 = Q$ and $X_t = \{z \in Q \mid f_1(z) = \dots = f_t(z) = 0\}$, $1 \le t \le s$. We set also $X_t^* = X_t \setminus \{0\}$.

The “strictly ordinary singularity” assumption implies that for any $1 \le t \le s$ then $X_t^* := (f_1 = \dots = f_t = 0) \setminus \{0\}$ has an isolated singularity at 0:

$$ \mathrm{d}f_1(z) \wedge \dots \wedge \mathrm{d}f_t(z) \ne 0, \forall z \in X_t^*. \quad (6.1) $$

From the above remark we get the following:

**LEMMA 6.2.** — There are representatives of $f_1, \dots, f_s$ in a polydisc $Q$ such that:

(a) $0$ is the unique singularity of $X_t$ in $Q$, $\forall 1 \le t \le s$. In particular, $X_t^*$ is smooth of codimension $t$, $\forall 1 \le t \le s$.

(b) For all $0 \le t \le s-1$ the function $f_{t+1}|_{X_t^*}$ is a submersion. In particular, $\mathrm{d}f_{t+1}(z) \ne 0$ for all $z \in X_t^*$.

With Lemma 6.2 the proof of Theorem 2.10 is reduced to the following:

**CLAIM 6.3.** — Let $Q \subset \mathbb{C}^n$ be a polydisc with $0 \in Q$. Let $X \subset Q$ be a connected complete intersection with a singularity $0 \in X$, defined by $X = (f_1 = \dots = f_{n-k} = 0)$. Assume $2 \le k \le n-1$ and:

(1) $f_j$ has an isolated singularity at $0 \in Q$, $\forall 1 \le j \le n-k$.

(2) $\forall I = (i_1, \dots, i_t)$, where $i_j \ne i_k$ if $j \ne k$, and $\forall z \in (f_{i_1} = \dots = f_{i_t} = 0) \setminus \{0\}$ then $\mathrm{d}f_{i_1}(z) \wedge \dots \wedge \mathrm{d}f_{i_t}(z) \ne 0$. In particular:

(a) $(f_{i_1} = \dots = f_{i_t} = 0) \setminus \{0\}$ is smooth of codimension $t$.

(b) $\dim_{\mathbb{C}}(X) = k$.

If $0 \le \ell \le k-2$ then any $\ell$-form $\omega_\ell \in \Omega^\ell(X \setminus \{0\})$ admits an extension $\tilde{\omega}_\ell \in \Omega^\ell(Q)$.

In order to prove Claim 6.3 we will consider the situation below:

Let $Y$ be a connected complex manifold of dimension $n \ge 3$ and $Z \subset Y$ be a codimension one complex codimension one submanifold defined by $f=0$, where $f \in \mathcal{O}(Y)$ and $0$ is a regular value of $f$. In particular, $Z$ is a smooth submanifold of $Y$. For simplicity, we will use the notations $\Omega^\ell$ for the sheaf of holomorphic $\ell$-forms on $Y$ and $Z$. Of course $\Omega^0 = \mathcal{O}$.
LEMMA 6.4. — In the above situation assume that $H^k(Y, \Omega^\ell) = 0$ for all $k$ and $\ell$ such that $k \ge 1$ and $1 \le k + \ell \le r + 1$. Then:

(a) If $k \ge 1$, $\ell \ge 0$ and $r \ge 1$ are such that $1 \le k + \ell \le r$ then $H^k(Z, \Omega^\ell) = 0$.

(b) If $r \ge 0$ and $0 \le \ell \le r$ then any $\ell$-form $\omega \in \Omega^\ell(Z)$ can be extended to a $\ell$-form $\tilde{\omega} \in \Omega^\ell(Y)$.

*Proof*. — We will use Leray's theorem (cf. [12, p. 43]). Let us consider Leray coverings $\mathcal{U} = (U_\alpha)_{\alpha \in A}$ and $\tilde{\mathcal{U}} = (\tilde{U}_{\tilde{\alpha}})_{\tilde{\alpha} \in \tilde{A}}$ of $Z$ and $Y$ by open sets, respectively, such that: $A \subset \tilde{A}$ and if $\alpha \in A$ then:

(i) $\tilde{U}_\alpha$ is the domain of a holomorphic chart $z^\alpha = (z_1, \dots, z_n) : \tilde{U}_\alpha \to \mathbb{C}^n$, such that $\tilde{U}_\alpha = \{z^\alpha | |z_j| < 1, j = 1, \dots, n\}$ and $f|_{\tilde{U}_\alpha} = z_1$

(ii) $U_\alpha = \tilde{U}_\alpha \cap Z$. In particular, $U_\alpha = \{z^\alpha \in \tilde{U}_\alpha | z_1 = 0\}$.

Note that $\tilde{U}_\alpha$ is biholomorphic to a polydisc of $\mathbb{C}^n$ and $U_\alpha$ to a polydisc of $\mathbb{C}^{n-1}$. We assume also that:

(iii) If $\alpha \in \tilde{A} \setminus A$ then $\tilde{U}_\alpha \cap Z = \emptyset$. This implies that $A = \{\alpha \in \tilde{A} | \tilde{U}_\alpha \cap Z \ne \emptyset\}$.

Given $J = (j_0, \dots, j_k) \in \tilde{A}^{k+1}$ (resp. $J \in A^{k+1}$) we set $\tilde{U}_J = \tilde{U}_{j_0} \cap \dots \cap \tilde{U}_{j_k}$ (resp. $U_J = U_{j_0} \cap \dots \cap U_{j_k}$). Note that by construction, if $J \in A^{k+1}$ is such that $U_J \ne \emptyset$ then $U_J = \tilde{U}_J \cap Z$. Moreover, if $z^{\alpha_0} = (z_1, \dots, z_n)$ is a chart as in (i) then $U_J \subset \{z_1 = 0\}$.

CLAIM 6.5. — Given $0 \le \ell \le n-1$ and $J = (j_0, \dots, j_k) \in A^{k+1}$ such that $U_J \ne \emptyset$ then any $\ell$-form $\omega$ on $U_J$ can be extended to a $\ell$-form on $\tilde{U}_J$. Moreover, if $\tilde{\omega}_1$ and $\tilde{\omega}_2$ are two extensions of $\omega$ to $\tilde{U}_J$ then:

(a) $\tilde{\omega}_2 - \tilde{\omega}_1 = g.f, g \in \mathcal{O}(\tilde{U}_J)$ if $\ell = 0$.

(b) $\tilde{\omega}_2 - \tilde{\omega}_1 = \alpha \wedge df + f.\beta$, where $\alpha \in \Omega^{\ell-1}(\tilde{U}_J)$ and $\beta \in \Omega^\ell(\tilde{U}_J)$, if $\ell \ge 1$.

*Proof of the Claim.* — Since $\tilde{U}_\alpha$ is biholomorphic to a polydisc, for any $\alpha \in \tilde{A}$ it follows that $\tilde{U}_J$ is a Stein open subset of $Y$. Since $U_J = f^{-1}(0) \cap \tilde{U}_J$ it follows that any holomorphic function $h \in \mathcal{O}(U_J)$ admits an extension $\tilde{h} \in \mathcal{O}(\tilde{U}_J)$ (cf. [12]). This proves the case $\ell = 0$. When $\ell \ge 1$, we consider the chart $z^{\alpha_0} = (z_1, \dots, z_n) : \tilde{U}_{\alpha_0} \to \mathbb{C}^n$ where $f|_{\tilde{U}_{\alpha_0}} = z_1$ and $U_{\alpha_0} = \{z^\alpha \in \tilde{U}_{\alpha_0} | z_1 = 0\}$, so that $U_J = \{z^{\alpha_0} \in \tilde{U}_J | z_1 = 0\}$. In particular, any $\ell$-form $\omega \in \Omega^\ell(U_J)$ can be written as

$$ \omega = \sum_{I=(2 \le i_1 < \dots < i_\ell \le n)} h_I \cdot dz_{i_1} \wedge \dots \wedge dz_{i_\ell}, \text{ where } h_I = h_I(z_2, \dots, z_n) \in \mathcal{O}(U_J). $$
LEMMA 6.4. — In the above situation assume that $H^k(Y, \Omega^\ell) = 0$ for all $k$ and $\ell$ such that $k \ge 1$ and $1 \le k + \ell \le r + 1$. Then:

(a) If $k \ge 1$, $\ell \ge 0$ and $r \ge 1$ are such that $1 \le k + \ell \le r$ then $H^k(Z, \Omega^\ell) = 0$.

(b) If $r \ge 0$ and $0 \le \ell \le r$ then any $\ell$-form $\omega \in \Omega^\ell(Z)$ can be extended to a $\ell$-form $\tilde{\omega} \in \Omega^\ell(Y)$.

*Proof*. — We will use Leray's theorem (cf. [12, p. 43]). Let us consider Leray coverings $\mathcal{U} = (U_\alpha)_{\alpha \in A}$ and $\tilde{\mathcal{U}} = (\tilde{U}_{\tilde{\alpha}})_{\tilde{\alpha} \in \tilde{A}}$ of $Z$ and $Y$ by open sets, respectively, such that: $A \subset \tilde{A}$ and if $\alpha \in A$ then:

(i) $\tilde{U}_\alpha$ is the domain of a holomorphic chart $z^\alpha = (z_1, \dots, z_n): \tilde{U}_\alpha \to \mathbb{C}^n$, such that $\tilde{U}_\alpha = \{z^\alpha | |z_j| < 1, j = 1, \dots, n\}$ and $f|_{\tilde{U}_\alpha} = z_1$

(ii) $U_\alpha = \tilde{U}_\alpha \cap Z$. In particular, $U_\alpha = \{z^\alpha \in \tilde{U}_\alpha | z_1 = 0\}$.

Note that $\tilde{U}_\alpha$ is biholomorphic to a polydisc of $\mathbb{C}^n$ and $U_\alpha$ to a polydisc of $\mathbb{C}^{n-1}$. We assume also that:

(iii) If $\alpha \in \tilde{A} \setminus A$ then $\tilde{U}_\alpha \cap Z = \emptyset$. This implies that $A = \{\alpha \in \tilde{A} | \tilde{U}_\alpha \cap Z \ne \emptyset\}$.

Given $J = (j_0, \dots, j_k) \in \tilde{A}^{k+1}$ (resp. $J \in A^{k+1}$) we set $\tilde{U}_J = \tilde{U}_{j_0} \cap \dots \cap \tilde{U}_{j_k}$ (resp. $U_J = U_{j_0} \cap \dots \cap U_{j_k}$). Note that by construction, if $J \in A^{k+1}$ is such that $U_J \ne \emptyset$ then $U_J = \tilde{U}_J \cap Z$. Moreover, if $z^{\alpha_0} = (z_1, \dots, z_n)$ is a chart as in (i) then $U_J \subset \{z_1 = 0\}$.

CLAIM 6.5. — Given $0 \le \ell \le n-1$ and $J = (j_0, \dots, j_k) \in A^{k+1}$ such that $U_J \ne \emptyset$ then any $\ell$-form $\omega$ on $U_J$ can be extended to a $\ell$-form on $\tilde{U}_J$. Moreover, if $\tilde{\omega}_1$ and $\tilde{\omega}_2$ are two extensions of $\omega$ to $\tilde{U}_J$ then:

(a) $\tilde{\omega}_2 - \tilde{\omega}_1 = g.f, g \in \mathcal{O}(\tilde{U}_J)$ if $\ell = 0$.

(b) $\tilde{\omega}_2 - \tilde{\omega}_1 = \alpha \wedge df + f.\beta$, where $\alpha \in \Omega^{\ell-1}(\tilde{U}_J)$ and $\beta \in \Omega^\ell(\tilde{U}_J)$, if $\ell \ge 1$.

*Proof of the Claim.* — Since $\tilde{U}_\alpha$ is biholomorphic to a polydisc, for any $\alpha \in \tilde{A}$ it follows that $\tilde{U}_J$ is a Stein open subset of $Y$. Since $U_J = f^{-1}(0) \cap \tilde{U}_J$ it follows that any holomorphic function $h \in \mathcal{O}(U_J)$ admits an extension $\tilde{h} \in \mathcal{O}(\tilde{U}_J)$ (cf. [12]). This proves the case $\ell = 0$. When $\ell \ge 1$, we consider the chart $z^{\alpha_0} = (z_1, \dots, z_n): \tilde{U}_{\alpha_0} \to \mathbb{C}^n$ where $f|_{\tilde{U}_{\alpha_0}} = z_1$ and $U_{\alpha_0} = \{z^\alpha \in \tilde{U}_{\alpha_0} | z_1 = 0\}$, so that $U_J = \{z^{\alpha_0} \in \tilde{U}_J | z_1 = 0\}$. In particular, any $\ell$-form $\omega \in \Omega^\ell(U_J)$ can be written as

$$ \omega = \sum_{I=(2 \le i_1 < \dots < i_\ell \le n)} h_I \cdot dz_{i_1} \wedge \dots \wedge dz_{i_\ell}, \quad \text{where } h_I = h_I(z_2, \dots, z_n) \in \mathcal{O}(U_J). $$
By the case $\ell = 0$ any function $h_I$ admits an extension $\tilde{h}_I \in \mathcal{O}(\tilde{U}_J)$. Therefore, $\omega$ admits the extension

$$ \tilde{\omega} = \sum_{I=(2 \le i_1 < \dots < i_{\ell} \le n)} \tilde{h}_I \cdot dz_{i_1} \wedge \dots \wedge dz_{i_{\ell}} \in \Omega^{\ell}(\tilde{U}_J). $$

If $\tilde{\omega}_2$ and $\tilde{\omega}_1$ are two extensions of $\omega$ to $\tilde{U}_J$ then $(\tilde{\omega}_2 - \tilde{\omega}_1)|_{z_1=0} = 0$. Therefore, if $\ell = 0$ then $\tilde{\omega}_2 - \tilde{\omega}_1 = g.z_1 = g.f$ as in (a), whereas if $\ell \ge 1$ then $\tilde{\omega}_2 - \tilde{\omega}_1 = \alpha \wedge dz_1 + z_1.\beta = \alpha \wedge df + f.\beta$ as in (b). $\square$

Since $\Omega^\ell$ is a holomorphic sheaf, by Leray's theorem we have $H^k(Z, \Omega^\ell) = H^k(\mathcal{U}, \Omega^\ell)$ and $H^k(Y, \Omega^\ell) = H^k(\tilde{U}, \Omega^\ell)$ for all $k \ge 1$ and $\ell \ge 0$. Of course, $H^0(Z, \Omega^\ell) = \Omega^\ell(Z)$ and $H^0(Y, \Omega^\ell) = \Omega^\ell(Y)$. Let us fix some notations (cf. [12]):

(1) $C^k(\mathcal{U}, \Omega^\ell)$ (resp. $C^k(\tilde{U}, \Omega^\ell)$) the $\mathcal{O}$-module of $k$-cochains of $\ell$-forms with respect to $\mathcal{U}$ (resp. with respect to $\tilde{U}$).

(2) $\delta = \delta_k: C^k(*, \Omega^\ell) \to C^{k+1}(*, \Omega^\ell)$ the coboundary operator, where $*=\mathcal{U}$ or $\tilde{\mathcal{U}}$. In this way, we have:

$$ H^k(*, \Omega^\ell) = \ker(\delta_k) / \operatorname{Im}(\delta_{k-1}), k \ge 0. $$

Recall that

$$ C^k(\mathcal{U}, \Omega^\ell) = \prod_{J \in A^{k+1}} \Omega^\ell(U_J), $$

where $J = (\alpha_0, \dots, \alpha_k) \in A^{k+1}$ and $U_J = U_{\alpha_0} \cap \dots \cap U_{\alpha_k}$. In particular, a cochain in $\omega_\ell^k \in C^k(\mathcal{U}, \Omega^\ell)$ is of the form

$$ \omega_\ell^k = (\omega_J)_{J \in A^{k+1}}, \omega_J \in \Omega^\ell(U_J). $$

When $U_J = \emptyset$ by convention we set $\omega_J = 0$. Anagolously, a cochain $\tilde{\omega}_\ell^k \in C^k(\tilde{U}, \Omega^\ell)$ is of the form

$$ \tilde{\omega}_{\ell}^{k} = (\tilde{\omega}_{J})_{J \in \tilde{A}^{k+1}}, \omega_{J} \in \Omega^{\ell}(\tilde{U}_{J}). $$

**Restriction of cochains.** — Given a cochain $\tilde{\omega}_\ell^k \in C^k(\tilde{U}, \Omega^\ell)$, where $\tilde{\omega}_\ell^k = (\tilde{\omega}_J)_{J \in \tilde{A}^{k+1}}$, its restriction to $Z$ is defined as

$$ \tilde{\omega}_{\ell}^{k}|_{Z} := (\tilde{\omega}_{J}|_{U_J})_{J \in A^{k+1}} \in C^{k}(\mathcal{U}, \Omega^{\ell}). $$

Recall that if $J \in A^{k+1}$ then $U_J = \tilde{U}_J \cap Z$.

**Remark 6.6.** — Let $\tilde{\omega}_\ell^k, \tilde{\eta}_\ell^k \in C^k(\tilde{U}, \Omega^\ell)$ be two cochains with the same restriction to $Z: (\tilde{\eta}_\ell^k - \tilde{\omega}_\ell^k)|_Z = 0$. It follows from Claim 6.5 that:

(a) If $\ell = 0$ then there exists a cochain $g_0^k = (g_J)_{J \in \tilde{A}^{k+1}} \in C^k(\tilde{U}, \mathcal{O})$ such that $\tilde{\eta}_J - \tilde{\omega}_J = g_J.f$, for all $J \in \tilde{A}^{k+1}$. In this case we will write $\tilde{\eta}_\ell^k - \tilde{\omega}_\ell^k = f.g_0^k$.
By the case $\ell = 0$ any function $h_I$ admits an extension $\tilde{h}_I \in \mathcal{O}(\tilde{U}_J)$. Therefore, $\omega$ admits the extension

$$ \tilde{\omega} = \sum_{I=(2 \le i_1 < \dots < i_{\ell} \le n)} \tilde{h}_I \cdot dz_{i_1} \wedge \dots \wedge dz_{i_{\ell}} \in \Omega^{\ell}(\tilde{U}_J). $$

If $\tilde{\omega}_2$ and $\tilde{\omega}_1$ are two extensions of $\omega$ to $\tilde{U}_J$ then $(\tilde{\omega}_2 - \tilde{\omega}_1)|_{z_1=0} = 0$. Therefore, if $\ell = 0$ then $\tilde{\omega}_2 - \tilde{\omega}_1 = g.z_1 = g.f$ as in (a), whereas if $\ell \ge 1$ then $\tilde{\omega}_2 - \tilde{\omega}_1 = \alpha \wedge dz_1 + z_1.\beta = \alpha \wedge df + f.\beta$ as in (b). $\square$

Since $\Omega^\ell$ is a holomorphic sheaf, by Leray's theorem we have $H^k(Z, \Omega^\ell) = H^k(\mathcal{U}, \Omega^\ell)$ and $H^k(Y, \Omega^\ell) = H^k(\tilde{U}, \Omega^\ell)$ for all $k \ge 1$ and $\ell \ge 0$. Of course, $H^0(Z, \Omega^\ell) = \Omega^\ell(Z)$ and $H^0(Y, \Omega^\ell) = \Omega^\ell(Y)$. Let us fix some notations (cf. [12]):

(1) $C^k(\mathcal{U}, \Omega^\ell)$ (resp. $C^k(\tilde{U}, \Omega^\ell)$) the $\mathcal{O}$-module of $k$-cochains of $\ell$-forms with respect to $\mathcal{U}$ (resp. with respect to $\tilde{U}$).

(2) $\delta = \delta_k: C^k(*, \Omega^\ell) \to C^{k+1}(*, \Omega^\ell)$ the coboundary operator, where $*=\mathcal{U}$ or $\tilde{\mathcal{U}}$. In this way, we have:

$$ H^k(*, \Omega^\ell) = \ker(\delta_k) / \operatorname{Im}(\delta_{k-1}), k \ge 0. $$

Recall that

$$ C^k(\mathcal{U}, \Omega^\ell) = \prod_{J \in A^{k+1}} \Omega^\ell(U_J), $$

where $J = (\alpha_0, \dots, \alpha_k) \in A^{k+1}$ and $U_J = U_{\alpha_0} \cap \dots \cap U_{\alpha_k}$. In particular, a cochain in $\omega_\ell^k \in C^k(\mathcal{U}, \Omega^\ell)$ is of the form

$$ \omega_{\ell}^{k} = (\omega_{J})_{J \in A^{k+1}}, \omega_{J} \in \Omega^{\ell}(U_{J}). $$

When $U_J = \emptyset$ by convention we set $\omega_J = 0$. Anagolously, a cochain $\tilde{\omega}_\ell^k \in C^k(\tilde{U}, \Omega^\ell)$ is of the form

$$ \tilde{\omega}_{\ell}^{k} = (\tilde{\omega}_{J})_{J \in \tilde{A}^{k+1}}, \omega_{J} \in \Omega^{\ell}(\tilde{U}_{J}). $$

**Restriction of cochains.** — Given a cochain $\tilde{\omega}_\ell^k \in C^k(\tilde{U}, \Omega^\ell)$, where $\tilde{\omega}_\ell^k = (\tilde{\omega}_J)_{J \in \tilde{A}^{k+1}}$, its restriction to $Z$ is defined as

$$ \tilde{\omega}_{\ell}^{k}|_{Z} := (\tilde{\omega}_{J}|_{U_J})_{J \in A^{k+1}} \in C^{k}(\mathcal{U}, \Omega^{\ell}). $$

Recall that if $J \in A^{k+1}$ then $U_J = \tilde{U}_J \cap Z$.

**Remark 6.6.** — Let $\tilde{\omega}_\ell^k, \tilde{\eta}_\ell^k \in C^k(\tilde{U}, \Omega^\ell)$ be two cochains with the same restriction to $Z: (\tilde{\eta}_\ell^k - \tilde{\omega}_\ell^k)|_Z = 0$. It follows from Claim 6.5 that:

(a) If $\ell = 0$ then there exists a cochain $g_0^k = (g_J)_{J \in \tilde{A}^{k+1}} \in C^k(\tilde{U}, \mathcal{O})$ such that $\tilde{\eta}_J - \tilde{\omega}_J = g_J.f$, for all $J \in \tilde{A}^{k+1}$. In this case we will write $\tilde{\eta}_\ell^k - \tilde{\omega}_\ell^k = f.g_0^k$.
(b) If $\ell \ge 1$ then there are cochains $\tilde{\alpha}_{\ell-1}^k = (\tilde{\alpha}_J)_{J \in \tilde{A}^{k+1}} \in C^k(\tilde{U}, \Omega^{\ell-1})$ and $\tilde{\beta}_{\ell}^k = (\tilde{\beta}_J)_{J \in \tilde{A}^{k+1}} \in C^k(\tilde{U}, \Omega^{\ell})$ such that $\tilde{\eta}_J - \tilde{\omega}_J = \tilde{\alpha}_J \wedge df + f.\tilde{\beta}_J$, for all $J \in \tilde{A}^{k+1}$. In this case, we will write $\tilde{\eta}_{\ell}^k - \tilde{\omega}_{\ell}^k = \tilde{\alpha}_{\ell-1}^k \wedge df + f.\tilde{\beta}_{\ell}^k$.

We leave the details to the reader.

*Extension of cochains.* — Claim 6.5 implies that given a cochain $\omega_{\ell}^k \in C^k(\tilde{U}, \Omega^{\ell})$ then there exists a cochain $\tilde{\omega}_{\ell}^k \in C^k(\tilde{U}, \Omega^{\ell})$ whoose restriction to $Z$ coincides with $\omega_{\ell}^k$. We leave the details to the reader. The cochain $\tilde{\omega}_{\ell}^k$ will be called an *extension* of the cochain $\omega_{\ell}^k$.

*Division of cochains.* — Given a cochain $\beta_{\ell}^k = (\beta_J)_{J \in \tilde{A}^{k+1}} \in C^k(\tilde{U}, \Omega^{\ell})$ we define the cochain $\beta_{\ell}^k \wedge df := (\beta_J \wedge df)_{J \in \tilde{A}^{k+1}} \in C^k(\tilde{U}, \Omega^{\ell+1})$. We would like to observe that, if $\ell \ge 1$ and $df \wedge \beta_{\ell}^k = 0$ then there exists a cochain $\beta_{\ell-1}^k \in C^k(\tilde{U}, \Omega^{\ell-1})$ such that $\beta_{\ell}^k = \beta_{\ell-1}^k \wedge df$. The proof is easy and is left to the reader.

Let us assume the hypothesis of Lemma 6.4: $H^k(Y, \Omega^\ell) = 0$ if $k \ge 1$ and $1 \le k + \ell \le r + 1$.

**CLAIM 6.7.** — In the above situation, if $k \ge 0$ and $\ell \ge 0$ are such that $k + \ell \le r$ then any cocycle $\omega_{\ell}^k \in C^k(\tilde{U}, \Omega^{\ell})$ such that $\delta\omega_{\ell}^k = 0$ admits an extension $\tilde{\omega}_{\ell}^k \in C^k(\tilde{U}, \Omega^{\ell})$ such that $\delta\tilde{\omega}_{\ell}^k = 0$.

*Proof.* — Let $\omega_{\ell}^k \in C^k(\tilde{U}, \Omega^{\ell})$ be such that $\delta\omega_{\ell}^k = 0$. As we have seen before, $\omega_{\ell}^k$ admits an extension $\tilde{\omega}_{\ell}^k \in C^k(\tilde{U}, \Omega^{\ell})$. Then $\delta\tilde{\omega}_{\ell}^k \in C^{k+1}(\tilde{U}, \Omega^{\ell})$ and so $\delta\tilde{\omega}_{\ell}^k|_Z = \delta\omega_{\ell}^k = 0$.

Let us assume first that $\ell = 0$, so that $k + \ell = k \le r$. In this case, from Remark 6.6 we obtain $\delta\tilde{\omega}_0^k = f.g_0^{k+1}$, where $g_0^{k+1} \in C^{k+1}(\tilde{U}, \mathcal{O})$. Now, since $\delta^2 = 0$, we have $f.\delta g_0^{k+1} = 0$, and so $\delta g_0^{k+1} = 0$. Since $k + 1 \le r + 1$ the hypothesis implies that $H^{k+1}(\tilde{U}, \mathcal{O}) = 0$ and so there exists a cochain $h_0^k \in C^k(\tilde{U}, \mathcal{O})$ with $g_0^{k+1} = \delta h_0^k$. Therefore,

$$ \delta\tilde{\omega}_0^k = f.\delta h_0^k \implies \delta(\tilde{\omega}_0^k - f.h_0^k) = 0. $$

If we set $\tilde{\omega}_0^k = \tilde{\omega}_0^k - f.h_0^k$ then $\tilde{\omega}_0^k|_Z = \omega_0^k$ and $\delta\tilde{\omega}_0^k = 0$, which proves in the case $\ell = 0$.

Let us assume now that $\ell \ge 1$. In this case, Remark 6.6 implies that

$$ \delta\tilde{\omega}_{\ell}^{k} = \hat{\alpha}_{\ell-1}^{k+1} \wedge df + f.\hat{\beta}_{\ell}^{k+1}, \quad (6.2) $$

where $\hat{\alpha}_{\ell-1}^{k+1} \in C^{k+1}(\tilde{U}, \Omega^{\ell-1})$ and $\hat{\beta}_{\ell}^{k+1} \in C^{k+1}(\tilde{U}, \Omega^{\ell})$. We assert that we can choose $\hat{\alpha}_{\ell-1}^{k+1} \in C^{k+1}(\tilde{U}, \Omega^{\ell-1})$ and $\hat{\beta}_{\ell}^{k+1} \in C^{k+1}(\tilde{U}, \Omega^{\ell})$ such that (6.2) is true and $\delta\hat{\beta}_{\ell}^{k+1} = 0$. Let us prove this assertion.
(b) If $\ell \ge 1$ then there are cochains $\tilde{\alpha}_{\ell-1}^k = (\tilde{\alpha}_J)_{J \in \tilde{A}^{k+1}} \in C^k(\tilde{U}, \Omega^{\ell-1})$ and $\tilde{\beta}_{\ell}^k = (\tilde{\beta}_J)_{J \in \tilde{A}^{k+1}} \in C^k(\tilde{U}, \Omega^{\ell})$ such that $\tilde{\eta}_J - \tilde{\omega}_J = \tilde{\alpha}_J \wedge df + f.\tilde{\beta}_J$, for all $J \in \tilde{A}^{k+1}$. In this case, we will write $\tilde{\eta}_{\ell}^k - \tilde{\omega}_{\ell}^k = \tilde{\alpha}_{\ell-1}^k \wedge df + f.\tilde{\beta}_{\ell}^k$.

We leave the details to the reader.

*Extension of cochains.* — Claim 6.5 implies that given a cochain $\omega_{\ell}^k \in C^k(\tilde{U}, \Omega^{\ell})$ then there exists a cochain $\tilde{\omega}_{\ell}^k \in C^k(\tilde{U}, \Omega^{\ell})$ whoose restriction to $Z$ coincides with $\omega_{\ell}^k$. We leave the details to the reader. The cochain $\tilde{\omega}_{\ell}^k$ will be called an *extension* of the cochain $\omega_{\ell}^k$.

*Division of cochains.* — Given a cochain $\beta_{\ell}^k = (\beta_J)_{J \in \tilde{A}^{k+1}} \in C^k(\tilde{U}, \Omega^{\ell})$ we define the cochain $\beta_{\ell}^k \wedge df := (\beta_J \wedge df)_{J \in \tilde{A}^{k+1}} \in C^k(\tilde{U}, \Omega^{\ell+1})$. We would like to observe that, if $\ell \ge 1$ and $df \wedge \beta_{\ell}^k = 0$ then there exists a cochain $\beta_{\ell-1}^k \in C^k(\tilde{U}, \Omega^{\ell-1})$ such that $\beta_{\ell}^k = \beta_{\ell-1}^k \wedge df$. The proof is easy and is left to the reader.

Let us assume the hypothesis of Lemma 6.4: $H^k(Y, \Omega^\ell) = 0$ if $k \ge 1$ and $1 \le k + \ell \le r + 1$.

**CLAIM 6.7.** — In the above situation, if $k \ge 0$ and $\ell \ge 0$ are such that $k + \ell \le r$ then any cocycle $\omega_{\ell}^k \in C^k(\tilde{U}, \Omega^{\ell})$ such that $\delta\omega_{\ell}^k = 0$ admits an extension $\tilde{\omega}_{\ell}^k \in C^k(\tilde{U}, \Omega^{\ell})$ such that $\delta\tilde{\omega}_{\ell}^k = 0$.

*Proof.* — Let $\omega_{\ell}^k \in C^k(\tilde{U}, \Omega^{\ell})$ be such that $\delta\omega_{\ell}^k = 0$. As we have seen before, $\omega_{\ell}^k$ admits an extension $\tilde{\omega}_{\ell}^k \in C^k(\tilde{U}, \Omega^{\ell})$. Then $\delta\tilde{\omega}_{\ell}^k \in C^{k+1}(\tilde{U}, \Omega^{\ell})$ and so $\delta\tilde{\omega}_{\ell}^k|_Z = \delta\omega_{\ell}^k = 0$.

Let us assume first that $\ell = 0$, so that $k + \ell = k \le r$. In this case, from Remark 6.6 we obtain $\delta\tilde{\omega}_0^k = f.g_0^{k+1}$, where $g_0^{k+1} \in C^{k+1}(\tilde{U}, \mathcal{O})$. Now, since $\delta^2 = 0$, we have $f.\delta g_0^{k+1} = 0$, and so $\delta g_0^{k+1} = 0$. Since $k + 1 \le r + 1$ the hypothesis implies that $H^{k+1}(\tilde{U}, \mathcal{O}) = 0$ and so there exists a cochain $h_0^k \in C^k(\tilde{U}, \mathcal{O})$ with $g_0^{k+1} = \delta h_0^k$. Therefore,

$$ \delta\tilde{\omega}_0^k = f.\delta h_0^k \implies \delta(\tilde{\omega}_0^k - f.h_0^k) = 0. $$

If we set $\tilde{\omega}_0^k = \tilde{\omega}_0^k - f.h_0^k$ then $\tilde{\omega}_0^k|_Z = \omega_0^k$ and $\delta\tilde{\omega}_0^k = 0$, which proves in the case $\ell = 0$.

Let us assume now that $\ell \ge 1$. In this case, Remark 6.6 implies that

$$ \delta\tilde{\omega}_{\ell}^{k} = \hat{\alpha}_{\ell-1}^{k+1} \wedge df + f.\hat{\beta}_{\ell}^{k+1}, \quad (6.2) $$

where $\hat{\alpha}_{\ell-1}^{k+1} \in C^{k+1}(\tilde{U}, \Omega^{\ell-1})$ and $\hat{\beta}_{\ell}^{k+1} \in C^{k+1}(\tilde{U}, \Omega^{\ell})$. We assert that we can choose $\hat{\alpha}_{\ell-1}^{k+1} \in C^{k+1}(\tilde{U}, \Omega^{\ell-1})$ and $\hat{\beta}_{\ell}^{k+1} \in C^{k+1}(\tilde{U}, \Omega^{\ell})$ such that (6.2) is true and $\delta\hat{\beta}_{\ell}^{k+1} = 0$. Let us prove this assertion.
First we construct by induction a sequence of cochains

$$ \beta_{\ell-j}^{k+j+1} \in C^{k+j+1}(\tilde{U}, \Omega^{\ell-j}), j = 0, \dots, \ell $$

such that $\beta_{\ell}^{k+1} = \tilde{\beta}_{\ell}^{k+1}$ and:

(i) $\delta\beta_{\ell-j}^{k+j+1} \wedge df = 0, \forall j = 0, \dots, \ell$.

(ii) $\delta\beta_{\ell-j}^{k+j+1} = \beta_{\ell-j-1}^{k+j+2} \wedge df, \forall j = 0, \dots, \ell-1$.

The construction is based in the division property. Since $\delta^2 = 0$, relation (6.2) implies that

$$ \begin{align*} \delta\hat{\alpha}_{\ell-1}^{k+1} \wedge df + f \cdot \delta\hat{\beta}_{\ell}^{k+1} &= 0 &\implies \delta\hat{\beta}_{\ell}^{k+1} \wedge df &= 0 &\implies \delta\hat{\beta}_{\ell}^{k+1} &= \beta_{\ell-1}^{k+2} \wedge df \\ \delta\beta_{\ell-1}^{k+2} \wedge df &= 0 &\implies \delta\beta_{\ell-1}^{k+2} &= \beta_{\ell-2}^{k+3} \wedge df &\implies \dots &\implies \delta\beta_{\ell-j}^{k+j+1} \wedge df &= 0 \\ \delta\beta_{\ell-j}^{k+j+1} &= \beta_{\ell-j-1}^{k+j+2} \wedge df &\implies \dots &\implies \delta\beta_{1}^{k+\ell} \wedge df &= 0 &\implies \delta\beta_{1}^{k+\ell} &= \beta_{0}^{k+\ell+1} \wedge df. \end{align*} $$

Next, we will see that the sequence can be constructed in such a way that $\delta\beta_j^{k+\ell-j+1} = 0, \forall j = 0, \dots, \ell$. This involves another induction argument.

1st step: $j = 0$. — From $\delta\beta_1^{k+\ell} = \beta_0^{k+\ell+1} \cdot df$ we get $\delta\beta_0^{k+\ell+1} = 0$. Hence $\beta_0^{k+\ell+1} \in \ker(\delta)$.

2nd step. — Assume that we have constructed the sequence satisfying (i), (ii) with $\delta\beta_i^{k+\ell-i+1} = 0$ for $i = 0, \dots, j-1 \le \ell-1$ and let us prove that can assume that $\delta\beta_j^{k+\ell-j+1} = 0$.

From (ii) we have $\delta\beta_j^{k+\ell-j+1} = \beta_{j-1}^{k+\ell-j+2} \wedge df$, where $\delta\beta_{j-1}^{k+\ell-j+2} = 0$ by the induction hypothesis. Since $(k+\ell-j+2)+(j-1) = k+\ell+1 \le r+1$ we have $H^{k+\ell-j+2}(\tilde{U}, \Omega^{j-1}) = 0$ and so there exists a cochain $\gamma_{j-1}^{k+\ell-j+1} \in C^{k+\ell-j+1}(\tilde{U}, \Omega^{j-1})$ such that $\beta_{j-1}^{k+\ell-j+2} = \delta\gamma_{j-1}^{k+\ell-j+1}$. Therefore, if we set $\tilde{\beta}_j^{k+\ell-j+1} = \beta_j^{k+\ell-j+1} - \gamma_{j-1}^{k+\ell-j+1} \wedge df$ then

$$ \delta\tilde{\beta}_j^{k+\ell-j+1} = \delta(\beta_j^{k+\ell-j+1} - \gamma_{j-1}^{k+\ell-j+1} \wedge df) = 0. $$

Moreover,

$$ \tilde{\beta}_j^{k+\ell-j+1} \wedge df = \beta_j^{k+\ell-j+1} \wedge df = \beta_{j+1}^{k+\ell-j}. $$

Hence, if we replace $\beta_j^{k+\ell-j+1}$ by $\tilde{\beta}_j^{k+\ell-j+1}$ in the sequence, then the new sequence still satisfies (i) and (ii).

The induction process implies that there exists a cochain $\gamma_{\ell-1}^{k+1} \in C^{k+1}(\tilde{U}, \Omega^{\ell-1})$ such that $\delta(\hat{\beta}_{\ell}^{k+1} - \gamma_{\ell-1}^{k+1} \wedge df) = 0$. Hence, if we set $\tilde{\beta}_{\ell}^{k+1} = \hat{\beta}_{\ell}^{k+1} - \gamma_{\ell-1}^{k+1} \wedge df$ and $\tilde{\alpha}_{\ell-1}^{k+1} = \tilde{\alpha}_{\ell-1}^{k+1} + f.h_{\ell-1}^{k+1}$ then (6.2) can be written as

$$ \delta\tilde{\omega}_{\ell}^{k} = \tilde{\alpha}_{\ell-1}^{k+1} \wedge df + f \cdot \tilde{\beta}_{\ell}^{k+1}, \quad \text{where } \delta\tilde{\beta}_{\ell}^{k+1} = 0. $$
First we construct by induction a sequence of cochains

$$ \beta_{\ell-j}^{k+j+1} \in C^{k+j+1}(\tilde{U}, \Omega^{\ell-j}), j = 0, \dots, \ell $$

such that $\beta_{\ell}^{k+1} = \tilde{\beta}_{\ell}^{k+1}$ and:

(i) $\delta\beta_{\ell-j}^{k+j+1} \wedge df = 0, \forall j = 0, \dots, \ell$.

(ii) $\delta\beta_{\ell-j}^{k+j+1} = \beta_{\ell-j-1}^{k+j+2} \wedge df, \forall j = 0, \dots, \ell-1$.

The construction is based in the division property. Since $\delta^2 = 0$, relation (6.2) implies that

$$ \begin{align*} \delta\hat{\alpha}_{\ell-1}^{k+1} \wedge df + f.\delta\hat{\beta}_{\ell}^{k+1} &= 0 &\implies \delta\hat{\beta}_{\ell}^{k+1} \wedge df &= 0 &\implies \delta\hat{\beta}_{\ell}^{k+1} &= \beta_{\ell-1}^{k+2} \wedge df \\ \delta\beta_{\ell-1}^{k+2} \wedge df &= 0 &\implies \delta\beta_{\ell-1}^{k+2} &= \beta_{\ell-2}^{k+3} \wedge df &\implies \dots &\implies \delta\beta_{\ell-j}^{k+j+1} \wedge df &= 0 \\ \delta\beta_{\ell-j}^{k+j+1} &= \beta_{\ell-j-1}^{k+j+2} \wedge df &\implies \dots &\implies \delta\beta_{1}^{k+\ell} \wedge df &= 0 &\implies \delta\beta_{1}^{k+\ell} &= \beta_{0}^{k+\ell+1} \wedge df. \end{align*} $$

Next, we will see that the sequence can be constructed in such a way that $\delta\beta_j^{k+\ell-j+1} = 0, \forall j = 0, \dots, \ell$. This involves another induction argument.

1st step: $j = 0$. — From $\delta\beta_1^{k+\ell} = \beta_0^{k+\ell+1} \wedge df$ we get $\delta\beta_0^{k+\ell+1} = 0$. Hence $\beta_0^{k+\ell+1} \in \ker(\delta)$.

2nd step. — Assume that we have constructed the sequence satisfying (i), (ii) with $\delta\beta_i^{k+\ell-i+1} = 0$ for $i = 0, \dots, j-1 \le \ell-1$ and let us prove that can assume that $\delta\beta_j^{k+\ell-j+1} = 0$.

From (ii) we have $\delta\beta_j^{k+\ell-j+1} = \beta_{j-1}^{k+\ell-j+2} \wedge df$, where $\delta\beta_{j-1}^{k+\ell-j+2} = 0$ by the induction hypothesis. Since $(k+\ell-j+2)+(j-1) = k+\ell+1 \le r+1$ we have $H^{k+\ell-j+2}(\tilde{U}, \Omega^{j-1}) = 0$ and so there exists a cochain $\gamma_{j-1}^{k+\ell-j+1} \in C^{k+\ell-j+1}(\tilde{U}, \Omega^{j-1})$ such that $\beta_{j-1}^{k+\ell-j+2} = \delta\gamma_{j-1}^{k+\ell-j+1}$. Therefore, if we set $\tilde{\beta}_j^{k+\ell-j+1} = \beta_j^{k+\ell-j+1} - \gamma_{j-1}^{k+\ell-j+1} \wedge df$ then

$$ \delta\tilde{\beta}_j^{k+\ell-j+1} = \delta(\beta_j^{k+\ell-j+1} - \gamma_{j-1}^{k+\ell-j+1} \wedge df) = 0. $$

Moreover,

$$ \tilde{\beta}_j^{k+\ell-j+1} \wedge df = \beta_j^{k+\ell-j+1} \wedge df = \beta_{j+1}^{k+\ell-j}. $$

Hence, if we replace $\beta_j^{k+\ell-j+1}$ by $\tilde{\beta}_j^{k+\ell-j+1}$ in the sequence, then the new sequence still satisfies (i) and (ii).

The induction process implies that there exists a cochain $\gamma_{\ell-1}^{k+1} \in C^{k+1}(\tilde{U}, \Omega^{\ell-1})$ such that $\delta(\hat{\beta}_{\ell}^{k+1} - \gamma_{\ell-1}^{k+1} \wedge df) = 0$. Hence, if we set $\tilde{\beta}_{\ell}^{k+1} = \hat{\beta}_{\ell}^{k+1} - \gamma_{\ell-1}^{k+1} \wedge df$ and $\tilde{\alpha}_{\ell-1}^{k+1} = \tilde{\alpha}_{\ell-1}^{k+1} + f.h_{\ell-1}^{k+1}$ then (6.2) can be written as

$$ \delta\tilde{\omega}_{\ell}^{k} = \tilde{\alpha}_{\ell-1}^{k+1} \wedge df + f.\tilde{\beta}_{\ell}^{k+1}, \quad \text{where } \delta\tilde{\beta}_{\ell}^{k+1} = 0. $$
Since $H^{k+1}(\tilde{U}, \Omega^\ell) = 0$, there exists a cochain $\gamma_\ell^k \in C^k(\tilde{U}, \Omega^\ell)$ such that $\tilde{\beta}_\ell^{k+1} = \delta\gamma_\ell^k$. In particular, if we set $\bar{\omega}_\ell^k = \hat{\omega}_\ell^k - f.\gamma_\ell^k$ then $\bar{\omega}_\ell^k|_Z = \hat{\omega}_\ell^k|_Z = \omega_\ell^k$ and

$$ \delta \bar{\omega}_{\ell}^{k} = \tilde{\alpha}_{\ell-1}^{k+1} \wedge df. \qquad (6.3) $$

If $\ell = 1$ then $\tilde{\alpha}_0^{k+1} \in H^{k+1}(\tilde{U}, \mathcal{O})$ and (6.3) implies that $\delta\alpha_{\ell-1}^{k+1} = 0$ and there exists a cochain $g_0^k \in C^k(\tilde{U}, \mathcal{O})$ such that $\tilde{\alpha}_0^{k+1} = \delta g_0^k$. In particular, the cochain $\tilde{\omega}_1^k = \bar{\omega}_1^k - g_0^k$. df satisfies $\delta\tilde{\omega}_1^k = 0$ and $\tilde{\omega}_1^k|_Z = \omega_1^k$, proving Claim 6.7 in this case.

Finally, when $\ell \ge 2$ using (6.3) and an induction argument similar to that used in the case of $\tilde{\beta}_\ell^{k+1}$ it is possible to obtain a cochain $\gamma_{\ell-2}^{k+1} \in C^{k+1}(\tilde{U}, \Omega^{\ell-2})$ such that $\delta(\tilde{\alpha}_{\ell-1}^{k+1} - \gamma_{\ell-2}^{k+1} \wedge df) = 0$. Since $(\ell-1)+k+1 = \ell+k \le r+1$ we have $H^{k+1}(\tilde{U}, \Omega^{\ell-1}) = 0$, so that $\tilde{\alpha}_{\ell-1}^{k+1} - \gamma_{\ell-2}^{k+1} \wedge df = \delta\eta_{\ell-1}^k$, where $\eta_{\ell-1}^k \in C^k(\tilde{U}, \Omega^{\ell-1})$. From (6.3) we get

$$ \delta \bar{\omega}_{\ell}^{k+1} = \tilde{\alpha}_{\ell-1}^{k+1} \wedge df = \delta \eta_{\ell-1}^{k} \wedge df \implies \delta (\bar{\omega}_{\ell}^{k} - \eta_{\ell-1}^{k} \wedge df) = 0. $$

Hence, if we set $\tilde{\omega}_\ell^k = \bar{\omega}_\ell^k - \eta_{\ell-1}^k \wedge df$ then $\delta\tilde{\omega}_\ell^k = 0$ and $\tilde{\omega}_\ell^k|_Z = \omega_\ell^k$, which proves Claim 6.7.

Let us finish the proof of Lemma 6.4.

*Proof of (a).* — By Leray's theorem it is sufficient to prove that $H^k(\tilde{U}, \Omega^\ell) = 0$, if $k \ge 1$ and $k + \ell \le r$. If $\omega_\ell^k \in C^k(\tilde{U}, \Omega^\ell)$ is such that $\delta\omega_\ell^k = 0$ then by Claim 6.7, $\omega_\ell^k$ admits an extension $\tilde{\omega}_\ell^k$ such that $\delta\tilde{\omega}_\ell^k = 0$. Since $k + \ell \le r < r+1$ then $H^k(\tilde{U}, \Omega^\ell) = 0$, so that $\tilde{\omega}_\ell^k = \delta\tilde{\eta}_\ell^{k-1}$ for some cochain $\tilde{\eta}_\ell^{k-1} \in C^{k-1}(\tilde{U}, \Omega^\ell)$. As the reader can check, this implies that $\omega_\ell^k = \delta(\tilde{\eta}_\ell^{k-1}|_Z)$, which proves the assertion.

*Proof of (b).* — Let $\omega_\ell \in \Omega^\ell(Z)$, where $\ell \le r$. We can associate to $\omega_\ell$ a 0-cochain $\omega_\ell^0 = (\omega_\ell|_{U_\alpha})_{\alpha \in A}$ with $\delta\omega_\ell^0 = 0$. By Claim 6.7, $\omega_\ell^0$ admits an extension $\tilde{\omega}_\ell^0 \in C^0(\tilde{U}, \Omega^\ell)$ such that $\delta\tilde{\omega}_\ell^0 = 0$. This is equivalent to say that there exists a section $\tilde{\omega}_\ell \in \Omega^\ell(Y)$ such that $\tilde{\omega}_\ell^0 = (\tilde{\omega}_\ell|_{\tilde{U}_\alpha})_{\alpha \in \tilde{A}}$. Hence, $\tilde{\omega}_\ell$ extends $\omega_\ell$ proving Lemma 6.4. $\square$

We are now in position to prove the statement of Theorem 2.10. Let $0 \in Q \subset \mathbb{C}^n$, $Q$ a polydisc, and $X = (f_1 = \cdots = f_{n-k} = 0)$ be as in the statement of Lemma 2.3. Define a sequence of analytic complete intersections $X_0 \supset X_1 \supset \cdots \supset X_{n-k}$, where $X_0 = Q$ and $X_q = (f_1 = \cdots = f_q = 0)$ if $1 \le q \le n-k$, and set $X_q^* := X_q \setminus \{0\}$, $0 \le q \le n-k$. The hypothesis implies the following:

$$ (i) \dim_{\mathbb{C}}(X_q) = k(q) := n - q \text{ and } X_q^* \text{ is smooth, } \forall 0 \le q \le k. $$
Since $H^{k+1}(\tilde{U}, \Omega^\ell) = 0$, there exists a cochain $\gamma_\ell^k \in C^k(\tilde{U}, \Omega^\ell)$ such that
$\tilde{\beta}_\ell^{k+1} = \delta\gamma_\ell^k$. In particular, if we set $\bar{\omega}_\ell^k = \hat{\omega}_\ell^k - f.\gamma_\ell^k$ then $\bar{\omega}_\ell^k|_Z = \hat{\omega}_\ell^k|_Z = \omega_\ell^k$
and

$$
\delta \bar{\omega}_{\ell}^{k} = \tilde{\alpha}_{\ell-1}^{k+1} \wedge df. \qquad (6.3)
$$

If $\ell = 1$ then $\tilde{\alpha}_0^{k+1} \in H^{k+1}(\tilde{U}, \mathcal{O})$ and (6.3) implies that $\delta\alpha_{\ell-1}^{k+1} = 0$ and there exists a cochain $g_0^k \in C^k(\tilde{U}, \mathcal{O})$ such that $\tilde{\alpha}_0^{k+1} = \delta g_0^k$. In particular, the cochain $\tilde{\omega}_1^k = \bar{\omega}_1^k - g_0^k$. df satisfies $\delta\tilde{\omega}_1^k = 0$ and $\tilde{\omega}_1^k|_Z = \omega_1^k$, proving Claim 6.7 in this case.

Finally, when $\ell \ge 2$ using (6.3) and an induction argument similar to
that used in the case of $\hat{\beta}_{\ell}^{k+1}$ it is possible to obtain a cochain $\gamma_{\ell-2}^{k+1} \in$
$C^{k+1}(\tilde{U}, \Omega^{\ell-2})$ such that $\delta(\tilde{\alpha}_{\ell-1}^{k+1} - \gamma_{\ell-2}^{k+1} \wedge df) = 0$. Since $(\ell-1)+k+1=$
$\ell+k \le r+1$ we have $H^{k+1}(\tilde{U}, \Omega^{\ell-1}) = 0$, so that $\tilde{\alpha}_{\ell-1}^{k+1} - \gamma_{\ell-2}^{k+1} \wedge df = \delta\eta_{\ell-1}^k$,
where $\eta_{\ell-1}^k \in C^k(\tilde{U}, \Omega^{\ell-1})$. From (6.3) we get

$$
\delta \bar{\omega}_{\ell}^{k+1} = \tilde{\alpha}_{\ell-1}^{k+1} \wedge df = \delta \eta_{\ell-1}^{k} \wedge df \implies \delta (\bar{\omega}_{\ell}^{k} - \eta_{\ell-1}^{k} \wedge df) = 0.
$$

Hence, if we set $\tilde{\omega}_{\ell}^{k} = \bar{\omega}_{\ell}^{k} - \eta_{\ell-1}^{k} \wedge df$ then $\delta\tilde{\omega}_{\ell}^{\ell} = 0$ and $\tilde{\omega}_{\ell}^{k}|_{Z} = \omega_{\ell}^{k}$, which
proves Claim 6.7.

Let us finish the proof of Lemma 6.4.

Proof of (a). — By Leray’s theorem it is sufficient to prove that $H^k(\mathcal{U}, \Omega^\ell) = 0$, if $k \ge 1$ and $k + \ell \le r$. If $\omega_\ell^k \in C^k(\mathcal{U}, \Omega^\ell)$ is such that $\delta\omega_\ell^k = 0$ then by Claim 6.7, $\omega_\ell^k$ admits an extension $\tilde{\omega}_\ell^k$ such that $\delta\tilde{\omega}_\ell^k = 0$. Since $k + \ell \le r < r + 1$ then $H^k(\tilde{U}, \Omega^\ell) = 0$, so that $\tilde{\omega}_\ell^k = \delta\tilde{\eta}_\ell^{k-1}$ for some cochain $\tilde{\eta}_\ell^{k-1} \in C^{k-1}(\tilde{U}, \Omega^\ell)$. As the reader can check, this implies that $\omega_\ell^k = \delta(\tilde{\eta}_\ell^{k-1}|_Z)$, which proves the assertion.

Proof of (b). — Let $\omega_\ell \in \Omega^\ell(Z)$, where $\ell \le r$. We can associate to $\omega_\ell$
a 0-cochain $\omega_\ell^0 = (\omega_\ell|_{U_\alpha})_{\alpha\in A}$ with $\delta\omega_\ell^0 = 0$. By Claim 6.7, $\omega_\ell^0$ admits an
extension $\tilde{\omega}_\ell^0 \in C^0(\tilde{U}, \Omega^\ell)$ such that $\delta\tilde{\omega}_\ell^0 = 0$. This is equivalent to say that
there exists a section $\tilde{\omega}_\ell \in \Omega^\ell(Y)$ such that $\tilde{\omega}_\ell^0 = (\tilde{\omega}_\ell|_{\tilde{U}_\alpha})_{\alpha\in \tilde{A}}$. Hence, $\tilde{\omega}_\ell$
extends $\omega_\ell$ proving Lemma 6.4. $\square$

We are now in position to prove the statement of Theorem 2.10. Let $0 \in Q \subset \mathbb{C}^n$, $Q$ a polydisc, and $X = (f_1 = \cdots = f_{n-k} = 0)$ be as in the statement of Lemma 2.3. Define a sequence of analytic complete intersections $X_0 \supset X_1 \supset \cdots \supset X_{n-k}$, where $X_0 = Q$ and $X_q = (f_1 = \cdots = f_q = 0)$ if $1 \le q \le n-k$, and set $X_q^* := X_q \setminus \{0\}$, $0 \le q \le n-k$. The hypothesis implies the following:

$$
(i) \dim_{\mathbb{C}}(X_q) = k(q) := n - q \text{ and } X_q^* \text{ is smooth, } \forall 0 \le q \le k.
$$
(ii) $X_q = f_q^{-1}(0) \cap X_{q-1}, \forall 1 \le q \le n-k$. Moreover, 0 is a regular value of $f_q|_{X_{q-1}^*}$.

Recall that $k \ge 2$, so that $k(q) \ge 3$ if $q \le n-3$.

CLAIM 6.8. — Let $p \ge 1$, $\ell \ge 0$ and $0 \le q \le n-k-1$ be such that $1 \le p+\ell \le k(q)-2$. Then $H^p(X_q^*, \Omega^\ell) = 0$.

*Proof*. — The proof is by induction on $q = 0, \dots, n-3$. The case $q=0$ is consequence of a generalization of Cartan's theorem (cf. [5]): since $X_0 = Q$ is Stein then ([12, p. 133]):

$$H^p(X_0^*, \Omega^\ell) = 0, \forall p = 1, \dots, n-2, \forall \ell \ge 0.$$

In particular, $H^p(X_0^*, \Omega^\ell) = 0$ if $p \ge 1$, $\ell \ge 0$ and $1 \le p+\ell \le n-2 = k(0)-2$. The induction step is consequence of Lemma 6.4(a): let us assume that Claim 6.8 is true for $q$, where $1 \le q \le n-k-2$. Set $Y = X_q^*$, $Z = X_{q+1}^*$ and $f = f_{q+1}|_{X_{q+1}^*}$ in Lemma 6.4. The induction hypothesis implies that, if $p \ge 1$ and $\ell \ge 0$ are such that $1 \le p+\ell \le k(q)-2$ then $H^p(X_q^*, \Omega^\ell) = 0$. In particular, Lemma 6.4(a) implies that $H^p(X_{q+1}^*, \Omega^\ell) = 0, \forall p \ge 1, \ell \ge 0$ such that $1 \le p+\ell \le k(q)-3 = k(q+1)-2$.

The extension property is consequence of Lemma 6.4(b). The idea is to use Claim 6.8 and Lemma 6.4(b) inductively. Let $\omega_\ell \in \Omega^\ell(X \setminus \{0\})$, where $\ell \le k-2$. In the first step we set $Z = X_{n-k}^* = X \setminus \{0\}$, $Y = X_{n-k-1}^*$ and $f = f_{n-k}|_{X_{n-k-1}^*}$. From Claim 6.8 we have $H^p(X_{n-k-1}^*, \Omega^\ell) = 0$ if $p \ge 1$ and $\ell \ge 0$ are such that $1 \le p+\ell \le k(n-k-1)-2 = k-1$. Hence, Lemma 6.4(b) implies that if $0 \le \ell \le k-2$ then any form $\omega_\ell \in \Omega(X_{n-k}^*)$ has an extension $\omega_\ell^1 \in \Omega^\ell(X_{n-k-1}^*)$. The induction step is similar: assume that $\omega_\ell$ has an extension $\omega_\ell^j \in \Omega^\ell(X_{n-k-j}^*)$, where $1 \le j \le n-k-1$. Since $\ell \le k-2 < k-2+j = k(n-k-j)-2$, Lemma 6.4(b) implies that $\omega_\ell^j$ has an extension $\omega_\ell^{j+1} \in \Omega^\ell(X_{n-k-j-1}^*)$. Finally, $\omega_\ell$ has an extension $\omega_\ell^{n-k} \in \Omega^\ell(X_0^*)$, which by Hartog's theorem has an extension $\tilde{\omega}_\ell \in \Omega^\ell(Q)$. $\square$

Bibliography

[1] V. I. ARNOL'D, *Chapitres supplémentaires de la théorie des équations différentielles ordinaires*, Mir, 1980.

[2] W. BARTH, “Fortsetzung meromorfer Funktionen in Tori und komplex-projektiven Räumen”, *Invent. Math.* **5** (1968), no. 1, p. 42-62.

[3] O. CALVO-ANDRADE, “Irreducible components of the space of holomorphic foliations”, *Math. Ann.* **299** (1994), no. 1, p. 751-767.

[4] C. CAMACHO, A. LINS NETO & P. SAD, “Foliations with algebraic limit sets”, *Ann. Math.* **136** (1992), no. 2, p. 429-446.
(ii) $X_q = f_q^{-1}(0) \cap X_{q-1}, \forall 1 \le q \le n-k$. Moreover, 0 is a regular value of $f_q|_{X_{q-1}^*}$.

Recall that $k \ge 2$, so that $k(q) \ge 3$ if $q \le n-3$.

**CLAIM 6.8.** — Let $p \ge 1$, $\ell \ge 0$ and $0 \le q \le n-k-1$ be such that $1 \le p+\ell \le k(q)-2$. Then $H^p(X_q^*, \Omega^\ell) = 0$.

*Proof.* — The proof is by induction on $q=0, \dots, n-3$. The case $q=0$ is consequence of a generalization of Cartan's theorem (cf. [5]): since $X_0 = Q$ is Stein then ([12, p. 133]):

$$H^p(X_0^*, \Omega^\ell) = 0, \forall p = 1, \dots, n-2, \forall \ell \ge 0.$$

In particular, $H^p(X_0^*, \Omega^\ell) = 0$ if $p \ge 1$, $\ell \ge 0$ and $1 \le p+\ell \le n-2 = k(0)-2$. The induction step is consequence of Lemma 6.4(a): let us assume that Claim 6.8 is true for $q$, where $1 \le q \le n-k-2$. Set $Y = X_q^*$, $Z = X_{q+1}^*$ and $f = f_{q+1}|_{X_{q+1}^*}$ in Lemma 6.4. The induction hypothesis implies that, if $p \ge 1$ and $\ell \ge 0$ are such that $1 \le p+\ell \le k(q)-2$ then $H^p(X_q^*, \Omega^\ell) = 0$. In particular, Lemma 6.4(a) implies that $H^p(X_{q+1}^*, \Omega^\ell) = 0, \forall p \ge 1, \ell \ge 0$ such that $1 \le p+\ell \le k(q)-3 = k(q+1)-2$.

The extension property is consequence of Lemma 6.4(b). The idea is to use Claim 6.8 and Lemma 6.4(b) inductively. Let $\omega_\ell \in \Omega^\ell(X \setminus \{0\})$, where $\ell \le k-2$. In the first step we set $Z = X_{n-k}^* = X \setminus \{0\}$, $Y = X_{n-k-1}^*$ and $f = f_{n-k}|_{X_{n-k-1}^*}$. From Claim 6.8 we have $H^p(X_{n-k-1}^*, \Omega^\ell) = 0$ if $p \ge 1$ and $\ell \ge 0$ are such that $1 \le p+\ell \le k(n-k-1)-2 = k-1$. Hence, Lemma 6.4(b) implies that if $0 \le \ell \le k-2$ then any form $\omega_\ell \in \Omega(X_{n-k}^*)$ has an extension $\omega_\ell^1 \in \Omega^\ell(X_{n-k-1}^*)$. The induction step is similar: assume that $\omega_\ell$ has an extension $\omega_\ell^j \in \Omega^\ell(X_{n-k-j}^*)$, where $1 \le j \le n-k-1$. Since $\ell \le k-2 < k-2+j = k(n-k-j)-2$, Lemma 6.4(b) implies that $\omega_\ell^j$ has an extension $\omega_\ell^{j+1} \in \Omega^\ell(X_{n-k-j-1}^*)$. Finally, $\omega_\ell$ has an extension $\omega_\ell^{n-k} \in \Omega^\ell(X_0^*)$, which by Hartog's theorem has an extension $\tilde{\omega}_\ell \in \Omega^\ell(Q)$. $\square$

Bibliography

[1] V. I. ARNOL'D, *Chapitres supplémentaires de la théorie des équations différentielles ordinaires*, Mir, 1980.

[2] W. BARTH, “Fortsetzung meromorpher Funktionen in Tori und komplex-projektiven Räumen”, *Invent. Math.* **5** (1968), no. 1, p. 42-62.

[3] O. CALVO-ANDRADE, “Irreducible components of the space of holomorphic foliations”, *Math. Ann.* **299** (1994), no. 1, p. 751-767.

[4] C. CAMACHO, A. LINS NETO & P. SAD, “Foliations with algebraic limit sets”, *Ann. Math.* **136** (1992), no. 2, p. 429-446.
[5] H. CARTAN, "Sur le premier problème de Cousin", C. R. Math. Acad. Sci. Paris **207** (1938), p. 558-560.

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[7] D. CERVEAU & J.-F. MATTEI, *Formes intégrables holomorphes singulières*, Astérisque, vol. 97, Société Mathématique de France, 1982.

[8] F. CUKIERMAN, J. V. PEREIRA & I. VAINSENCHER, "Stability of foliations induced by rational maps", Ann. Fac. Sci. Toulouse, Math. **18** (2009), no. 4, p. 685-715.

[9] A. S. DE MEDEIROS, "Singular foliations and differential p-forms", Ann. Fac. Sci. Toulouse, Math. **9** (2000), no. 3, p. 451-466.

[10] P. DELIGNE, "Théorie de Hodge II", Publ. Math., Inst. Hautes Étud. Sci. **40** (1971), no. 1, p. 5-57.

[11] J. GARGIULO ACEA, "Logarithmic forms and singular projective foliations", Ann. Inst. Fourier **70** (2020), no. 1, p. 171-203.

[12] H. GRAUERT & R. REMMERT, *Theory of Stein spaces*, Springer, 1979.

[13] H. A. HAMM, "Lokale topologische Eigenschaften komplexer Räume", Math. Ann. **191** (1971), no. 3, p. 235-252.

[14] —, "Lefschetz theorems for singular varieties", in *Singularities, Part 1 (Ar-cata, Calif., 1981)*, Proceedings of Symposia in Pure Mathematics, vol. 40, American Mathematical Society, 1983, p. 547-557.

[15] M. W. HIRSCH, C. C. PUGH & M. SHUB, *Invariant manifolds*, Lecture Notes in Mathematics, vol. 583, Springer, 1977.

[16] A. LINS NETO, "Componentes Irredutíveis dos Espacos de Folheacões", 2007, 26º Colóquio Brasileiro de Matemática, Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro. Available at: http://www.impa.br/opencms/pt/biblioteca/cbm/26CBM/26CBM02.pdf.

[17] —, "Germs of complex two dimensional foliations", Bull. Braz. Math. Soc. (N.S.) **46** (2015), no. 4, p. 645-680.

[18] A. LINS NETO & W. COSTA E SILVA, "Pull-back components of the space of foliations of codimension ≥ 2", Trans. Am. Math. Soc. **371** (2019), no. 2, p. 949-969.

[19] J. MARTINET, "Normalisation des champs de vecteurs holomorphes", in *Séminaire Bourbaki vol. 1980/81*, Lecture Notes in Mathematics, vol. 901, Springer, 1981, p. 55-70.

[20] J. MILNOR, *Singular points of complex hypersurfaces*, no. 61, Princeton University Press, 1968.

[21] G. DE RHAM, "Sur la division de formes et de courants par une forme linéaire", Comment. Math. Helv. **28** (1954), no. 1, p. 346-352.

[22] H. ROSSI, "Continuation of subvarieties of projective varieties", Am. J. Math. **91** (1969), no. 2, p. 565-575.

[23] J. SEADE, *On the topology of isolated singularities in analytic spaces*, Progress in Mathematics, vol. 241, Birkhäuser, 2006.

[24] Y.-T. SIU, *Techniques of extension of analytic objects*, Lecture Notes in Pure and Applied Mathematics, vol. 8, Marcel Dekker, 1974.
[5] H. CARTAN, "Sur le premier problème de Cousin", C. R. Math. Acad. Sci. Paris 207 (1938), p. 558-560.

[6] D. CERVEAU, "Distributions involutives singulières", Ann. Inst. Fourier 29 (1979), no. 3, p. 261-294.

[7] D. CERVEAU & J.-F. MATTEI, *Formes intégrables holomorphes singulières*, Astérisque, vol. 97, Société Mathématique de France, 1982.

[8] F. CUKIERMAN, J. V. PEREIRA & I. VAINSENCHER, "Stability of foliations induced by rational maps", Ann. Fac. Sci. Toulouse, Math. 18 (2009), no. 4, p. 685-715.

[9] A. S. DE MEDEIROS, "Singular foliations and differential $p$-forms", Ann. Fac. Sci. Toulouse, Math. 9 (2000), no. 3, p. 451-466.

[10] P. DELIGNE, "Théorie de Hodge II", Publ. Math., Inst. Hautes Étud. Sci. 40 (1971), no. 1, p. 5-57.

[11] J. GARGIULO ACEA, "Logarithmic forms and singular projective foliations", Ann. Inst. Fourier 70 (2020), no. 1, p. 171-203.

[12] H. GRAUERT & R. REMMERT, *Theory of Stein spaces*, Springer, 1979.

[13] H. A. HAMM, "Lokale topologische Eigenschaften komplexer Räume", Math. Ann. 191 (1971), no. 3, p. 235-252.

[14] ——, "Lefschetz theorems for singular varieties", in *Singularities, Part 1 (Ar-cata, Calif., 1981)*, Proceedings of Symposia in Pure Mathematics, vol. 40, American Mathematical Society, 1983, p. 547-557.

[15] M. W. HIRSCH, C. C. PUGH & M. SHUB, *Invariant manifolds*, Lecture Notes in Mathematics, vol. 583, Springer, 1977.

[16] A. LINS NETO, "Componentes Irredutíveis dos Espacos de Folheacões", 2007, 26º Colóquio Brasileiro de Matemática, Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro. Available at: http://www.impa.br/opencms/pt/biblioteca/cbm/26CBM/26CBM02.pdf.

[17] ——, "Germs of complex two dimensional foliations", Bull. Braz. Math. Soc. (N.S.) 46 (2015), no. 4, p. 645-680.

[18] A. LINS NETO & W. COSTA E SILVA, "Pull-back components of the space of foliations of codimension $\ge 2$", Trans. Am. Math. Soc. 371 (2019), no. 2, p. 949-969.

[19] J. MARTINET, "Normalisation des champs de vecteurs holomorphes", in *Séminaire Bourbaki vol. 1980/81*, Lecture Notes in Mathematics, vol. 901, Springer, 1981, p. 55-70.

[20] J. MILNOR, *Singular points of complex hypersurfaces*, no. 61, Princeton University Press, 1968.

[21] G. DE RHAM, "Sur la division de formes et de courants par une forme linéaire", Comment. Math. Helv. 28 (1954), no. 1, p. 346-352.

[22] H. ROSSI, "Continuation of subvarieties of projective varieties", Am. J. Math. 91 (1969), no. 2, p. 565-575.

[23] J. SEADE, *On the topology of isolated singularities in analytic spaces*, Progress in Mathematics, vol. 241, Birkhäuser, 2006.

[24] Y.-T. SIU, *Techniques of extension of analytic objects*, Lecture Notes in Pure and Applied Mathematics, vol. 8, Marcel Dekker, 1974.