Proof. Let \(f: \mathcal{M}\hookrightarrow \mathbb{C}^n\) be any embedding. Let \(\Phi_f: \mathcal{M}\rightarrow\operatorname{Hom}_{\mathbb{C}\!}(P,Q)\) be the associated Gauss map as defined in 1 . We will use the parametric transversality theorem; see [17]. Let \(Y\) denote the set of all affine transformations \({\mathbf{A}}(z)=A(z)+a\), where \(A\in \operatorname{Gl}(2n,{\mathbb{R}})\), \(a\in{\mathbb{R}}^{2n}\) and \(z\) is viewed as an element in \({\mathbb{R}}^{2n}\). Then \[({\mathbf{A}},p)\mapsto \Phi_{{\mathbf{A}}\circ f}(p)=d({\mathbf{A}}\circ f)_\mathbb{C}(p)=(A\circ df)_\mathbb{C}(p)\] defines a smooth map \(G:Y\times \mathcal{M}\rightarrow \operatorname{Hom}_{\mathbb{C}\,} (P,Q)\). We claim that \(G\) is transverse to \(\operatorname{Hom}_{\mathbb{C}}^\nu(P,Q)\) for all \(\nu\leq\lfloor m/2\rfloor\).

Let \(G({\mathbf{A}}, p)\in \operatorname{Hom}_\mathbb{C}^\nu(P,Q)\) for some fixed \(\nu\in\{1,...,\lfloor m/2\rfloor\}\). Relabelling \({\mathbf{A}}\circ f\) as \(f\), it suffices to check that at \(u=(I,p)\), where \(I\) is the identity map on \({\mathbb{R}}^{2n}\), \[T_{G(u)}\operatorname{Hom}_\mathbb{C}^\nu(P,Q)+T_{G(u)}\left(G(Y\times \mathcal{M})\right)=T_{G(u)}\operatorname{Hom}_{\mathbb{C}\!}\,(P,Q).\] Let \(N_{df_\mathbb{C}(p)}\operatorname{Hom}_\mathbb{C}^\nu(P_p,Q_p)\) denote the normal space of \(\operatorname{Hom}_\mathbb{C}^\nu(P_p,Q_p)\) in \(\operatorname{Hom}_{\mathbb{C}\!}(P_p,Q_p)\) at \(df_\mathbb{C}(p)\). Then, \[\begin{align} T_{G(u)}\operatorname{Hom}_{\mathbb{C}\!}\,(P,Q)&=& T_p \mathcal{M}\oplus T_{df_\mathbb{C}(p)}\operatorname{Hom}_\mathbb{C}^\nu(P_p,Q_p)\oplus N_{df_\mathbb{C}(p)}\operatorname{Hom}_\mathbb{C}^\nu(P_p,Q_p)\\ &=&T_{G(u)}\operatorname{Hom}_\mathbb{C}^\nu(P,Q)\oplus N_{df_\mathbb{C}(p)}\operatorname{Hom}_\mathbb{C}^\nu(P_p,Q_p). \end{align}\] We will show that \(N_{df_\mathbb{C}(p)}\operatorname{Hom}_\mathbb{C}^\nu(P_p,Q_p)\subset T_{G(u)}(G(Y\times\{p\}))=DG_u\left(T_{u}(Y\times \{p\})\right)\), which establishes the transversality claim.

Since we work entirely within the fiber over \(p\), we make the following identifications:

• \(T_u(Y\times\{p\})\cong\text{Gl}(2n;{\mathbb{R}})\),

• \(\operatorname{Hom}_{\mathbb{C}\!}(P_p,Q_p)\cong \operatorname{Mat}_{n\times m}(\mathbb{C})\), and

• \(\operatorname{Hom}_\mathbb{C}^\nu(P_p,W_p)\cong \operatorname{Mat}^\nu_{n\times m}(\mathbb{C})\),

where \(\operatorname{Mat}^\nu_{n\times m}(\mathbb{C})\) denotes the space of \(n\times m\) complex matrices of rank \(m-\nu\). After a holomorphic change of coordinates, one may assume that \[\begin{align} \label{E:dfC} G(u)=df_\mathbb{C}(p)= \left(\begin{array}{c | c} \begin{array}{c|c} \mathbf{I}_{\nu\times\nu} & i\mathbf{I}_{\nu\times\nu} \\ \hline \mathbf{0}_{(m-2\nu)\times\nu} & \mathbf{0}_{(m-2\nu)\times\nu} \end{array} & \begin{array}{c} \mathbf{0}_{\nu\times(m-2\nu)} \\ \hline \mathbf{I}_{(m-2\nu)\times(m-2\nu)} \end{array}\\ \hline \mathbf{0}_{(n-m+\nu)\times 2\nu} & \mathbf{0}_{(n-m+\nu)\times(m-2\nu)} \end{array}\right), \end{align}\tag{2}\] where \(\mathbf{0}_{r\times s}\) and \(\mathbf{I}_{r\times r}\) denote the \(r\times s\) zero matrix and \(r\times r\) identity matrix, respectively. Above, we have used the fact that \(f\) is an embedding, i.e., \(\operatorname{rank}_{{\mathbb{R}}\!}df=m\), and \(\operatorname{rank}_{\mathbb{C}\!}df_\mathbb{C}=m-\nu\). We use the coordinates \(W=(w_{jk})_{j=1,...,n}^{k=1,...,m}\) for a matrix \(W\in \operatorname{Mat}_{n\times m}(\mathbb{C})\). For a fixed \(W\), \(r\in\{m-\nu+1,...,n\}\) and \(s\in\{1,...,\nu\}\), let \(W^{[rs]}\) denote the \((m-\nu+1)\times (m-\nu+1)\) matrix obtained from \(W\) by

• omitting all the lower \(n-m+\nu\) rows, except for the \(r\)-th one, and

• omitting all the leftmost \(\nu\) columns, except for the \(s\)-th one.

Then, locally near \(G(u)=df_\mathbb{C}(p)\), \[\begin{align} \operatorname{Mat}^\nu_{n\times m}(\mathbb{C})=\left\{W\in\operatorname{Mat}_{n\times m}(\mathbb{C}) : \phi_{rs}(W):=\det W^{[rs]}=0, r=m-\nu+1,...,n, s=1,...,\nu \right\},\\ N_{G(u)}\operatorname{Mat}^\nu_{n\times m}(\mathbb{C})=\text{span}_{\mathbb{C}\!}\left\{E^{[rs]}=\left(\overline{\dfrac{\partial{\phi_{rs}}}{\partial{ w_{jk}}}}(G(u))\right)_{j=1,...,n}^{k=1,...,m}:r=m-\nu+1,...,n, s=1,...,\nu\right\}. \end{align}\] We compute the matrix \(E^{[rs]}\) for \(r=m-\nu+1\) and \(s=1\). The computation goes along similar lines for all other values of \(r\) and \(s\). For a fixed \(W=(w_{jk})\), let \(W^{[(m-\nu+1)1]}=(\widetilde{w}_{jk})_{j,k=1,...,m-\nu+1}\). Then, for \(j=1,...,m-\nu+1\), \[\begin{align} \widetilde{w}_{jk}=\begin{cases} w_{j1},& \text{if }k=1,\\ w_{j(k+\nu-1)},& \text{if }k=2,...,m-\nu+1. \end{cases} \end{align}\] Furthermore, \[\phi_{(m-\nu+1)1}(W)=\det W^{[(m-\nu+1)1]}=\sum_{k=1}^{m-\nu+1}(-1)^{m-\nu+k+1}\widetilde{w}_{(m-\nu+1)k}\Delta_{(m-\nu+1)k},\] where \(\Delta_{jk}\) is the \((j,k)\) minor of \(W^{[(m-\nu+1)1]}\). Now, for \(W=G(u)=df_\mathbb{C}(p)\), \[\begin{align} \widetilde{w}_{(m-\nu+1)k}&=&0, \, k=1,...,m-\nu+1,\\ \Delta_{(m-\nu+1)k}&=&\begin{cases} i^\nu,& \text{if }k=1,\\ i^{\nu-1},& \text{if }k=2,\\ 0,& \text{if }k=3,...,m-\nu+1. \end{cases} \end{align}\] Thus, \[\dfrac{\partial{\phi_{(m-\nu+1)1}}}{\partial{ w_{jk}}}(G(u))=\begin{cases} (-1)^{m-\nu}i^\nu,& \text{if }j=m-\nu+1, k=1,\\ (-1)^{m-\nu+1}i^{\nu-1},& \text{if }j=m-\nu+1, k=2,\\ 0,& \text{for all other values of j and k}. \end{cases}\] Repeating this computation for all \(r\in\{m-\nu+1,...,n\}\) and \(s\in\{1,...,\nu\}\), we obtain that \[E^{[rs]}=\left(\overline{\dfrac{\partial{\phi_{rs}}}{\partial{ w_{jk}}}}(G(u))\right)_{j=1,...,n}^{k=1,...,m} =\left(\begin{array}{c c c} (-1)^{m}i^\nu\boldsymbol{\delta}_{rs} & (-1)^mi^{\nu-1}\boldsymbol{\delta}_{rs} & \mathbf{0}_{n\times(m-2\nu)} \end{array}\right),\] where \(\boldsymbol{\delta}_{rs}\) is the \(n\times \nu\) matrix whose \((r,s)\) entry is \(1\) and all other entries are \(0\).

Next, we compute the space \(DG_u(T_u(Y\times\{p\}))\). Given \(A\in \text{Gl}(2n;{\mathbb{R}})\), we write \[A=\left(\begin{array}{c c c} \begin{array}{c c} a_{11} & b_{11}\\ c_{11} & d_{11} \end{array} & \cdots & \begin{array}{c c} a_{1n} & b_{1n}\\ c_{1n} & d_{1n} \end{array}\\ \vdots & \cdots & \vdots \\ \begin{array}{c c} a_{n1} & b_{n1}\\ c_{n1} & d_{n1} \end{array} & \cdots & \begin{array}{c c} a_{nn} & b_{nn}\\ c_{nn} & d_{nn} \end{array}\\ \end{array}\right).\] Then, invoking 2 , we have that for \(\mathbf{A}(z)=A(z)+a\), \(a\in{\mathbb{R}}^{2n}\), \[d(\mathbf{A}\circ f)(p)=(A\circ df)(p)=\left(\begin{array}{c c c c c c| c } a_{11} & \cdots & a_{1\nu} & b_{11} &\cdots & b_{1\nu}& \\ c_{11} & \cdots & c_{1\nu} & d_{11} & \cdots & d_{1\nu} & \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \mathbf{0}_{2n\times (m-2\nu)} \\ a_{n1} & \cdots & a_{n\nu} & b_{n1} & \cdots & b_{n\nu} & \\ c_{n1} & \cdots & c_{n\nu} & d_{n1} & \cdots & d_{n\nu} & \end{array}\right).\\\] and \[G({\mathbf{A}},p)=\left(A\circ df\right)_\mathbb{C}(p)= \left(\begin{array}{c c c c c c| c } a_{11}+i c_{11} & \cdots & a_{1\nu}+ic_{1\nu} & b_{11}+id_{11} & \cdots & b_{1\nu}+id_{1\nu}& \\ \vdots & \vdots & \vdots &\vdots & \vdots & \vdots & \mathbf{0}_{n\times (m-2\nu)} \\ a_{n1}+ic_{n1} & \cdots & a_{n\nu}+ic_{n\nu} & b_{n1}+id_{n1} & \cdots & b_{n\nu}+id_{n\nu} & \\ \end{array}\right).\] Now note that for \(r\in\{n-\nu+1,...,n\}\) and \(s\in\{1,...,\nu\}\), we have \[(-1)^mi^{\nu-1}\left(\dfrac{\partial{G}}{\partial{c_{rs}}}(u)+\dfrac{\partial{G}}{\partial{a_{rs}}}(u)\right)=E^{[rs]},\] and thus \(N_{G(u)}\operatorname{Hom}_\mathbb{C}^\nu(P_p,Q_q)\subset DG_u(T_uG(Y\times p))\), which establishes the transversality claim. The result now follows from the parametric transversality theorem, and the fact that \[\text{codim}_{\mathbb{C}} (\operatorname{Mat}^\nu_{n\times m}(\mathbb{C}))=\nu(n-m+\nu).\] ◻

Proof. We recall two facts about fibre bundles over manifolds. Given two complex vector bundles \(P\) and \(Q\) over \(\mathcal{M}\) of rank \(m\leq n\), it is known that \(\operatorname{Hom}^\nu_\mathbb{C}(P,Q)\) is a fiber subbundle of \(\operatorname{Hom}_\mathbb{C}(P,Q)\) of complex codimension \(\nu(n-m+\nu)\). Moreover, the fiber of the normal bundle to \(\operatorname{Hom}_\mathbb{C}^1(P,Q)\) in \(\operatorname{Hom}(P,Q)\) at \(A\) is isomorphic to \(\operatorname{Hom}_\mathbb{C}(\ker A,{\rm coker\,} A)\). See ([19], Chapter VI §1).

Next, if \(g:X\rightarrow Y\) is a smooth map between differentiable manifolds, and \(g\) is transverse to some submanifold \(Z\subset Y\) with \({\rm codim\,}Z\leq \dim X\), then the normal bundle of \(W=g^{-1}(Z)\) in \(X\), denoted by \(N(W,X)\), is the pullback of the normal bundle \(N(Z,Y)\) of \(Z\) in \(Y\). Applying these facts to \(g=\Phi\), \(Y=\operatorname{Hom}_\mathbb{C}(T^{\mathbb{C}\!}\mathcal{M},\mathcal{M}\times\mathbb{C}^n)\), and \(Z=\operatorname{Hom}_\mathbb{C}^1(T^{\mathbb{C}\!}\mathcal{M},\mathcal{M}\times\mathbb{C}^n)\) yields the result. ◻

Proof. Since \(\ker (df_\mathbb{C})=\overline{\mathcal{H}_\mathbb{C}}\) for any any embedding \(f:\mathcal{M}\hookrightarrow\mathbb{C}^n\), we solve for a vector field \(\mathcal{V}=a_1\dfrac{\partial{}}{\partial{x_1}}+b_1\dfrac{\partial{}}{\partial{y_1}}+\sum\limits_{s=2}^{m-1}a_s\dfrac{\partial{}}{\partial{x_s}}\), for smooth \(\mathbb{C}\)-valued \(a_1,b_1,a_s\), so that \[(df_\mathbb{C})\overline{\mathcal{V}}\equiv 0\quad \text{on }\mathcal{S}_1,\] for \(df_\mathbb{C}\) as computed in 8 . ◻

Proof. First, we prove (a). Since our computations are local, we identify \(\operatorname{Hom}_{\mathbb{C}\!}(T^{\mathbb{C}\!} \mathcal{M},\mathcal{M}\times\mathbb{C})\) with \(\mathcal{M}\times \operatorname{Mat}_{n\times m}(\mathbb{C})\) and \(\operatorname{Hom}^1_\mathbb{C}(T^{\mathbb{C}\!} \mathcal{M},\mathcal{M}\times\mathbb{C}^n)\) with \(\mathcal{M}\times \operatorname{Mat}^1_{n\times m}(\mathbb{C})\), where \(\operatorname{Mat}_{n\times m}(\mathbb{C})\) denotes the space of \(n\times m\) complex matrices, and \(\operatorname{Mat}^1_{n\times m}(\mathbb{C})\) denotes the space of \(n\times m\) complex matrices of rank \(m-1\). The transversality of \(\Phi=\Phi_f\) to \(\operatorname{Hom}_\mathbb{C}^1(P,Q)\) at \(p\) is equivalent to \[(d\Phi)_p(T_p\mathcal{M})+T_{df_\mathbb{C}(p)}\operatorname{Mat}^1_{n\times m}(\mathbb{C})=T_{df_\mathbb{C}(p)}\operatorname{Mat}_{n\times m}(\mathbb{C}),\] as real subspaces, or that the projection of \((d\Phi)_p(T_p \mathcal{M})\) onto \(N_{df_\mathbb{C}(p)}\operatorname{Mat}^1_{n\times m}(\mathbb{C})\) is surjective.

As noted in the proof of Proposition 2.a, \[df_\mathbb{C}(0) = \left(\begin{array}{c | c} \begin{array}{c|c} 1 & i\\ \hline \mathbf{0}_{(m-2)\times 1} & \mathbf{0}_{(m-2)\times 1} \end{array} & \begin{array}{c} \mathbf{0}_{1\times(m-2)} \\ \hline \mathbf{I}_{(m-2)\times (m-2)} \end{array}\\ \hline \mathbf{0}_{(n-m+1)\times 2} & \mathbf{0}_{(n-m+1)\times(m-2)} \end{array}\right) ,\] and \[\label{E:sigma} N_{df_\mathbb{C}(0)}\operatorname{Mat}^1_{n\times m}(\mathbb{C})=\text{span}_{\mathbb{C}\!}\left\{\sigma_r=\left(\begin{array}{c | c} \mathbf{0}_{(m-1)\times 2} & \mathbf{0}_{(m-1)\times (m-2)}\\ \hline \begin{array}{c|c} i\boldsymbol{\delta}_{r} & \boldsymbol{\delta}_{r} \end{array} & \mathbf{0}_{(n-m+1)\times(m-2)} \end{array}\right):r=1,...,n-m+1\right\},\tag{11}\] where \(\boldsymbol{\delta}_r\) is the \((n-m+1)\) column matrix whose \(r\)-th entry is \(1\), and the rest are \(0\).

Next, note that \[\label{E:push} (d\Phi)_0(T_0 \mathcal{M})=\text{span}_{\mathbb{R}} \left\{\frac{\partial \Phi}{\partial x_1}(0), \frac{\partial \Phi}{\partial y_1}(0), \frac{\partial \Phi}{\partial x_2}(0),\dots, \frac{\partial \Phi}{\partial x_{m-1}}(0)\right\}.\tag{12}\] The surjectivity of the projection of \((d\Phi)_0(T_0 \mathcal{M})\) onto \(N_{df_\mathbb{C}(0)}\operatorname{Mat}^1_{n\times m}(\mathbb{C})\) is equivalent to the matrix \[\Lambda=\begin{pmatrix}[1.5] \operatorname{Re}\Big\langle \frac{\partial \Phi}{\partial x_1}(0),\overline{\sigma}_1\Big\rangle & \operatorname{Im}\Big\langle \frac{\partial \Phi}{\partial x_1}(0),\overline{\sigma}_1\Big\rangle & \dots & \operatorname{Re}\Big\langle \frac{\partial \Phi}{\partial x_1}(0),\overline{\sigma}_{\ell}\Big\rangle &\operatorname{Im}\Big\langle \frac{\partial \Phi}{\partial x_1}(0),\overline{\sigma}_{\ell}\Big\rangle \\ \operatorname{Re}\Big\langle \frac{\partial \Phi}{\partial y_1}(0),\overline{\sigma}_1\Big\rangle & \operatorname{Im}\Big\langle \frac{\partial \Phi}{\partial y_1}(0),\overline{\sigma}_1\Big\rangle & \dots & \operatorname{Re}\Big\langle \frac{\partial \Phi}{\partial y_1}(0),\overline{\sigma}_{\ell}\Big\rangle &\operatorname{Im}\Big\langle \frac{\partial \Phi}{\partial y_1}(0),\overline{\sigma}_{\ell}\Big\rangle \\ \operatorname{Re}\Big\langle \frac{\partial \Phi}{\partial x_2}(0),\overline{\sigma}_1\Big\rangle & \operatorname{Im}\Big\langle \frac{\partial \Phi}{\partial x_2}(0),\overline{\sigma}_1\Big\rangle & \dots & \operatorname{Re}\Big\langle \frac{\partial \Phi}{\partial x_2}(0),\overline{\sigma}_{\ell}\Big\rangle &\operatorname{Im}\Big\langle \frac{\partial \Phi}{\partial x_2}(0),\overline{\sigma}_{\ell}\Big\rangle \\\operatorname{Re}\Big\langle \frac{\partial \Phi}{\partial x_{m-1}}(0),\overline{\sigma}_1\Big\rangle & \operatorname{Im}\Big\langle \frac{\partial \Phi}{\partial x_{m-1}}(0),\overline{\sigma}_1\Big\rangle & \dots & \operatorname{Re}\Big\langle \frac{\partial \Phi}{\partial x_{m-1}}(0),\overline{\sigma}_{\ell}\Big\rangle &\operatorname{Im}\Big\langle \frac{\partial \Phi}{\partial x_{m-1}}(0),\overline{\sigma}_{\ell}\Big\rangle \\ \end{pmatrix}\] having (real) rank \(2\ell\), where \(\ell=n-m+1\). Here \(\Big\langle \cdot,\cdot\Big\rangle\) is the standard complex bilinear pairing on \(\operatorname{Mat}_{n\times m}(\mathbb{C})\). From the description of \(\sigma_r\) in 11 , we only need to compute the \((u,1)\) and \((u,2)\) entries of the matrices in 12 , where \(u=m,\dots,n\). A direct computation using 3 , 5 and 6 yields that, if \(\Phi(\mathbf{x})=df_\mathbb{C}(\mathbf{x})=\Big(\Phi_{jk}(\mathbf{x})\Big)_{j=1,...,n}^{k=1,...,m}\), then \[\begin{align} \Phi_{u1}(\mathbf{x})& =& (2\beta_u +2 \operatorname{Re}\gamma_u)x_1 + 2 \operatorname{Im}\gamma_u y_1 + \sum_{s=2}^{m-1} (\operatorname{Re}\varepsilon_u^s) x_s + \notag\\ && \qquad i\left(2\operatorname{Im}\gamma_u x_1 - 2\operatorname{Re}\gamma_u y_1 + \sum_{s=2}^{m-1} (\operatorname{Im}\varepsilon_u^s) x_s \right) + O(2) \tag{13},\\ \Phi_{u2}(\mathbf{x}) &=& (2\beta_u -2\operatorname{Re}\gamma_u)y_1 +2 \operatorname{Im}\gamma_u x_1 + \sum_{s=2}^{m-1} (\operatorname{Im}\varepsilon_u^s) x_s \notag \\ && \qquad -i\left( 2\operatorname{Im}\gamma_u y_1 + 2\operatorname{Re}\gamma_u x_1+ \sum_{s=2}^{m-1}(\operatorname{Re}\varepsilon_u^s )x_s\right)+ O(2)\tag{14}, \end{align}\] for \(u=m,...,n\). Now, using 11 , 13 and 14 to compute \(\Lambda\), we obtain that \[\Lambda=\begin{pmatrix} 4 \operatorname{Im}\gamma_m & -2\beta_m-4 \operatorname{Re}\gamma_m & \cdots & 4 \operatorname{Im}\gamma_n & -2\beta_n - 4 \operatorname{Re}\gamma_n \\ 2\beta_m - 4 \operatorname{Re}\gamma_m& -4 \operatorname{Im}\gamma_m & \cdots & 2\beta_n - 4 \operatorname{Re}\gamma_n& -4 \operatorname{Im}\gamma_n \\ 2\operatorname{Im}\varepsilon_{m}^2 & -2\operatorname{Re}\varepsilon_m^2 & \cdots & 2\operatorname{Im}\varepsilon_{n}^2 & -2\operatorname{Re}\varepsilon_n^2 \\ 2\operatorname{Im}\varepsilon_{m}^3 & -2\operatorname{Re}\varepsilon_m^3 & \cdots & 2\operatorname{Im}\varepsilon_{n}^3 & -2\operatorname{Re}\varepsilon_n^3 \\ \vdots &\vdots &\ddots &\vdots &\vdots \\ 2\operatorname{Im}\varepsilon_{m}^{m-1} & -2\operatorname{Re}\varepsilon_m^{m-1} & \cdots & 2\operatorname{Im}\varepsilon_{n}^{m-1} & -2\operatorname{Re}\varepsilon_n^{m-1} \\ \end{pmatrix}.\] Thus, the transversality of \(\Phi_f\) to \(\operatorname{Hom}_\mathbb{C}^1(P,Q)\) at \(p=0\) is equivalent to the condition that the rank of the above matrix is \(2(n-m+1)\), which in turn is equivalent to the condition ?? .

Next, we establish ?? and ?? . Let \(\mathcal{V}\) be a section of \(\mathcal{H}_\mathbb{C}|_{\mathcal{S}_1}\) such that \(\mathcal{H}_\mathbb{C}=\operatorname{span}_{\mathbb{C}\!}\{\mathcal{V}\}\) on \(S_1\). Assume that condition ?? holds. Then \(dB_m\wedge d\overline{B_m}\wedge\cdots\wedge B_n\wedge d\overline{B_n}\neq 0\) in \(U\) (shrinking further, if necessary). This implies that the \(2(n-m+1)\) vectors \[\nabla(\operatorname{Im}B_m)(\mathbf{x}),-\nabla(\operatorname{Re}B_m)(\mathbf{x}), ...,\nabla(\operatorname{Im}B_n)(\mathbf{x}),-\nabla(\operatorname{Re}B_n)(\mathbf{x}),\] form a real basis of \(N_{\mathbf{x}}(\mathcal{S}_1,\mathcal{M})\) at any \(\mathbf{x}\in\mathcal{S}_1\cap U\) in that order. Furthermore, \[\mathcal{H}_{\mathbf{x}}=\text{span}_{{\mathbb{R}}}\left\{\operatorname{Re}\mathcal{V}(\mathbf{x}),\operatorname{Im}\mathcal{V}(\mathbf{x})\right\} .\] Thus, the projection map \(\pi:\mathcal{H}_{\mathbf{x}}\rightarrow N_{\mathbf{x}}(\mathcal{S}_1, \mathcal{M})\) with respect to this choice of bases is given by \(\Pi(\mathbf{x})\), where \[\Pi=\begin{pmatrix} (\operatorname{Re}\mathcal{V})(\operatorname{Im}B_m) & (\operatorname{Im}\mathcal{V})(\operatorname{Im}B_m)\\ (\operatorname{Re}\mathcal{V})(-\operatorname{Re}B_m) & (\operatorname{Im}\mathcal{V})(-\operatorname{Re}B_m)\\ \vdots &\vdots\\ (\operatorname{Re}\mathcal{V})(\operatorname{Im}B_n) & (\operatorname{Im}\mathcal{V})(\operatorname{Im}B_n)\\ (\operatorname{Re}\mathcal{V})(-\operatorname{Re}B_n) & (\operatorname{Im}\mathcal{V})(-\operatorname{Re}B_n)\\ \end{pmatrix},\] and the complexification \(\pi_\mathbb{C}: \mathcal{H}_{\mathbf{x}}\otimes\mathbb{C}\rightarrow N_{\mathbf{x}}(\mathcal{S}_1, \mathcal{M})\) is given by \[\begin{align} \Pi_\mathbb{C}&=& \begin{pmatrix} (\operatorname{Re}\mathcal{V})(\operatorname{Im}B_m-i\operatorname{Re}B_m) & (\operatorname{Im}\mathcal{V})(\operatorname{Im}B_m-i\operatorname{Re}B_m)\\ \vdots &\vdots\\ (\operatorname{Re}\mathcal{V})(\operatorname{Im}B_n-i\operatorname{Re}B_n) & (\operatorname{Im}\mathcal{V})(\operatorname{Im}B_n-i\operatorname{Re}B_n) \end{pmatrix} =_{\operatorname{CR}} \begin{pmatrix} \mathcal{V}B_m(\mathbf{x}) & \overline{\mathcal{V}} B_m(\mathbf{x})\\ \vdots &\vdots\\ \mathcal{V}B_n(\mathbf{x}) & \overline{\mathcal{V}} B_n(\mathbf{x})\\ \end{pmatrix}, \end{align}\] where “\(=_{\operatorname{CR}}\)" denotes equality up to elementary column operations (over \(\mathbb{C}\)).

By Definitions 3 and 5, respectively, \(p=0\in W_j\) if and only if \(\operatorname{rank}_{{\mathbb{R}}\!}\Pi(0)=j\), and \(p=0\in C_j\) if and only if \(\operatorname{rank}_{\mathbb{C}\!}\Pi_\mathbb{C}(0)=j\). Choosing \(\mathcal{V}\) as granted by Lemma 3, and using the formulas for \(B_j\)’s in 10 , we obtain that the former condition is equivalent to condition ?? , and the latter is equivalent to condition ?? (after elementary column and row reductions). ◻

Proof. The proof of 15 is implicit in the proof of Proposition 7. Now, assume that \(0\in\mathcal{C}_1\). Let \[\begin{align} A(\mathbf{x})&=&\begin{pmatrix} \mathcal{V}B_m(\mathbf{x}) & \overline{\mathcal{V}} B_m(\mathbf{x})\\ \vdots &\vdots\\ \mathcal{V}B_n(\mathbf{x}) & \overline{\mathcal{V}} B_n(\mathbf{x})\\ \end{pmatrix} \end{align}\] for the choice of \(\mathcal{V}\) as granted by Lemma 3. Then, \[\begin{align} \mathcal{V}B_j(\mathbf{x}) &=&\beta_j+(2\theta_j-i\sum_{s=2}^m\varepsilon_j^s\beta_j)\overline{z}_1+2\kappa_jz_1+\sum_{s=2}^{m-1}\varphi_j^sx_s+O(2),\\ \overline{\mathcal{V}}B_j(\mathbf{x}) &=&2\gamma_j+(2\theta_j-i\sum_{s=2}^m\varepsilon_j^s\beta_j)z_1+6\pi_j\overline{z}_1 +2\sum_{s=2}^{m-1}\psi_j^sx_s+O(2). \end{align}\] Since \(0\in\mathcal{C}_1\), we have that the rank of \[A(0)=\begin{pmatrix} \beta_m & 2\gamma_m\\ \vdots &\vdots\\ \beta_n & 2\gamma_n \end{pmatrix}\;\text{is }1.\] Thus, one of the rows in \(A\) must be non-zero. Suppose, the row corresponding to index \(j\) is non-zero. Then, by composing \(f\) with a biholomorphism of \(\mathbb{C}^n\) that swaps \(z_m\) with \(z_j\), we have that \((\beta_m,2\gamma_m)\neq (0,0)\). Next, by the rank condition, we have the existence of \(\lambda_j\in\mathbb{C}\) such that \[(\beta_j,2\gamma_j)=\lambda_j(\beta_m,2\gamma_m),\quad m+1\leq j\leq n.\] Then, by composing \(f\) with a biholomorphism of \(\mathbb{C}^n\) that maps \(z_j\mapsto z_j-\lambda_jz_m\), \(m+1\leq j\leq n\), we can ensure that \((\beta_j,\gamma_j)= (0,0)\) for all \(m+1\leq j\leq n\).

Finally, by continuity of the matrix-valued function \(A\), there is a \(\delta>0\) such that the first row of \(A(\mathbf{x})\) is non-zero for \(\mathbf{x}\in\mathbb{B}(0,\delta)\). Thus, \[\mathcal{C}_1\cap \mathbb{B}(0,\delta)= \left\{\mathbf{x}\in \mathcal{S}_1\cap\mathbb{B}(0,\delta): \mathcal{V} B_m(\mathbf{x})\overline{\mathcal{V}} B_k(\mathbf{x})-\overline{\mathcal{V}} B_m(\mathbf{x})\mathcal{V} B_k(\mathbf{x})=0, m+1\leq k\leq n\right\},\] which yields 16 , since \((\beta_j,\gamma_j)= (0,0)\) for all \(m+1\leq j\leq n\). ◻

Proof. It suffices to show that if condition ?? holds with \(j=2\), then 18 and ?? are equivalent conditions. In order to check 18 , one first performs a local biholomorphic change of coordinates so that 17 holds. On the other hand, condition ?? is invariant under local biholomorphic changes of coordinates at \(p\) since it is equivalent to the fact that \(\Phi_f: \mathcal{M}\rightarrow\operatorname{Hom}_\mathbb{C}(P,Q)\) is transverse to \(\operatorname{Hom}^1_\mathbb{C}(P,Q)\) at \(p\).

Thus, we may assume that, the coefficients in 5 and 6 satisfy 17 . Then, referring to ?? , \[d\mathscr S(0)= \begin{pmatrix} 0 & 0 & \operatorname{Re}\varepsilon_m^2 & \cdots & \operatorname{Re}\varepsilon_m^{m-1} \\ \vdots & \vdots &\vdots &\ddots & \vdots\\ 0 & 0 & \operatorname{Im}\varepsilon_{n-2}^2 & \cdots & \operatorname{Im}\varepsilon_{n-2}^{m-1} \\ 2 & 0 & 0 &\cdots & 0\\ 0& 2 & 0 & \cdots & 0\\ 0 & 0 & \operatorname{Re}\varepsilon_n^2 & \cdots & \operatorname{Re}\varepsilon_n^{m-1} \\ 0& 0 & \operatorname{Im}\varepsilon_n^2 & \cdots & \operatorname{Im}\varepsilon_n^{m-1} \end{pmatrix}.\] The rank of the above \(m\times 2(n-m+1)\) matrix being \(2(n-m+1)\) is clearly equivalent to 18 . ◻

Proof. As in the proof of Proposition 2, we use the parametric transversality theorem. Let \(f\) be as granted by Proposition 9. Assume that \(\mathcal{S}_\nu=\emptyset\) for all \(\nu\geq 2\). We denote \(\mathcal{S}\) by \(\mathcal{S}^f\) and \(\mathcal{S}_\nu\) by \(\mathcal{S}_\nu^f\) in this proof.

Let \(U(n)\) and \(\mathbf{P}_2(\mathbb{C}^n)\) denote the space of \(n\times n\) unitary matrices and the space of homogeneous degree \(2\) polynomial maps on \(\mathbb{C}^n\), respectively. Consider the following family of transformations: \[\widetilde{Y}=\{\mathbf{A}(z)=a+A' z+A''(z):a\in\mathbb{C}^n, A'\in \text{U}(n), A''\in\mathbf{P}_2(\mathbb{C}^n)\}.\] For any \(\mathbf{A}\in \widetilde{Y}\), if \(A''\) is sufficiently small, then

• \(\mathcal{S}_\nu^{\mathbf{A}\circ f}=\emptyset\) for all \(\nu\geq 2\), i.e., \(\mathcal{S}^{\mathbf{A}\circ f}=\mathcal{S}_1^{\mathbf{A}\circ f}\),

• \(\mathcal{S}^{\mathbf{A}\circ f}\) is a compact \((3m-2n-2)\)-submanifold of \(\mathcal{M}\),

• viewing any small tubular neighbourhood \(\mathcal{T}\) of \(\mathcal{S}^f\) in \(\mathcal{M}\) as a disk bundle over \(\mathcal{S}^f\), there is a smooth regular section \(\eta_{\mathbf{A}}: \mathcal{S}^f\rightarrow \mathcal{T}\) whose image is precisely \(\mathcal{S}^{\mathbf{A}\circ f}\).

Let \(Y\subset\widetilde{Y}\) denote the set of those \(\mathbf{A}\in \widetilde{Y}\) whose corresponding \(A''\) is sufficiently small so that conditions (i)-(iii) hold. Let \(E(P,Q)\) be the bundle over \(\operatorname{Hom}_\mathbb{C}^1(P,Q)\) defined in 19 .

Now, consider the smooth map \(G:Y\times \mathcal{S}^f\rightarrow E(P,Q)\) given by \[G(\mathbf{A}, p)=\left(\Psi_{\mathcal{A}\circ f}\circ \eta_{\mathbf{A}}\right)(p).\] We show that \(G\) is transverse to \(Z\subset E(P,Q)\), where \(Z\) is any one of the submanifolds \(E_0(P,Q)=F_0(P,Q)\), \(E_1(P,Q)\) and \(F_1(P,Q)\). The result then follows from the parametric transversality theorem and the codimension count given in 21 .

We first make some simplifications. By relabelling \(\mathbf{A}\circ f\) as \(f\), verifying the transversality of \(G\) to \(Z\) at \(G(\mathbf{A},p)\in Z\) is equivalent to verifying the transversality of \(G\) to \(Z\) at \(G(I,p)\in Z\), where \(I\) is the identity map on \(\mathbb{C}^n\). Moreover, for any translation map \(T\) and unitary map \(U\) on \(\mathbb{C}^n\), \(G(U\circ T\circ \mathbf{A},p)\) is transverse to \(Z\) if and only if \(G(\mathbf{A},p)\) is transverse to \(Z\). So, we assume that \(f(p)=0\) and \(M\) is given near \(0\) by 3 . Then, \[\label{E:difff} df_\mathbb{C}(p) = \left(\begin{array}{c | c} \begin{array}{c|c} 1 & i\\ \hline \mathbf{0}_{(m-2)\times 1} & \mathbf{0}_{(m-2)\times 1} \end{array} & \begin{array}{c} \mathbf{0}_{1\times(m-2)} \\ \hline \mathbf{I}_{(m-2)\times (m-2)} \end{array}\\ \hline \mathbf{0}_{(n-m+1)\times 2} & \mathbf{0}_{(n-m+1)\times(m-2)} \end{array}\right) .\tag{22}\] Thus, \(G(I,p)\) lies in \[E(P,Q)|_{df_\mathbb{C}(p)}=\text{the fiber of E(P,Q) over df_\mathbb{C}(p)\in\operatorname{Hom}_\mathbb{C}^1(P,Q)}.\]

Next, we consider \(G\) restricted to the following submanifold \(Y'\subset Y\): \[\begin{align} Y'=\left\{\mathbf{A}\in Y:\mathbf{A}(z_1,...,z_n)= \left(z_1,...,z_{m-1},z_m+b_m|z_1|^2+c_m\overline{z}_1^2,...,z_n+b_n|z_1|^2+c_n\overline{z}_1^2\right),\right.\\ \left. \text{for some }b_m,...,b_n,c_m,...,c_n\in\mathbb{C}\right\}, \end{align}\] where we give \(Y'\) the coordinates \((b,c)=(b_m,...,b_n,c_m,...,c_n)\in\mathbb{C}^{n-m+1}\). With this choice, \(\eta_{\mathbf{A}}(p)=p\) and \(G(\mathbf{A},p)\in E(P,Q)|_{df_\mathbb{C}(p)}\) for all \(\mathbf{A}\in Y'\). Since we work in this fixed fiber, it can be identified with \(\operatorname{Mat}_{(2n-2m+2)\times 2}({\mathbb{R}})\). From the computations in the proof of Proposition 7.b, we obtain that \[\label{E:G40A44p41} \Psi_{\mathbf{A}\circ f}= G(\mathbf{A},p)=\begin{pmatrix} 2\operatorname{Im}\gamma_m +\operatorname{Im}b_m+2\operatorname{Im}c_m & \beta_m+2\operatorname{Re}\gamma_m+\operatorname{Re}b_m-2\operatorname{Re}c_m\\ \beta_m-2\operatorname{Re}\gamma_m+\operatorname{Re}b_m+2\operatorname{Re}c_m & 2\operatorname{Im}\gamma_m-\operatorname{Im}b_m+2\operatorname{Im}c_m\\ \vdots &\vdots\\ 2\operatorname{Im}\gamma_n+\operatorname{Im}b_n+2\operatorname{Im}c_n & \beta_n+2\operatorname{Re}\gamma_n+\operatorname{Re}b_n-2\operatorname{Re}c_n\\ \beta_n-2\operatorname{Re}\gamma_n+\operatorname{Re}b_n+2\operatorname{Re}c_n & 2\operatorname{Im}\gamma_n-\operatorname{Im}b_n+2\operatorname{Im}c_n \end{pmatrix},\tag{23}\] when \(\mathbf{A}=(b,c)\in Y'\). It follows that \[\begin{align} DG_{(I,p)}\left(T_{(I,p)}(Y'\times\{p\})\right) &=&\text{span}_{\mathbb{R}}\left\{\dfrac{\partial{G}}{\partial{\operatorname{Re}b_j}},\dfrac{\partial{G}}{\partial{\operatorname{Im}b_j}},\dfrac{\partial{G}}{\partial{\operatorname{Re}c_j}},\dfrac{\partial{G}}{\partial{\operatorname{Im}c_j}}:j=m,...,n\right\}\Big|_{(I,p)}\\ &=& \operatorname{Mat}_{(2n-2m+2)\times 2}({\mathbb{R}})\\ &=& T_{G(I,p)}E(P,Q). \end{align}\] Thus, \(G\) is transverse to \(Z\) at \((I,p)\) for each of the relevant choices of \(Z\).

The last statement of the proposition follows from Hirsch [17]. ◻

Proof. This is essentially a restatement of Lemma 3.3 in [14], which under the assumptions of the lemma gives \(c>0\) such that \(||f-\tilde{f}||_{\mathcal{C}^1(\mathcal{M})}<c\) implies \(h({K\cup \tilde{f}(\mathcal{M})})\subset U\). Here \(h(X) = \overline{\widehat X \setminus X}\) is the essential hull. From this Lemma 7 follows. ◻

Proof. We set \(M = f(\mathcal{M})\). Since \(C\) is polynomially convex, there exists (see [1]) a smooth nonnegative plurisubharmonic function \(\phi(z)\) on \(\mathbb{C}^n\) such that \(\{\phi^{-1}(0)\}=C\) and \(\phi\) is strictly plurisubharmonic on \(\mathbb{C}^n \setminus C\). It follows that for any \(b>0\), the sublevel set \(\{\phi \le b\}\) is a compact polynomially convex neighbourhood of \(C\) (we may restrict consideration to a connected component of \(\{\phi \le b\}\) containing \(C\) if necessary). By applying Sard’s theorem to the restriction of \(\phi\) to \(M\), for almost all \(b>0\), the set \(\{\phi=b \}\cap M\) is a closed smooth submanifold of \(M\) of dimension \(m-1\). Thus, by choosing an appropriate sequence \(b_j \searrow 0\), we may construction an open neighbourhood basis \(\{U_j=\{\phi < b_j\}\}_{j\in\mathbb{N}}\) in \(\mathbb{C}^n\) such that for each \(j\in\mathbb{N}\) we have

• \(U_{j+1}\Subset U_{j}\),

• \(\overline{U_j}\) is polynomially convex in \(\mathbb{C}^n\),

• \(M\setminus U_j\) is a smooth totally real manifold with boundary.

Let \(\varepsilon>0\) and \(k \ge 1\) be given. The idea of the proof is the following: using Lemma 6, for each \(j \in \mathbb{N}\) construct a \({\mathcal{C}^{k+j}(\mathcal{M})}\)-small perturbation \(f_j\) of \(f\) such that the polynomially convex hull of \(M_j=f_j(\mathcal{M})\) is contained in \(U_j \cup M_j\). Further, in the inductive procedure, the map \(f_{j+1}\) is chosen from a neighbourhood of \(f_j\) in the space of embeddings that satisfy 24 with \(f=f_j\) and \(\tilde{f} = f_{j+1}\). The required map \(f_\varepsilon\) is then obtained as a pointwise limit of \(\{f_j\}\).

Set \(\mathcal{M}_j=\mathcal{M}\setminus f^{-1}\left(U_{j+1}\right)\), \(K_j =\overline{U_j}\). By Lemma 6 applied to \(\mathcal{M}=\mathcal{M}_1\), \(f=f|_{\mathcal{M}_1}\), \(K=K_1\) and choosing some \(U_0\) that satisfies the assumption on \(U\), there exists a totally real \(\mathcal{C}^\infty\) embedding \(f_1: \mathcal{M}_1\rightarrow\mathbb{C}^n\) such that

1. \(f_1=f\) on \(f^{-1}(K_1)\),

2. \(||f_1-f||_{\mathcal{C}^k(\mathcal{M}_1)} = ||f_1-f||_{\mathcal{C}^k(\mathcal{M})}<\varepsilon/4\),

3. \(\widehat{K_1\cup f_1(\mathcal{M}_1)}\subset U_0\cup f_1(\mathcal{M}_1)\).

Now, by Lemma 7 there exists a constant \(c_1 = c (\mathcal{M}_1, f_1,K_1,U_0)\) such that for any smooth \(\tilde{f}\) we have \[\label{e46cnb1} ||f_1-\tilde{f}||_{\mathcal{C}^1(\mathcal{M})}<c_1 \;\;\Longrightarrow \;\; \widehat{K_1\cup \tilde{f}(\mathcal{M}_1)} \subset U_0 \cup \tilde{f}(\mathcal{M}_1).\tag{25}\] Chose \(\varepsilon_2 = \min\{\varepsilon_1/2, c_1/2\}\), and after shrinking \(\varepsilon_2\) further we may also assume that \[K_2+\mathbb{B}(0,\varepsilon_2)\subset U_{1}.\] Again, by Lemma 6 there exists \(f_2: \mathcal{M}\to \mathbb{C}^n\) such that

1. \(f_1=f_2\) on \(f^{-1}(K_2)\),

2. \(||f_2-f_1||_{\mathcal{C}^{k+1}(\mathcal{M}_2)} = ||f_2-f_1||_{\mathcal{C}^{k+1}(\mathcal{M})} <\varepsilon_2\),

3. \(\widehat{K_2\cup f_2(\mathcal{M}_2)}\subset U_1\cup f_2(\mathcal{M}_2)\).

Note that by construction, the map \(f_2\) satisfies 25 , and therefore, \(\widehat{K_1\cup f_2(\mathcal{M}_1)} \subset U_0 \cup f_2(\mathcal{M}_1)\).

Suppose now that there exist \(\varepsilon_r>0\) and maps \(f_r: \mathcal{M}\to \mathbb{C}^n\), \(r = 1, 2, \dots, j-1\), such that for \(r>1\) we have

• \(\varepsilon_r <\min\{\varepsilon_{r-1}/2, c_{r-1}/2\}\) and \({K_r}+\mathbb{B}(0,\varepsilon_r)\subset U_{r-1}\).

• \(f_r=f_{r-1}\) on \(f^{-1}\left({K_r}\right)\),

• \(||f_r-f_{r-1}||_{\mathcal{C}^{k+r}(\mathcal{M}_r)}<\varepsilon_r\),

• \(\widehat{{K_r}\cup f_r(\mathcal{M}_r)}\subset U_{r-1}\cup f_r(\mathcal{M}_r)\).

Then by Lemma 6, there exist \(\varepsilon_{j}\) and an embedding \(f_{j}: \mathcal{M}\to \mathbb{C}^n\) such that

1. \(\varepsilon_{j} <\min\{\varepsilon_{j-1}/2, c_{j-1}/2\}\) and \({K_{j}}+\mathbb{B}(0,\varepsilon_j)\subset U_{j-1}\).

2. \(f_{j-1}=f_{j}\) on \({f}^{-1}\left({K_{j}}\right)\),

3. \(||f_{j-1}-f_j||_{\mathcal{C}^{k+j}(\mathcal{M}_{j})} = ||f_{j-1}-f_j||_{\mathcal{C}^{k+j}(\mathcal{M})} <\varepsilon_{j}\),

4. \(\widehat{{K_{j}}\cup f_{j}(\mathcal{M}_{j})}\subset U_{j-1}\cup f_{j}(\mathcal{M}_{j})\).

Note that for any \(1<r<j\) we have \(||f _r - f_j ||_{\mathcal{C}^1(\mathcal{M})} < c_r\), and therefore, by Lemma 7, \[\widehat{K_r\cup f_j(\mathcal{M})}\subset U_{r-1} \cup f_j (\mathcal{M}) .\]

Consider now the limit of the sequence \(\{f_j\}\). We claim that (1) the limit exists and is a smooth function, call it \(f_\varepsilon\), and (2) \(f_\varepsilon\) satisfies conditions (a)-(c) of the lemma.

(1) The sequence \(\{f_j\}\) is Cauchy in \(\mathcal{C}^{\tilde{\eta}}(\mathcal{M})\) for any \(\tilde{\eta}\ge 1\). To prove that assume, without loss of generality, that \(\tilde{\eta} = k + \eta\), \(\eta>0\). Given any \(\delta>0\), choose \(N>0\) such that \(N > \eta\) and \(||f_{N-1}-f_N||_{\mathcal{C}^{k +N}(\mathcal{M})}<\varepsilon_N <\delta\) (condition \((ii_N)\)). Let \(\nu, \mu > N\) with \(\nu > \mu\). Then \[||f_\nu - f_\mu||_{\mathcal{C}^{k+\eta}(\mathcal{M})} \le \sum_{l=0}^{\nu-\mu-1} ||f_{\mu+l} - f_{\mu+l+1}||_{\mathcal{C}^\eta(\mathcal{M}_{k+\nu+l+1})} \le \sum_{l=0}^{\nu-\mu-1} \frac{\varepsilon_{\mu}}{2^l} \le \varepsilon_{\mu} \sum_{l=0}^{\infty} \frac{1}{2^{l+1}} = \varepsilon_{\mu} <\varepsilon_N < \delta,\] where we used the estimate \[||f_{\mu+l} - f_{\mu+l+1}||_{\mathcal{C}^{k+\eta}(\mathcal{M})} \le ||f_{\mu+l} - f_{\mu+l+1}||_{\mathcal{C}^{k+\mu+l+1}(\mathcal{M})} \le \varepsilon_{\mu+l+1} < \frac{\varepsilon_{\mu+l}}{2} \le \dots \le \frac{\varepsilon_{\mu}}{2^{l+1}}.\] This shows that \(\{f_j\}\) is a Cauchy sequence and therefore it converges in any \(\mathcal{C}^{\tilde{\eta}}(\mathcal{M})\). Thus, \(f_\varepsilon\in \mathcal{C}^{\infty}(\mathcal{M})\).

(2) Let \(\nu\in \mathbb{N}\) be such that \(||f _\infty -f_\nu||_{\mathcal{C}^k(\mathcal{M})} < \varepsilon/2\). Then \[\begin{align} ||f_\infty -f||_{\mathcal{C}^k(\mathcal{M})} \le ||f_\infty - f_\nu||_{\mathcal{C}^k} + ||f_\nu - f||_{\mathcal{C}^k} \le \\ \varepsilon/2 + \sum_{j=2}^\nu ||f_j - f_{j-1}||_{\mathcal{C}^k} + ||f_1 - f||_{\mathcal{C}^k}\le \varepsilon/2 + \sum_{j=2}^\nu \varepsilon/ 2^j < \varepsilon. \end{align}\] This verifies condition (a). Conditions (b) holds by construction. To prove that \(f_\varepsilon(\mathcal{M})\) is polynomially convex we argue as follows. Note that by Lemma 7 any perturbation of \(f\) in a neighbourhood of \(f\) controlled by \(c_1\) satisfies  25 . By the choice of \(\varepsilon_j\) in condition \((0_j)\) for all \(j\) we conclude that \[\widehat{K_1\cup f_\varepsilon(\mathcal{M})} \subset U_0 \cup f_\varepsilon(\mathcal{M}) .\] The same argument can used to show that \(\widehat{K_j \cup f_\varepsilon(\mathcal{M}) }\subset U_{j-1} \cup f_\varepsilon(\mathcal{M})\) for any \(j>0\), this again follows from the choice of \(\varepsilon_j\) in condition \((0_j)\). Since \(U_j\) form a neighbourhood basis of \(C\), we conclude that \(\widehat{f_\varepsilon(\mathcal{M})} = f_\varepsilon(\mathcal{M})\). This completes the proof of the proposition. ◻

Proof. This follows from the transversality of \(\Phi_f\) to \(\text{Hom}_\mathbb{C}^1(T^\mathbb{C}\mathcal{M},\mathcal{M}\times\mathbb{C}^n)\) and the continuity of the map \(f\mapsto \Phi_f\). ◻

Proof. We produce \(\tau,\delta>0\) and \(c>1\), corresponding to \(f'=f\) and fixed \(p\in\mathcal{L}\) so that (i)-(iv) hold for a \(g\) satisfying (a) and (b), and observe that all the steps in this proof depend continuously on the choice of \(f'\) and \(p\).

Since \(p\in\mathcal{C}_2\), by Proposition 8, there is a neighborhood \(V_p\) of \(f(p)\) in \(\mathbb{C}^n\) and a biholomorphism \(Q:V_p\rightarrow \mathbb{B}^{m}(0,\eta)\times \mathbb{B}^{2n-m}(0,\eta)\), for some \(\eta\in(0,1)\), such that \(Q(p)=0\), and \[\begin{align} Q(M\cap V_p)=\left\{z\in \mathbb{B}^{m}(0,\eta)\times \mathbb{B}^{2n-m}(0,\eta): \;\; \begin{aligned} & y_s=0,\;2\leq s\leq m-1,\\ & z_{m}=\overline{z}_1^2,\\ &z_{m+1}=|z_1|^2+\overline{z}_1(x_2+ix_3),\\ &z_u =\overline{z}_1 (x_{2(u-m)}+ix_{2(u-m)+1}),\;m+2\leq u\leq n\\ \end{aligned} \right\}. \end{align}\]

Thus, for the rest of the proof, it suffices to assume that \(U_p(\tau)=\mathbb{B}^m(0,\eta)\subset{\mathbb{R}}^m\), which has coordinates \(\mathbf{x}=(x_1,y_1,x_2,...,x_{m-1})\), and \(f=(f_1,...,f_n):\mathbb{B}^m(0,\eta)\rightarrow \mathbb{C}^n\) is given by \[\begin{align} f_1(\mathbf{x})&=&x_1+iy_1,\\ f_s(\mathbf{x})&=& x_s,\quad s=2,...,m-1,\\ f_m(\mathbf{x})&=&(x_1 -iy_1)^2,\\ f_{m+1}(\mathbf{x})&=&(x_1 - iy_1)(x_1+iy_1 + x_2 + i x_3),\\ f_u(\mathbf{x})&=&(x_1-iy_1)(x_{2(u-m)}+ix_{2(u-m)+1}),\quad m+2\leq u\leq n. \end{align}\] Referring to 9 , we obtain that \[\mathcal{S}\cap U_p(\tau) =\left\{\mathbf{x}\in \mathbb{B}^m(0,\eta): x_1=y_1=x_2=\cdots=x_{2(n-m)+1}=0\right\}.\] Let \(\mathbf{x}'=(x_{2n-2m+2},...,x_{m-1})\), and \(f|_{\mathcal{S}}\) be denoted by \(f_{\mathcal{S}}\). Then, \[f_{\mathcal{S}}:(0,\mathbf{x}')\mapsto (0,...,0), \quad \mathbf{x}'\in\mathbb{B}^{3m-2n-2}(0,\eta).\] Given \(h\in\mathcal{C}^{\infty}_c(\mathbb{B}^{3m-2n-2}(0,\eta);\mathbb{C}^n)\), let \(g(0,\mathbf{x}')=f_{\mathcal{S}}(0,\mathbf{x}')+h(\mathbf{x}')\), \(\mathbf{x}'\in \mathbf{B}^{3m-2n-2}(0,\eta)\) (and \(f_{\mathcal{S}}\) outside this ball). Assume that \(\Vert h\Vert_{\mathcal{C}^k(\mathbb{B}^{3m-2n-2}(0,\eta))}<\delta\), where \(\delta\) is chosen so that \(g(\mathcal{S}\cap U_p(\tau))\subset \mathbb{B}^{2n}(0,\eta)\). Then, \(h\) takes the form \[h(\mathbf{x}')=\left(\phi^1(\mathbf{x}')+i\psi^1(\mathbf{x}'),...,\phi^n(\mathbf{x}')+i\psi^n(\mathbf{x}')\right),\] for some \(\phi^j,\psi^j\in\mathcal{C}_c^\infty\left(\mathbb{B}^{3m-2n-2}(0,\eta)\right)\), with \(\Vert\phi^j\Vert_{\mathcal{C}^k},\Vert\psi^j\Vert_{\mathcal{C}^k}<\delta\).

Now, let \(G=(G_1,...,G_n):\mathbb{B}^{m}(0,\eta)\rightarrow\mathbb{C}^n\) be given by \[G:\mathbf{x}\rightarrow f(\mathbf{x})+\left(\phi^1(\mathbf{x}')+i\psi^1(\mathbf{x}'),...,\phi^n(\mathbf{x}')+i\psi^n(\mathbf{x}')\right).\] Claims (ii) and (iv) clearly hold for \(G\) (for an appropriate choice of \(c>1\)). To compute \(\mathcal{S}^{G}\) (to confirm (iii)), we seek points in \(\mathbf{x}\in \mathbb{B}^{m}(0,\eta)\) at which the rank of the matrix \(dG_\mathbb{C}(\mathbf{x})\) is \(m-1\). After some elementary row reductions, the matrix becomes For sufficiently small \(\varepsilon_p\), \(\mathbf{I}_{3m-2n-2}+\mathbf{E}\) is invertible, and thus \(\text{rank}DG(\mathbf{x})\geq m-2\) for all \(\mathbf{x}\in V_p\). Thus, we seek those \(\mathbf{x}\in V_p\) for which \(\text{rank}_{\mathbb{C}!}dG_\mathbb{C}(\mathbf{x})=m-1\). After some simple row reductions (and shrinking \(\delta\), if necessary), the matrix \(dG_\mathbb{C}(\mathbf{x})\) becomes \[\left(\begin{array}{c c|c c c} 1 & i & 0 & \cdots & 0 \\ & & & & \\ \hline & & & &\\ 0 & 0 & & & \\ \vdots & \vdots & & \mathbf{I}_{m-2} & \\ 0 & 0 & & & \\ & & & &\\ \hline & & & & \\ 2(x_1-iy_1)& -i2(x_1-iy_1) & & & \\ 2x_1+x_2+ix_3 & -2y_1-i(x_2+ix_3)& & & \\ x_4+ix_5& -i(x_4+ix_5) & & \mathbf{0}_{(n-m+1)\times(m-2)} & \\ \vdots & \vdots & & \\ x_{2n-2m}+ix_{2n-2m+1}& -i(x_{2n-2m}+ix_{2n-2m+1}) & &\\ \end{array}\right),\] which has rank \(m-1\) precisely on the set \[\mathcal{S}\cap U_p(\tau) =\left\{\mathbf{x}\in \mathbb{B}^m(0,\eta): x_1=y_1=x_2=\cdots=x_{2(n-m)+1}=0\right\}.\] Finally, for (i), we note that \(G\), as defined, is not compactly supported in the directions normal to the \(\mathbf{x}'\) coordinates. However, \(G\) is totally real off of the \(\mathbf{x}'\)-axes, and thus, we can replace \(G\) by \[G(\mathbf{x})\mapsto f(\mathbf{x})+\chi(\mathbf{x}'')h(\mathbf{x}'),\] where \(\mathbf{x}''=(x_1,y_1,x_2,...,x_{2(n-m)}+1)\) and \(\chi\) is a radially symmetric cutoff function that is a constant \(1\) in a neighborhood of \(\mathbf{x}''\), and is chosen such that \(G\) remains totally real off of the \(\mathbf{x}'\)-axes. ◻

Proof. Since \(K\) is polynomially convex in \(\mathbb{C}^n\), there exists a smooth plurisubharmonic function \(\rho\) on \(\mathbb{C}^n\) such that \(\rho\) vanishes on \(K\), and is positive and strictly plurisubharmonic on \(\mathbb{C}^n \setminus K\). For \(z\in\mathbb{C}^n\), let \(\delta(z)\) denote the squared Euclidean distance of \(z\) from \(T\). Then, \(\delta\) is a strictly plurisubharmonic function on some neighbourhood \(W\) of \(T\). Given \(\varepsilon>0\), let \[\begin{align} U_\varepsilon&=&\{z\in\mathbb{C}^n:\rho(z)<\varepsilon\},\\ W_\varepsilon&=&\{z\in\mathbb{C}^n:\delta(z)<\varepsilon\}. \end{align}\] Choose \(\varepsilon_1, \varepsilon_2 >0\) such that \(U_{\varepsilon_2}\Subset W_{\varepsilon_1}\Subset W\). Let \(\chi\geq 0\) be a cutoff function which is 1 on \(W_{\varepsilon_1}\) and vanishes outside \(W\). Let \(R>0\) be large enough so that \(\chi \delta < R(\rho -\varepsilon_2)\) on \(\mathbb{C}^n \setminus W_{\varepsilon_1}\). Now consider the function \[\sigma (z) = \max \{R(\rho(z) - \epsilon_2), \chi(z) \delta(z)\},\qquad z\in\mathbb{C}^n.\] By construction, \(\sigma\) is a continuous function on \(\mathbb{C}^n\) that vanishes on \(\overline{U_{\varepsilon_2}}\cap T= \{ z\in T : \rho(z) \le \varepsilon_2\}\), and is positive everywhere else. Furthermore, \(\sigma\) is plurisubharmonic on \(\mathbb{C}^n\). Indeed, on \(W_{\varepsilon_1}\), it is the maximum of two plurisubharmonic functions, and outside a compact subset of \(W_{\varepsilon_1}\), it agrees with \(R(\rho-\varepsilon_2)\). Thus, we obtain that \(\overline{U}_{\varepsilon_2}\cap T\) is polynomially convex.

By Sard’s theorem, for almost all \(c\in \mathbb{R}\), the level set \(\{(\rho|_T)^{-1}(c)\}\) is a smooth submanifold of \(T\). Therefore, since \(K\) is a compact subset of \(\overline{U}_{\varepsilon_2}\cap T\), there exists a choice of \(\varepsilon_3 >0\) such that \[\overline{U}_{\varepsilon_3}\cap T \Subset \overline{U}_{\varepsilon_2}\cap T ,\] and \(\{\rho = \varepsilon_3\}\cap T\) is a smooth manifold. The set \(\overline{U}_{\varepsilon_3}\) is polynomially convex, and therefore, \[\overline{U}_{\varepsilon_3} \cap T = \overline{U}_{\varepsilon_3} \cap (\overline{U}_{\varepsilon_2}\cap T)\] is polynomially convex as the intersection of two polynomially convex compacts (see [1]). Thus, \(U=U_{\varepsilon_3}\) is the required neighbourhood of \(K\). ◻

Proof. After a small perturbation, we may assume that \(f:\mathcal{M}\rightarrow\mathbb{C}^n\) is real-analytic. Denote \(\mathcal{S}^f\) by \(\mathcal{S}\) and \(\mathcal{C}_2^f\) by \(\mathcal{C}_2\). Let \(\mu = \dim_{{\mathbb{R}}\!} \mathcal{S}\). As per our convention, set \(M = f(\mathcal{M})\), \(S=f(\mathcal{S})\), \(C_2=f(\mathcal{C}_2)\), and \(K=f(\mathcal{K})\).

Let \(\mathcal{L}=\overline{\mathcal{S}\setminus\mathcal{K}}\), and \(\widetilde{\varepsilon}>0\) as granted by Lemma 9 corresponding to this choice of \(\mathcal{L}\). Let \(\varepsilon\in(0,\widetilde{\varepsilon})\). Let \(\tau_\varepsilon,\delta_\varepsilon>0\) and \(c_\varepsilon>1\) be as granted by Lemma 9 corresponding to this choice of \(\varepsilon\). The idea of the proof is to incrementally increase the size of the polynomially convex compact set in the set of CR singularities by adding to \(\mathcal{K}\) a smooth image \(\mathcal{R}\) in \(\mathcal{S}\) of a standard cube \(I^\mu=[0,1]^\mu \subset \mathbb{R}^\mu\) and then producing a smooth perturbation \(g\) of \(f|_{\mathcal{S}}\) so that \(g(\mathcal{K}\cup \mathcal{R})\) is polynomially convex, followed by a matching smooth perturbation \(f^\sharp\) of \(f\) on \(\mathcal{M}\). This process is then repeated for (a real-analytic perturbation of) \(f^\sharp\). The size of \(\mathcal{R}\) will depend on \(\tau_\varepsilon\), and will be such that finitely many such cubes cover \(\mathcal{L}\). After applying this procedure finitely many times, we obtain a perturbation of \(f\) for which the whole set of CR-singularities is polynomially convex. A detailed construction is as follows.

Let \(\mathcal{K}\subset\mathcal{S}\) be as given. Let \(\mathcal{R}=\phi(I^\mu) \subset S\), where \(\phi\) is a diffeomorphism from a neighbourhood of \(I^\mu\) in \(\mathbb{R}^\mu\) onto its image in \(\mathcal{S}\). Assume that \(\mathcal{R}\cap \mathcal{K}\ne \varnothing\). Set \(R=f(\mathcal{R})\). The principal step is to obtain a polynomially convex perturbation of \(K \cup R\). For this, we induct over the skeleta of \(\mathcal{R}\). Let \(\mathcal{R}_k\) denote the \(k\)-dimensional skeleton of \(\mathcal{R}\), for \(k=0,1,...,\mu\). Set \(R_k=f(\mathcal{R}_k)\), \(k=0,1,...,\mu\).

As a base of the induction, note that since \(R_0\) is a finite set of points, by [3], the set \(K \cup R_0\) is polynomially convex. By Lemma 10, there exist compact polynomially convex neighbourhoods \(V_0\Subset V_0'\subset S\) of \(K \cup R_0\) that are smooth manifolds with boundary. Set \(Q_1=R_1\setminus V_0\) and \({Q_1'}=R_1\setminus V_0'\). Then, \(Q_1'\subset Q_1\subset R_1\) are smooth (totally real) manifolds with boundary of dimension \(1\) lying entirely in \(C_2\), and such that \[K \cup R_0 \cup R_1\subset V_0 \cup Q_1= V_0' \cup Q_1'.\] Set \(g_0=f\). Set \(\mathcal{Q}_1=g_0^{-1}(Q_1)\) and \(\mathcal{Q}_1'=g_0^{-1}(Q_1')\). Note that \(\mathcal{Q}_1'\subset\mathcal{Q}_1\subset \mathcal{R}_1\).

By [14], applied to \(\mathcal{Q}_1'\) as \(M\) and \(V_0\) as \(K\), for any \(\delta>0\), there exists a perturbation of \(g_0|_{\mathcal{Q}_1}\) of the form \(g_0+h\), so that \(h:\mathcal{Q}_1\rightarrow g_0(\mathcal{C}_2)\)

• is compactly supported on \(\mathcal{Q}_1\),

• has \(\delta\)-small \(\mathcal{C}^k\)-norm,

• vanishes on \(\mathcal{Q}_1\setminus \mathcal{Q}_1'\),

and \(V_0\cup (g_0+h)(\mathcal{Q}_1)\) is polynomially convex. We claim that given \(\delta >0\) there exists a perturbation \(g_1: \mathcal{S}\to \mathbb{C}^n\) of \(g_0\) satisfying

• \(g_1-g_0\) is compactly supported in the \(\delta\)-neighbourhood of \(\mathcal{Q}_1\) in \(\mathcal{S}\),

• \(\Vert g_1-g_0\Vert_{\mathcal{C}^k(\mathcal{S})}<\delta\)

• \(g_1\left(\mathcal{K}\cup \mathcal{R}_0\cup\mathcal{R}_1\right)\) is polynomially convex in \(\mathbb{C}^n\).

Using a partition of unity, we may assume that the compact support of \(h\) that lies in \(\mathcal{Q}_1\subset \mathcal{S}\), is in fact contained in a coordinate chart of \(\mathcal{S}\), which does not intersect \(\mathcal{K}\cup \mathcal{R}_0\cup(\mathcal{R}_1\setminus \mathcal{Q}_1)\). In this chart, we may assume that \(\mathcal{S}= \mathbb{R}^{1}_x \times \mathbb{R}^{\mu-1}_y\), \(\mathcal{Q}_1=\mathbb{R}^{1}_x\times\{0\}\), \(h=h(x)\), and \(\mathbb{R}^1_x\times [-\eta,\eta]^{\mu-1}\) is contained in the \(\delta\)-neighbourhood of \(\mathcal{Q}_1\) in \(\mathcal{S}\). Let \(\sigma\) be a radial bump function supported on \([-\eta,\eta]^{\mu-1}\) with \(\sigma(0)=1\). Choose \(A>0\) such that \(\Vert\sigma\Vert_{\mathcal{C}^k(\mathcal{S})} < A\), and choose \(\delta_1 < \delta/A\). Let \(h\) be the map produced above, but for \(\delta=\delta_1\). Then, the function \(h(x)\sigma(y)\) has compact support in the coordinate chart of \(\mathcal{S}\), and can be extended as 0 to the rest of \(\mathcal{S}\). We denote this extension by \(h_0\). It agrees with \(h\) on \(\mathcal{Q}_1\), vanishes on \(\mathcal{K}\cup \mathcal{R}_0\cup(\mathcal{R}_1\setminus \mathcal{Q}_1)\), and satisfies \(\Vert h_0\Vert_{\mathcal{C}^k(\mathcal{S})}=\Vert h(x) \sigma(y)\Vert_{\mathcal{C}^k(\mathcal{S})} <\delta\). This implies that \(g_1 = g_0+ h_0\) is a perturbation of \(\mathcal{S}\) that satisfies (1)-(3). This proves the claim.

We repeat this procedure inductively. Finally, we have that for any \(\delta>0\), there exists a smooth embedding \(g:\mathcal{S}\rightarrow\mathbb{C}^n\) such that \(f-g\) is compactly supported on the \(\delta\)-neighbourhood of \(\mathcal{R}\), \(f|_K=g|_K\), \(\Vert f-g\Vert_{\mathcal{C}^k(\mathcal{S})}<\delta\), and \(g(\mathcal{K}\cup\mathcal{R})\) is polynomially convex in \(\mathbb{C}^n\).

We now combine this procedure with Lemma 9. Let \(\varepsilon>0\) be as given. Set \(\delta_1=\min\{\delta_\varepsilon,\varepsilon/4c_{\varepsilon}\}\). Let \(p_1\in \mathcal{K}\setminus \operatorname{int}{\mathcal{K}}\) so that \(p_1 \in \mathcal{C}_2\). Let \(\mathcal{R}^1\) be a diffeomorphic image of the cube \(I^\mu \subset \mathbb{R}^\mu\) in \(\mathcal{S}\) such that the \(\delta_1\)-neighbourhood of \(\mathcal{R}^1\) is contained in \(U_{p_1}(\tau_\varepsilon)\) and \(p_1\in\mathcal{R}^1\). By the principal step outlined above, there exists a perturbation \(g_1: \mathcal{S}\to \mathbb{C}^n\) of \(f|_{\mathcal{S}}\) such that \(g_1\) coincides with \(f\) outside a compact set in \(\mathcal{S}\cap U_{p_1}(\tau_\varepsilon)\), satisfies \(\Vert f-g_1\Vert_{\mathcal{C}^k(\mathcal{S})}<\delta_1\), and \(g_1(\mathcal{K}\cup\mathcal{R}^1)\) is polynomially convex. By Lemma 9, (applied to \(f'=f\) itself), there exists a smooth perturbation \(G_1: \mathcal{M}\to \mathbb{C}^n\) of \(f\), which agrees with \(f\) outside a compact subset of \(U_{p_1}(\tau_\varepsilon)\), coincides with \(g_1\) on \(\mathcal{S}\), satisfies \(\mathcal{S}^{G_1}=\mathcal{S}^{f}=\mathcal{S}\), and \[\Vert f-G_1\Vert_{\mathcal{C}^k(\mathcal{M})}<c_\varepsilon\delta_1\leq\frac{\varepsilon}{4}.\] We wish to continue by induction, but \(G_1\) itself may not be real-analytic. We will replace \(G_1\) by a carefully chosen real-analytic approximant of \(G_1\) as follows. Recall that \(P=G_1(\mathcal{K}\cup \mathcal{R}^1)\) is a polynomially convex compact subset of the totally real manifold \(G_1(\mathcal{S})\). By Lemma 10, there is a bounded totally real manifold with smooth boundary \(\Omega\subset G_1(\mathcal{S})\) that contains \(P\) and whose closure is polynomially convex. By [7], there is a \(\delta_{\Omega}>0\) such that any smooth perturbation of \(G_1:G_1^{-1}(\Omega)\rightarrow\mathbb{C}^n\) that is \(\delta_{\Omega}\) close to it in the \(\mathcal{C}^{k-1}\)-norm (on \(G_1^{-1}(\Omega)\)) must be a totally real and polynomially convex embedding of \(G_1^{-1}(\Omega)\). Recall that \(\mathfrak U\subset\mathcal{C}^\infty_{\operatorname{emb}}(\mathcal{M};\mathbb{C}^n)\) is the neighborhood of \(f\) as granted by Lemma 8. Since \((f,G_1)\in\mathfrak U\times\mathfrak U\) (by our choice of \(\widetilde{\varepsilon}\)) and \(\mathcal{S}^f=\mathcal{S}^{G_1}\) (by construction), there exists an \(\varepsilon_\Omega>0\) such that \[\text{if }\Vert G_1-h\Vert_{\mathcal{C}^k}<\varepsilon_\Omega,\quad \text{then } \Vert G_1-h\circ \theta^{f}_{h}\Vert_{\mathcal{C}^{k-1}}<\delta_\Omega.\] Here, we have used that \(\theta^{G_1}_h=\theta^f_{h_1}\) (see (c) in Lemma 8). Now, let \(f_1:\mathcal{M}\rightarrow\mathbb{C}^n\) be a real-analytic embedding of \(\mathcal{M}\) such that \[\Vert f_1-G_1\Vert_{\mathcal{C}^k(\mathcal{M})}<\min\left\{\varepsilon_\Omega,\frac{\varepsilon}{4}\right\}.\] Then, the diffeomorphism \(\theta^{f}_{f_1}=\theta^f_{f_1}:\mathcal{S}^f\rightarrow\mathcal{S}^{f_1}\) satisfies \(\Vert G_1-f_1\circ \theta^{f}_{f_1}\Vert_{\mathcal{C}^{k-1}(\mathcal{M})}<\delta_{\Omega}\). Thus \(\left(f_1\circ\theta^{f}_{f_1}\right)(\mathcal{K}\cup\mathcal{R}^1)\) is polynomially convex in \(\mathbb{C}^n\). We have produced a real-analytic embedding \(f_1:\mathcal{M}\rightarrow\mathbb{C}^n\) such that the following hold:

• \(\Vert f-f_1\Vert_{\mathcal{C}^k(\mathcal{M})}<\varepsilon/2\),

• \((f_1\circ \theta_1)(\mathcal{K}\cup\mathcal{R}^1)\) is polynomially convex in \(\mathbb{C}^n\), for some diffeomorphism \(\theta_1:\mathcal{S}^{f}\rightarrow\mathcal{S}^{f_1}\).

We demonstrate the induction procedure by outlining the next step. Set \(\delta_2=\min\{\delta_\varepsilon,\varepsilon/8c_\varepsilon\}\). Let \(\delta_2^*>0\) be such that \[\label{E:theta} \text{if}\left\Vert f_1\circ \theta^{f}_{f_1}-g_2\circ \theta^{f}_{f_1}\right\Vert_{\mathcal{C}^k(\mathcal{S}^f)}<\delta_2^*,\quad \text{then} \Vert f_1-g_2\Vert_{\mathcal{C}^k(\mathcal{S}^{f_1})}<\delta_2.\tag{26}\] Let \(p_2\in (\mathcal{K}\cup\mathcal{R}^1)\setminus\operatorname{int}(\mathcal{K}\cup\mathcal{R}^1)\). Let \(\mathcal{R}^2\) be a cube in \(\mathcal{S}\) such that the \(\delta_2^*\)-neighbourhood of \(\mathcal{R}^2\) is contained in \(U_{p_2}(\tau_\varepsilon)\) and \(p_2\in\mathcal{R}^2\). As before, by the principal step outlined above, there is a perturbation \(\widetilde{g}_2: \mathcal{S}\to \mathbb{C}^n\) of \(\widetilde{f}_1=(f_1\circ \theta^{f}_{f_1})|_{\mathcal{S}}\) such that \(\widetilde{g}_2\) coincides with \(\widetilde{f}_1\) outside a compact set in \(\mathcal{S}\cap U_{p_2}(\tau_\varepsilon)\), satisfies \(\Vert \widetilde{f}_2-\widetilde{g}_2\Vert_{\mathcal{C}^2(\mathcal{S})}<\delta^*_2\), and \(\widetilde{g}_2(\mathcal{K}\cup\mathcal{R}^1\cup\mathcal{R}^2)\) is polynomially convex. Set \(g_2=\widetilde{g}_2\circ {\theta^{f}_{f_1}}^{-1}\) on \(\mathcal{S}^{f}\). Since \(\Vert f-f_1\Vert_{\mathcal{C}^2(\mathcal{S})}<\varepsilon\), we invoke Lemma 9 for \(f'=f_1\) to obtain a smooth perturbation \(G_2:\mathcal{M}\rightarrow\mathbb{C}^n\) of \(f_1\), that coincides with \(g_2\) on \(\mathcal{S}^{f_1}\), satisfies \(\mathcal{S}^{G_2}=\mathcal{S}^{f_1}\), and \[\Vert f_1-G_2\Vert_{\mathcal{C}^k(\mathcal{M})}<c_\varepsilon\delta_2\leq\frac{\varepsilon}{8}.\] To replace \(G_2\) with a real-analytic approximant, we repeat the idea above for \(P=(G_2\circ\theta^{f}_{f_1})(\mathcal{K}\cup\mathcal{R}^1\cup\mathcal{R}^2)\), to produce a real-analytic embedding \(f_2:\mathcal{M}\rightarrow\mathbb{C}^n\) so that \(\Vert f_1-f_2\Vert_{\mathcal{C}^k(\mathcal{M})}<\varepsilon/4\), and \((f_2\circ\theta_2):(\mathcal{K}\cup\mathcal{R}^1\cup\mathcal{R}^2)\) is polynomially convex for some diffeomorphism \(\theta_2:\mathcal{S}^f\rightarrow\mathcal{S}^{f_2}\).

Repeating this procedure \(N\) times, we obtain an embedding \(f_N:\mathcal{M}\rightarrow\mathbb{C}^n\) and a diffeomorphism \(\theta_N:\mathcal{S}^f\rightarrow\mathcal{S}^{f_N}\) such that \((f_N\circ\theta_N)(\mathcal{K}\cup\mathcal{R}^1\cup\cdots\cup\mathcal{R}^N)\) polynomially convex in \(\mathbb{C}^n\), and \[\Vert F-f\Vert_{\mathcal{C}^2({\mathcal{M}})}\le\Vert f_{N} - f_{N-1}\Vert_{\mathcal{C}^2({\mathcal{M}} )} +\cdots +\Vert f_{1} - f\Vert_{\mathcal{C}^2({\mathcal{M}} )} < \sum_{j=1}^\infty \varepsilon/2^j = \varepsilon.\] By the compactness of \(S\), we can ensure that there is some \(N\) and choice of cubes \(\mathcal{R}^j\)’s so that \(\mathcal{K}\cup\mathcal{R}^1\cup\cdots\cup\mathcal{R}^N=\mathcal{S}\). This proves the proposition. ◻

Proof. Since \(K=\Gamma\cup \gamma\) is not polynomially convex in \(\mathbb{C}^n\), there exists a purely \(1\)-dimensional analytic variety \(X\subset\mathbb{C}^n\setminus K\) such that \[\widehat K=X\cup K.\] Due to the regularity of \(\gamma\) and analyticity of \(\gamma^\sharp\), there is a compact set of zero length \(\varsigma\subset\gamma\) such that for each \(p\in(\gamma^\sharp\setminus \varsigma)\) there is some neighbourhood \(U_p\subset U\) of \(p\), so that \(X\cup(\gamma^\sharp\cap U_p)\) extends past \(\gamma^\sharp\cap U_p\) as a complex curve \(Y_p\) that is regular along \((\gamma^\sharp\setminus \varsigma)\cap U_p\), and in particular, at \(p\). Fix such a \(p\), and let \(\ell_p\) denote the (complex) tangent space of \(Y_p\) at \(p\).

Now, let \(h:U_p\rightarrow\mathbb{C}^n\) be a compactly supported map satisfying (a)-(c). Then, \(\widetilde{\gamma}=F(\gamma)\) differs from \(\gamma\) only in \(U_p\) by a smooth arc \(\sigma\) such that \((\widetilde{\gamma}\setminus\sigma)\cap U_p\) contains a real-analytic subarc, say \(\tau\). Since \(K=\Gamma\cup\widetilde{\gamma}\) is not polynomially convex, there is a one-dimensional variety \(\widetilde{X}\subset\mathbb{C}^n\setminus\widetilde{\gamma}\) such that \[\widehat {\widetilde{K}}= \widetilde{X}\cup {\widetilde{K}}.\] By the regularity of \(\widetilde{\gamma}\), there is a compact set of zero length \(T\subset\widetilde{\gamma}\) such that \(\widetilde{X}\cup\widetilde{\gamma}\) is a smooth manifold with boundary near each point in \(\widetilde{\gamma}\setminus T\). In particular, this is true along some subarc \(\tau'\) of \(\tau\setminus(S\cup T)\). By the reflection principle, \(\widetilde{X}\cup\tau'\) extends analytically past \(\tau'\) as a complex curve, say \(Z_{\tau'}\), that is regular along \(\tau'\). Let \(\widetilde{X}_1\) be the global branch of \(\widetilde{X}\) that continues as \(Z_{\tau'}\) past \(\tau'\). But \(Y_p\) also contains the arc \(\tau'\) in its regular set, as \(\tau'\subset (\gamma^\sharp\setminus S)\cap U_p\). Thus, by the identity principle for analytic curves, it must be that \(Y_p\) is the analytic continuation past \(p\) of \(\widetilde{X}_1\). However, since \(\Gamma\) is polynomially convex, every global branch of \(\widetilde{X}\) must contain \(\gamma\) in its closure, and in particular, the real-analytic subarc \(\tau'\), which implies that \(\widetilde{X}\) has only one global branch. Thus, the regular curve \(\widetilde{\gamma}\) lies in \(Y_p\) and must be tangential to \(\ell_p\) at \(p\). ◻

Proof. First, we prove (i). After removing the last row and the last column in \(\boldsymbol{R}\), we obtain an \((m-2) \times (m-1)\) submatrix, which by 27 , has full rank. Therefore, there exists a column \(j_0\) such that after also removing that column, the remaining square minor is nonzero.

For (ii), observe that the same argument as in (i) shows that there exists a \(k'\in\{1,...,m-1\}\) such that the submatrix obtained by removing the \(k'\)-th and \(m\)-th columns and the \((m-2)\)-th row of \(\boldsymbol{R}\) is nonsingular. If \(k' \ne j_0\) we are done. Suppose that \(k'=j_0\). Let \(T_j\in \mathbb{R}^{m-2}\), \(j=1,\dots, m-1\), be the columns of the matrix \(\boldsymbol{R}\) with the \(m\)-th column and \((m-2)\)-th row removed. By construction, \(\{T_j, j\ne k'\}\) are linearly independent, and therefore, \(T_{k'}\) is a linear combination of \(\{T_j, j\ne k'\}\). By a small modification of \(r^{m-1}_{k'}\), if necessary, we may assume that \(T_{k'} \ne 0\), and therefore, we may find \(k_0\ne k'\) such that \(\{T_j, j\ne k_0\}\) are linearly independent, and this proves (ii). ◻