Proof. A real monomorphism $B:T_pN\to E_p$ is totally real if and only if

$$ \begin{align*}B(e_1)\wedge_{\mathbb{C}}B(e_2)\wedge_{\mathbb{C}}B(e_3)\ne 0.\end{align*} $$

Thus, the subbundle $\mathcal {T}\subset \mathcal {M}$ is exactly the inverse image of the complement of the zero section under the determinant map. If a positive framing of $TN$ is changed by a map to ${\mathrm {GL}}^+(3,\mathbb {R})$ , then $\Delta _A$ is multiplied by the positive real determinant of this change of frame. Hence, the zero set $Z_A$ , its induced orientation and homology class $[Z_A]$ are independent of the chosen positive framing.

We apply obstruction theory to the bundle pair $(\mathcal {M},\mathcal {T})\to N.$ After choosing local positive frames of $TN$ and local complex frames of E, the fibre pair is identified with

$$ \begin{align*}(V,W)=(\mathrm{Mon}_{\mathbb{R}}(\mathbb{R}^3,\mathbb{C}^3),{\mathrm{GL}}(3,\mathbb{C})).\end{align*} $$

The fibre V-deformation retracts to the real Stiefel manifold $V_{6,3}$ , and W-deformation retracts to ${\mathrm {U}}(3)$ . Since $V_{6,3}$ is $2$ -connected, the relative homotopy exact sequence

$$ \begin{align*} \begin{aligned} &\cdots \longrightarrow \pi_3(U(3)) \longrightarrow \pi_3(V_{6,3}) \longrightarrow \pi_3(V_{6,3},U(3)) \overset{\partial}{\longrightarrow} \pi_2(U(3)) \longrightarrow \pi_2(V_{6,3})\\ & \quad \longrightarrow \pi_2(V_{6,3},U(3)) \overset{\partial}{\longrightarrow} \pi_1(U(3)) \longrightarrow \pi_1(V_{6,3}) \longrightarrow \pi_1(V_{6,3},U(3)) \longrightarrow 0 \end{aligned} \end{align*} $$

gives

$$ \begin{align*}\pi_1(V,W)=0, \quad \pi_2(V,W)\cong \pi_1(W)\cong \pi_1({\mathrm{U}}(3))\cong \mathbb{Z}.\end{align*} $$

Under the connecting isomorphism $\pi _2(V,W)\cong \pi _1(W)$ , the generator is detected by the determinant. Indeed,

$$ \begin{align*}\det\nolimits_{\mathbb{C}}:W={\mathrm{GL}}(3,\mathbb{C})\to \mathbb{C}^*\end{align*} $$

induces an isomorphism on $\pi _1$ , since ${\mathrm {SL}} (3,\mathbb {C})\simeq {\mathrm {SU}}(3)$ is simply connected. Equivalently, the preferred generator of $\pi _2(V,W)$ is represented by a small positively oriented normal disk to the codimension-two determinant locus $\{\det \nolimits _{\mathbb {C}}=0\}\subset V$ ; its boundary is a positive meridian.

There is no secondary obstruction in degree $3$ . Indeed, the map $\pi _3(W)\to \pi _3(V)$ is surjective. To see this, write $V\simeq \mathrm {SO}(6)/{\mathrm {SO}}(3)$ . The standard map ${\mathrm {U}}(3)\to \mathrm {SO}(6)$ , obtained by forgetting the complex structure, is an isomorphism on $\pi _3$ ; for instance, this follows from the fibration

$$ \begin{align*}{\mathrm{U}}(3)\longrightarrow \mathrm{SO}(6)\longrightarrow {\mathrm{SO}}(6)/U(3) \cong \mathbb{C} P^3\end{align*} $$

and the fact that $\pi _k(\mathbb {C}P^3)=0$ for $k=3,4$ . The projection ${\mathrm {SO}}(6)\to {\mathrm {SO}}(6)/{\mathrm {SO}}(3)$ is surjective on $\pi _3$ , by the fibration

$$ \begin{align*}{\mathrm{SO}}(3)\longrightarrow {\mathrm{SO}}(6)\longrightarrow {\mathrm{SO}}(6)/{\mathrm{SO}}(3)\end{align*} $$

and $\pi _2(\mathrm {SO}(3))=0$ . Hence, $\pi _3(V,W)=0 .$

We apply obstruction theory to address the question of when it is possible to deform A, relative to $\partial N$ , through sections of $\mathcal {M}$ to a section of $\mathcal {T}$ . Fixing a structure of a CW complex on N, it follows, using the triviality of $\pi _k(V,W)$ for $k=0, 1$ , that A is homotopic to a section that maps the 1-skeleton into $\mathcal {T}$ . Since for all $x\in N$ , $\pi _2(\mathcal {M}_x,\mathcal {T}_x) \cong \pi _2(V,W) \cong \mathbb {Z}$ , there is an obstruction class in

$$ \begin{align*}H^2(N,\partial N;\underline{\pi_2(\mathcal{M}_x,\mathcal{T}_x)})\end{align*} $$

whose vanishing is equivalent to the existence of a section homotopic to A that maps the 2-skeleton into $\mathcal {T}$ . If this obstruction vanishes, it follows from $\pi _3(V,W) = 0$ that A can be homotoped, relative to $\partial N$ , to a section of $\mathcal {T}$ .

Note that the local coefficient system $\underline {\pi _2(\mathcal {M}_x,\mathcal {T}_x)}$ is trivial. The determinant locus in $\mathcal {M}_x$ is independent of the choice of trivialisations of the bundles and, hence, defines a global codimension-two locus in $\mathcal {M}$ . Transporting the generator of $\pi _2(\mathcal {M}_x,\mathcal {T}_x)$ , represented by a positively oriented disk normal to the determinant locus around any loop, will result in a positively oriented normal disk representing the same generator. This follows, since a change of positive frame in $TN$ and a change of complex frame in E preserves the orientations. Alternatively, mapping the fibre pair by the determinant homomorphism $\det _{\mathbb {C}} : (V,W) \to (\mathbb {C},\mathbb {C}^*)$ induces an isomorphism on $\pi _2$ . A change in frames multiplies the determinant by a nowhere-zero complex-valued function and, hence, preserves the generator of $\pi _2(\mathbb {C},\mathbb {C}^*)$ , showing that the local coefficient group $\underline {\pi _2(\Lambda ^3_{\mathbb {C}}E_x,\Lambda ^3_{\mathbb {C}}E_x\setminus \{0\})}$ is trivial. It follows that the only obstruction to deforming A, relative to $\partial N$ , through sections of $\mathcal {M}$ into $\mathcal {T}$ lies in $H^2(N,\partial N;\mathbb {Z}).$

Next, we identify this obstruction. By naturality of obstruction theory, the determinant map sends the primary obstruction for the pair $(\mathcal {M},\mathcal {T})$ to the primary obstruction for the pair $(\Lambda ^3_{\mathbb {C}}E,\Lambda ^3_{\mathbb {C}}E\setminus \{0\}).$ Since the induced map

$$ \begin{align*}(\det\nolimits_{\mathbb{C}})_*:\pi_2(V,W)\to \pi_2(\mathbb C,\mathbb C^*)\end{align*} $$

is an isomorphism, these obstruction classes are identified. The latter obstruction is precisely the obstruction to deforming the determinant section $\Delta _A$ , fixed on $\partial N$ , to a nowhere-zero section of the complex line bundle $\Lambda ^3_{\mathbb C}E$ . Hence, it is the relative first Chern class

$$ \begin{align*}c_1(\Lambda^3_{\mathbb C}E,\Delta_A|_{\partial N})\in H^2(N,\partial N;\mathbb Z),\end{align*} $$

where the nonzero boundary section $\Delta _A|_{\partial N}$ is used as the boundary trivialisation. Since $\Delta _A$ is transverse to the zero section, the usual zero-locus representative of the relative Chern class gives

$$ \begin{align*}c_1(\Lambda^3_{\mathbb{C}}E,\Delta_A|_{\partial N}) =\mathrm{PD}_N[Z_A]\in H^2(N,\partial N;\mathbb{Z}),\end{align*} $$

with $Z_A$ oriented as the zero set of the section $\Delta _A$ of the complex line bundle $\Lambda ^3_{\mathbb {C}}E$ . Hence, the obstruction vanishes if and only if $ [Z_A]=0.$

Finally, since $\partial N$ is compact and $\mathcal {T}$ is open, A takes values in $\mathcal {T}$ on a (smooth) collar of $\partial N$ . Applying the same relative obstruction argument to the complement of this collar and keeping the new boundary fixed, and then extending the resulting homotopy by the constant homotopy on the collar, gives a homotopy fixed on a collar of $\partial N$ .

Proof of Theorem 1.1.

We now take a generic embedding $f:M\hookrightarrow X$ and apply the preceding lemma with

$$ \begin{align*}N=U,\quad E=f^*TX|_U,\quad A=df|_U.\end{align*} $$

Choose a positive global frame $(e_1,e_2,e_3)$ of $TU$ and define

$$ \begin{align*}\Delta=\Delta_{df}=df(e_1)\wedge_{\mathbb{C}}df(e_2)\wedge_{\mathbb{C}}df(e_3) \in \Gamma(U,\Lambda^3_{\mathbb{C}}f^*TX|_U).\end{align*} $$

By genericity, $\Delta $ is transverse to the zero section and

$$ \begin{align*} \Delta^{-1}(0)=L'.\end{align*} $$

Moreover, $\Delta $ is nonzero on $\partial U$ , since f is totally real there. The obstruction to homotoping the real differential $df|_U$ , relative to $\partial U$ , to a totally real section is the class

$$ \begin{align*}[\Delta^{-1}(0)] = [L'] \in H_1(U;\mathbb{Z}).\end{align*} $$

Since $L'=\partial S$ in U, this class vanishes. Hence, by Lemma 2.1, there exists a homotopy of sections

$$ \begin{align*}A_t\in \Gamma(U,\mathcal{M}),\quad t\in[0,1], \end{align*} $$

fixed near $\partial U$ , such that

$$ \begin{align*}A_0=df|_U,\quad A_1\in \Gamma(U,\mathcal T),\end{align*} $$

where now,

$$ \begin{align*}\mathcal{M}=\mathrm{Mon}_{\mathbb{R}}(TU,f^*TX|_U),\quad \mathcal{T}\subset \mathcal M \end{align*} $$

is the subbundle of totally real monomorphisms. By the relative h-principle for totally real embeddings [Reference Gromov7, Section 2.4.5], [Reference Eliashberg and Mishachev4, Section 19.4], this formal homotopy, fixed near $\partial U$ , gives an arbitrarily $\mathcal {C}^0$ -small isotopy supported in U. After this isotopy, the embedding is totally real on U. Since f is unchanged on $M\setminus U$ , and since $ U\cap L=L', $ the CR singular link of the final embedding is precisely $L\setminus L'$ . This proves the theorem.