$M \subset {\mathbb C}^m$ real-analytic submanifold.

$T^{0,1}_p M = \operatorname{span} \left\{ \frac{\partial}{\partial \bar{z}_1} \Big|_p,\ldots,\frac{\partial}{\partial \bar{z}_m}\Big|_p \right\} \cap {\mathbb C} \otimes T_p M$

If $\dim T^{0,1}_p M$ constant $\Rightarrow$ $M$ is CR.

Else $M$ is CR singular.

Points where $\dim T^{0,1}_p M$ jumps are CR singularities.

CR singular manifolds were first studied by Bishop in '65, and then followed a long list of papers among others in ${\mathbb{C}}^2$ Moser-Webster, Moser, Kenig-Webster, Gong Huang-Krants, Huang-Yin, many many others. In higher dimensions, Dolbeault-Tomassini-Zaitsev, Coffman, Huang, Huang-Yin, Fang-Huang, Gong-Stolovich, Gong-L., L.-Noell-Ravisankar, etc.

Definition: If $p$ is a CR singularity and $\exists$ a real-analytic vector bundle $\mathcal V$ near $p$ on $M$ such that for every CR point $q$, ${\mathcal V}_q = T^{0,1}_q M$, then we say $p$ is a removable CR singularity.

If $M$ has only removable CR singularities, then $M$ is a CR image.

Suppose $M$ is a CR image. Then $(M,{\mathcal V})$ is an abstract CR manifold.

Why "CR image"?

The inclusion map $\iota \colon (M,{\mathcal V}) \to {\mathbb C}^m$ is a CR map that is a diffeomorphism onto its image.

As $M$ is real-analytic, then $(M,{\mathcal V})$ is (locally) integrable, so:

For every $q \in M$, $\exists$ a generic $N \subset {\mathbb C}^n$, a neighborhood $U$ of $N$, $W$ a neighborhood of $q$, and a holomorphic $F \colon U \to {\mathbb C}^m$ such that $F(N) = M \cap W$, $F|_N$ is a diffeomorphism, and the rank of $DF$ is $n$ generically.

Recall $N \subset {\mathbb C}^n$ is generic means that $N$ is CR and $\operatorname{CR}\dim N + \operatorname{codim} N = n$.

Observation: The points of $M$ where $F$ is not of rank $n$ are exactly the CR singularities of $M$.

A Bishop surface is a CR image for example:

$F(x,y) = (x+iy,x^2+y^2 + \lambda (x+iy)^2 + \lambda (x-iy)^2)$

takes $N = {\mathbb R}^2$ to $M: w = |z|^2 + \lambda (z^2+\bar{z}^2)$.

When CR dimension is positive, most CR singular manifolds are not CR images.

CR images were first noticed in Ebenfelt-Rothschild '07,

studied more generally in L.-Minor-Shroff-Son-Zhang '14,

The extended structure is the CR Nash blowup, see Garrity '00.

First question is stability: Can a CR singularity can be perturbed away by keeping the CR structure $N$ fixed and perturbing the embedding $F$?

Let $H_p N = T_p N \cap i(T_p N)$.

Let $Z_f$ be the zero set of a function.

Let $C_p Z_f$ be tangent cone of $Z_f$ at $p$.

Suppose first that $n = m$.

The CR singularities of $M$ are the points $F(p)$ for $p \in N$ where $\det DF|_p = 0$.

Theorem: If $Z_{\det DF}$ is "transverse" to $N$ in the sense that

\[ H_p N \not\subset C_p Z_{\det DF} , \]

then a CR singularity at $F(p)$ cannot be eliminated by perturbing $F$.

That is, for all $G$ close enough to $F$, $G(N)$ is CR singular.

Example: $N={\mathbb R}^2 \subset {\mathbb C}^2$, $F(x,y) = (x+iy,x^2)$.

$F(N)$ is the parabolic Bishop surface $w = {(\operatorname{Re} z)}^2$,

$G(x,y) = (x+iy,x^2+i\epsilon x)$ so $\det DG = -i(2x+i\epsilon)$, never zero on $N$ so $G(N)$ is CR.

"Transversality" not satisfied: $H_0 N = \{ 0 \}$

Example: "Transversality" is only sufficient: $N={\mathbb R}^2 \subset {\mathbb C}^2$, $F(x,y) = (x+iy,x^2+y^2)$.

$F(N)$ is the elliptic Bishop surface $w = {|z|}^2$, and the CR singularity cannot be removed.

"Transversality" not satisfied: $H_0 N = \{ 0 \}$

The theorem follows from a lemma that may be of independent interest.

Lemma: Suppose $N \subset U \subset {\mathbb C}^n$ is generic, $\varphi \colon U \to {\mathbb C}$ holomorphic, $\varphi(p)=0$ for some $p \in N$ and

\[ H_p N \not\subset C_p Z_{\varphi} . \]

Then every holomorphic $\psi$ sufficiently close to $\varphi$ must vanish on $N$.

Difficulties: $Z_{\varphi}$ could be singular, or $d\varphi$ may vanish at $p$.

If $N$ would be one dimensional complex manifold instead of a generic submanifold, the correct statement would be Rouché and the "transversality" would be that $N \not\subset Z_\varphi$.

Example: The lemma is not true if $\varphi$ not holomorphic. Let $N = {\mathbb R} \subset {\mathbb C}$ and $\varphi(z) = {(\operatorname{Re} z)}^2$.

$N$ and $Z_\varphi$ are "transverse".

$\psi(z) = {(\operatorname{Re} z)}^2 + \epsilon$ is close but never zero.

Returning to CR images $N \subset {\mathbb C}^n$, $F(N) = M \subset {\mathbb C}^m$.

Example: Define $M$ by $w_1 = {|z|}^6 , w_2 = {|z|}^4$.

$M$ is an image of $N = {\mathbb R}^2$. $n=2$, $m=3$.

$M$ does not lie in a 2-dimensional complex submanifold.

Let $k$ be the real codimension of $N$.

If \[ 4n-k < 2(m+1) , \]

then near every point of $N$ there exists $G$ arbitrarily close to $F$ such that $G(N)$ is CR.

The inequality is sharp: For any $n$, $k$, and $m$ such that $4n-k \geq 2(m+1)$, there exist $N$ and $F$ such that $F(N)$ is CR singular and $G(N)$ is CR singular for all small enough perturbations $G$ of $F$.

In fact, for such dimensions, such an example exists near any CR image.

Our second question: Invariants?

Every codimension 2 CR image $M \subset {\mathbb C}^n$ is biholomorphic to exactly one of the following forms:

1) $w = \bar{z}_1 z_2 + \bar{z}_1^2 + O({\|z\|}^3)$,

2) $w = \bar{z}_1 z_2 + O({\|z\|}^3)$,

3) $w = {|z_1|}^2 + a \bar{z}_1^2 + O({\|z\|}^3)$, $a \geq 0$,

4) $w = \bar{z}_1^2 + O({\|z\|}^3)$,

5) $w = O({\|z\|}^3)$.

(Proved in a previous paper)

In fact, for every $N$, there exists an $F$ such that $F(M)$ is in any one of the above forms.

There exist higher order invariants of $M$ that are not invariants of $N$.

Example:

$M_1 : w = \bar{z}_1 z_2$

$M_2 : w = \bar{z}_1 z_2 + \bar{z}_1^3$

are not bioholomorphically equivalent, but have the same quadratic form.

Our last question is identifying removable CR singularities intrinsically.

The answer is not difficult in ${\mathbb C}^3$, but ... we really don't have time ...

Thank you for your attention!