Suppose that $M \subset {\mathbb{C}}^N$ is a real submanifold of real dimension $n.$ Take $p \in M,$ then let $T_p({\mathbb{C}}^N)$ be the tangent vectors of ${\mathbb{C}}^N$ at the point $p.$ If we identify ${\mathbb{C}}^N$ with ${\mathbb{R}}^{2N}$ by $z_j = x_j + i y_j,$ we can take the following vectors as our basis \begin{equation*} \frac{\partial}{\partial x_1} \Bigg\rvert_p, \frac{\partial}{\partial y_1} \Bigg\rvert_p, \ldots, \frac{\partial}{\partial x_N} \Bigg\rvert_p, \frac{\partial}{\partial y_N} \Bigg\rvert_p . \end{equation*} We define a real linear mapping $J\colon T_p({\mathbb{C}}^N) \to T_p({\mathbb{C}}^N)$ such that for any $1 \leq j \leq N$ we have \begin{equation*} J \left( \frac{\partial}{\partial x_1} \Bigg\rvert_p \right) = \frac{\partial}{\partial y_1} \Bigg\rvert_p \qquad \text{ and } J \left( \frac{\partial}{\partial y_1} \Bigg\rvert_p \right) = - \frac{\partial}{\partial x_1} \Bigg\rvert_p . \end{equation*}Where $J$ is referred to as the complex structure on $T_p({\mathbb{C}}^N).$ Note that $J^2 = -I,$ that is applying $J$ twice we just negate the vector.

Let $T_p(M)$ be the tangent space of $M$ at the point $p$ (that is, those vectors of $T_p({\mathbb{C}}^N)$ which are tangent to $M$ .

Do note that the complex tangent space is a real (not complex) vector space, despite its rather unfortunate name.

Let ${\mathbb{C}} T_p(M)$ and ${\mathbb{C}} T_p({\mathbb{C}}^N)$ be the complexified vector spaces, by just allowing the coefficents of the vectors to be complex numbers. That is for $X = \sum a_j \frac{\partial}{\partial x_1} \Big\rvert_p + b_j \frac{\partial}{\partial x_1} \Big\rvert_p$ we allow $a_j$ and $b_j$ to be complex numbers. Next we can extend the mapping $J$ to be ${\mathbb{C}}$ linear on these new vector spaces and still get that $J^2 = -I$ as before. We notice that the operator $J$ has two eigenvalues, $i$ and $-i$

An example of a CR submanifold is for example a hyperplane defined by $\operatorname{Im} z_N = 0$ where the CR dimension is $N-1.$ Another less trivial example is the Lewy hypersurface.

Note that sometimes ${\mathcal{V}}_p$ is written as $T_p^{0,1} (M)$ and referred to as the space of antiholomorphic vectors, where an antiholomorphic vector is a tangent vector which can be written in terms of the basis \begin{equation*} \frac{\partial}{\partial \bar{z}_j} \Bigg\rvert_p := \frac{1}{2} \left( \frac{\partial}{\partial x_j} \Bigg\rvert_p + i \frac{\partial}{\partial y_j} \Bigg\rvert_p \right) . \end{equation*} The CR in the name refers to Cauchy-Riemann and that is because the vector space ${\mathcal{V}}_p$ corresponds to differentiating with respect to $\bar{z}_j.$

Bibliography

1

M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. Real Submanifolds in Complex Space and Their Mappings, Princeton University Press, Princeton, New Jersey, 1999.

2

Albert Boggess. CR Manifolds and the Tangential Cauchy Riemann Complex, CRC, 1991.