Abstract:
Let a smooth real manifold $M$ be embedded in a complex space $X$. The complex structure of the ambient space $X$ will interact with the smooth structure of $M$. If we trace this interaction at the level of a 1-jet, we note that at each point $p \in M$ in the tangent space there is now a maximal complex subspace $T^{c}_p M$ — a complex tangent. If the dimension of the complex tangent is constant (let's denote it by $n$), then we say that $n$ is the $CR$-dimension of $M$. Let $k$ be the real codimension of a complex tangent in tangent space. If this value coincides with the codimension of $M$ in $X$, then we say that $M$ is a generic submanifold of $X$ whose $CR$-type is $(n,k)$. It is clear that in this case the dimension of $M$ is $(2n+k)$. It is not difficult to give a coordinate description to this picture. Let $X$ be a complex linear space of dimension $n+k$ with two groups of coordinates $z\in \mathbb{C}^n$ and $w\in \mathbb{C}^k$. Then the simplest model of a generic real submanifold of $CR$-type $(n,k)$ — is the real subspace $ \{\mathrm{Im} \, w=0\}$.
Turning to the analysis of the interaction of smooth and complex structures at the level of a 2-jet, we see that the simplest coordinate model is a real algebraic surface, which in the same coordinate space is given by the relation $\mathrm{Im}\, w=<z,\bar{z}>$, where $<z,\bar{z}>$ is a $\mathbb{R}^k$-valued Hermitian form on $\mathbb{C}^n$. If the coordinate Hermitian forms are linearly independent, then this set defines a $k$-dimensional subspace in the space of all Hermitian forms on $\mathbb{C}^n$, which has dimension $n^2$ (we assume that $1\leq k\leq n^2$). Traditionally, such a subspace is called a $k$-dimensional pencil of matrices. There are many interesting questions related to this object, generated by the problems of $CR$ geometry. And if you can get answers for extremal codimensions, namely $k=1, 2, n^2-2, n^2-1, n^2$, then the range of $3\leq k\leq n^2-3$ is basically terra incognita.
Author plans to discuss some of the issues in this area and approaches to their solution.
