what does following matrix says geometrically

Question

Let $M\subset \mathbb C^2$ be a hypersurface defined by $F(z,w)=0$. Then for some point $p\in M$, I've $$\text{ rank of }\left( \begin{array}{ccc} 0 &\frac{\partial F}{\partial z} &\frac{\partial F}{\partial w} \\ \frac{\partial F}{\partial z} &\frac{\partial^2 F}{\partial ^ 2z} &\frac{\partial^2 F}{\partial z\partial w} \\ \frac{\partial F}{\partial w} &\frac{\partial^2 F}{\partial w\partial z} & \frac{\partial^2 F}{\partial w^2} \\ \end{array} \right)_{\text{ at p}}=2.$$

What does it mean geometrically? Can anyone give a geometric picture near $p$?

Any comment, suggestion, please.

Edit: Actually I was reading about Levi flat points and Pseudo-convex domains. I want to understand the relation between these two concepts. A point p for which the rank of the above matrix is 2 is called Levi flat. If the surface is everywhere Levi flat then it is locally equivalent to $(0,1)\times \mathbb{C}^n$, so I have many examples....but what will happen for others for example take the three sphere in $\mathbb{C}^2$ given by $F(z,w)=|z|^2+|w|^2−1=0$. This doesn't satisfy the rank 2 condition. Can I have precisely these two situations?

Answer 1

Let $p=(z_0,w_0)$ and define $G(z,w)=F(z,w)-(z_0,w_0)$. Then the matrix is $$ \left( \begin{matrix} G & G_z & G_w \cr G_z & (G_z)_z & (G_z)_w \cr G_w & (G_w)_z & (G_w)_w \cr \end{matrix} \right)_{\text{at }p} $$ Since $G(p)=0$. Is that any help?

Answer 2

Here is a partial answer: I will givea geometric interpretation of Levi flatness/pseudoconvexity. To fix some notation, let $j$ be endomorphism of the tangent bundle to $\mathbb{C}^2$ induced by its complex structure. (I'm being a bit pedantic, normally we say it is the complex structure, but I want to make it very clear what I am describing.)

If you have a real hypersurface $\Sigma$ in $\mathbb{C}^2$, its tangent bundle has a preferred complex line bundle inside of it. This consists of those vectors in $TM$ such that $j v$ is also in $TM$. Let $\xi$ be this subbundle. We say that $\xi$ is Levi-flat if this distribution is (locally) integrable in the sense of Frobenius.

So what does this mean, geometrically? Suppose that $\Sigma$ is Levi-flat in an open neighbourhood of $p \in \Sigma$. Then, by the Frobenius integrability theorem, you can find a local function $G \colon \Sigma \to \mathbb{R}$ whose level sets have $j$ invariant tangent spaces, i.e. the level set is a complex (local) submanifold of $\mathbb{C}^2$. Again, since we are working locally, this allows you to describe the neighbourhood of $p$ as being of the form $(-\epsilon, \epsilon) \times D^2(\epsilon)$, where $D^2$ is the disk in $\mathbb{C}$.

Levi convexity is a bit harder to explain without appealing to the Levi form. See the reference I gave in the comments above for some definitions and discussion of the concept. In particular, a convex hypersurface in $\mathbb{C}^2$ is Levi convex.

The key fact about flatness/convexity has to do with holomorphic disks whose boundaries are in $\Sigma$. If $\Sigma$ is flat, you can foliate $\Sigma$ locally by such disks. If $\Sigma$ is strictly pseudoconvex, then only the boundary of the disk touches $\Sigma$, the interior of the disk is forced to lie in the interior region bounded by $\Sigma$. (For instance, think of the unit sphere $S^3$ as the typical example of a pseudoconvex hypersurface. Any holomorphic disk with boundary in $S^3$ lives inside the unit ball -- furthermore, only its boundary is allowed to touch the $S^3$.)

In an example like the one you gave, the complex line is $\ker dF \cap \ker dF \circ j$. You then want to compute the two form $\omega := -d (dF \circ j)$ on a pair of (nonzero) vectors $v, jv$, $v \in \xi$. If this is positive, then it is pseudoconvex (at this point). If it is zero, it is Levi-flat.