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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?

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Hint: try some examples, e.g. $F=z^2+w^2-2$ at $(1,1)$. If one first derivative is zero and the other is nonzero, you get another example, e.g. $F=e^{zw}-1$ at $(0,1)$. –  bgins Feb 28 '12 at 7:31
@bgins, hmm.. thanks.. But actually i was reading Levi flat point and Pseudo-convex domain, I want to understand relation of these two concept.... The point $p$ for which above matrix rank is 2 is called is levi flat....If surface is everywhere leviflat 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 two sphere in $\mathbb C^2$ that is $F(z,w)= |z|^2+|w|^2-1=0$ this doesn't satisfy rank 2 condition.... can i have precisely these two situation .. –  zapkm Feb 28 '12 at 7:57
So you got an answer to…, but not to, I see. –  bgins Feb 28 '12 at 8:06
@PradipMishra: I edited the question slightly. I hope that I didn't change the meaning of your last sentence (or of anything else for that matter). I don't have time to think about the answer at the moment, but the definition I am familiar with involves the Levi form. A hypersurface in $\mathbb{C}^2$ has a complex line in its tangent bundle (i.e. the subspace of its tangent space that is invariant under multiplication by $i$). If this is an integrable distribution, then we say it is Levi flat. –  Sam Lisi Mar 2 '12 at 23:29
Levi flat and convex are not the only possibilities. I think you might find the following unpublished book of Cieliebak and Eliashberg to be of interest: Look especially at Chapter 2. I think section 2.4 will relate to your question, but I am not entirely sure. –  Sam Lisi Mar 2 '12 at 23:36

2 Answers 2

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?

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I think you mean $G(z,w)= F(z,w)- F(z_0,w_0)$... I will reply after thinking if this gives some fruitful. thanks. –  zapkm Feb 28 '12 at 8:17
the person who has upvoted this answer, and @bgins, will you please explain this... I will be very grateful... –  zapkm Mar 1 '12 at 11:49
$G(p)$ is just shorthand $G(z_0,w_0)$. I'm afraid my comment was only very elementary, I don't have the background to answer your question (sorry if saying "Hint" was misleading, this was before you had mentioned Levi flatness & pseudoconvexity). However, writing the matrix this way, it seems to show that the (affine/projective) tangent approximations to $G$, $G_z$ & $G_w$ at $p$ are linearly dependent (rank $<3$). Do you have any good references for this stuff? The best I've found so far are some MIT OCW lecture notes. –  bgins Mar 1 '12 at 12:13

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.

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