Convexity, an explicit example:
2010-10-11
OK, I think some of what I wrote yesterday is wrong, but now I'm going to try to carry out an explicit calculation to point out some subtleties. I'm interested in $X$ being a neighborhood of a symplectic sphere $S$ of square $-1$, and I want to look at three things:
- A Lefschetz fibration $f$ of $X$ over $B^2$ with $S$ contained in the single singular fiber over $0$.
- A symplectic structure on $X$ making $S$ symplectic and all fibers of the $f$ symplectic.
- A Liouville vector field transverse to $\partial X$ and, ideally, defined on all of $X \setminus S$ and behaving well with respect to $f$, whatever that means.
We'll see how far we get - the hard part is number 3, of course.
So we write $X$ as a $0$-handle $H$ and a $2$-handle $H'$, but $H$ is not your ordinary round $0$-handle. Both are subsets of $\R^2 \times \R^2$ with polar coordinates $(r_1,\theta_1,r_2,\theta_2)$ on $H$ and $(r_1',\theta_1',r_2',\theta_2')$ on $H'$. The precise definitions are: \[ H = \{ r_1 r_2 \leq 1, r_1 \leq a, r_1 \leq b \}, \; a,b>1 \] and \[ H' = \{ r_1' \leq a', r_2' \leq 1/a \} \]
Also use polar coordinates $(r,\theta)$ on the base $B^2$ of the Lefschetz fibration.
- The Lefschetz fibration $f$ is defined on $H$ by $f(r_1,\theta_1,r_2,\theta_2) = (r_1 r_2, \theta_1 + \theta_2)$. The $2$-handle $H'$ is glued to the $0$-handle $H$ along $\{r_1' = a'\}$ via the glueing map \[ (\theta_1' = -\theta_1, r_2' = r_2, \theta_2' = \theta_1 + \theta_2 \] Then $f$ transforms appropriately and, on $H'$, is defined by $f(r_1',\theta_1',r_2',\theta_2') = (a r_2,\theta_2)$. This is because, at the boundary where the gluing is happening, $r_1 = a$. But notice that I am doing something fishy here, and I think this is actually important: I'm just describing how to glue along the boundary, not actually giving a gluing of overlapping collar neighborhoods. So in some sense this is just a piecewise smooth manifold, which, I guess, has a canonical smoothing? To properly specify things, I should say that $r_1' = a+a'-r_1$ and $r_2' = r_1 r_2/a$, I think, but this makes the following calculations much messier, if not intractable.
- The symplectic structure $\omega$ on $H$ is the standard $\omega = r_1 dr_1 d\theta_1 + r_2 dr_2 d\theta_2$. This restricts to the region where $H'$ is attached, $\{r_1 = a\}$, as the form $a dr_1 d\theta_1 + r_2 dr_2 d\theta_2$, which then transforms to the attaching region $\{r_1' = a'\}$ as $r_2' dr_2' d\theta_1' + r_2' dr_2' d\theta_2'$, which looks problematic for extending across $H'$. Hmmm... I'm already feeling stuck here... this is surprisingly different than the construction in my paper with Andras Stipsicz. But it's supposed to extend... grrr.
- Well, I may already be out of luck with the Liouville vector field since I can't even get the symplectic structure right here. But here are the calculations I have. I will rather work with the contact forms. On $H$ we have the form $\alpha = \frac{1}{2}(r_1^2 d\theta_1 + r_2^2 d\theta_2)$, with $d\alpha = \omega$. Note also that $dr_1 \wedge \alpha \wedge \omega >0$, which means that $\alpha$ restricts to the region $\{r_1 = a\}$ of $\partial H$ as a positive contact form. Thus, when we get the Liouville vector field $V_\alpha$ such that $\imath_{V_\alpha} \omega = \alpha$, we know it points out along that part of the boundary. Now, pull $\alpha$ back to the attaching region of $\partial $H$ (and use here that $r_1' = a+a'-r_1$ and we get $\alpha' = \ldots$.
Gotta go, will finish this later...