# Tuesday, April 05

## Symplectic Fillings

:::{.example title="Properties of the standard contact structure on $S^3$"}
Consider $(S^3, \xi_\std) \subseteq \CC^2$; some things that are true:

- There is a symplectic form $\omega = dx_1 \wedge dy_1 + dx_2 \wedge dy_2$ where $d\omega = 0$ and $\omega \wedge \omega =2 d\mathrm{Vol} > 0$.
  Write $\phi = \sum x_i^2 + \sum y_i^2$, then $S^3 = \phi\inv(1)$.
- Letting $\rho = \sum x_i \del x_i + \sum y_i \del y_i = {1\over 2}\grad \phi$ in the standard metric yields a contact form $\alpha = \omega(\rho, \wait)$.
- Since $\ro{\omega}{\xi_\std} > 0$, this yields an area form on contact planes.
- There is also a complex structure $J: \T_p \CC^2 \to \T_p \CC^2$ where $J(\del x_i) = \del y_i$ and $J(\del y_i) = - \del x_i$ with a compatibility $g(x, y) = \omega(x, Jy)$.
:::

:::{.definition title="Fillings"}
A complex symplectic manifold $(X^4, \omega, J)$ is a filling of $(Y^3, \xi)$ if $Y = \bd X$, 

- **Stein filling**: $(X^4, J)$ is a Stein manifold, and $\xi = \T Y \intersect J(\T Y)$.
- **Strong filling**: if there is an outward pointing (Liouville) vector field $\rho$ with $\mcl_\rho \omega = \omega$ with $\xi \ker (\omega(\rho, \wait))$ (which is always contact).
  Note $\mcl_\rho \omega = d(\iota_\rho \omega) + \iota_\rho(d\omega)$ where the 2nd term vanishes for a symplectic form.
- **Weak filling**: $\ro{\omega}{\xi} > 0$.

> Note that we aren't defining what "Stein" means here.

:::

:::{.theorem title="?"}
There are strict implications 

- Stein $\implies$
- Strong $\implies$
- Weak $\implies$
- Tight.

Note that the last implication is the harder part of the theorem.
:::

:::{.problem title="?"}
Given $(Y, \xi)$, classify all fillings.
:::

:::{.example title="?"}
Consider $(T^3, \xi_n)$ -- if $n=1$, this is Stein fillable, and for $n\geq 2$ these are weakly fillable but not strongly fillable.
In this case, all of the filling manifolds are $T^2 \times \BB^2$.
:::

:::{.example title="?"}
For lens spaces $L_{p, q}$ all tight contact structures are Stein fillable with the same smooth filling.
Take the linear plumbing $X$ of copies of $S^2$ corresponding to $-p/q = \tv{r_1, r_2, \cdots, r_k}$ as a continued fraction expansion.
They're distinguished by Chern classes $c_1(T_X, J)$.
:::

:::{.example title="?"}
Brieskorn spheres are examples of fillings, related to Milnor fibers.
For $p,q,r \geq 2$, define
\[
\Sigma(p,q,r) \da \ts{F_{p,q,r}(x,y,z) = x^p + y^q + z^r = \eps} \intersect S^5 \subseteq \CC^3 = \spanof_\CC\ts{x,y,z}
.\]

In this case, we have:

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![](figures/2022-04-05_11-58-26.png)

Note that $\eps = 0$ yields a singular variety, while $\eps>0$ small yields a smooth manifold.
:::

:::{.exercise title="?"}
Show $\Sigma_{p,q,r}$ is the $r\dash$fold cyclic branched cover of $S^3$ over the torus knot $T_{p, q}$.
:::

:::{.remark}
Let $J: \T X\to \T X$ with $J^2 = -\id$, so the eigenvalues are $\pm i$.
So consider complexifying to $\T_\CC X \da \T X \tensor_\RR \CC$, so e.g. $\del x_k \mapsto (a_k + i b_k) \del x_k$.
This splits into positive (holomorphic) and negative (antiholomorphic) eigenspaces $\T^{1, 0}_\CC X \oplus \T^{0, 1}_\CC X$.
Take a change of basis $\tv{x_1, y_1, x_2, y_2} \mapsto \tv{z_1, \bar{z}_1, z_2, \bar{z}_2}$ which yields $\del z = {1\over 2}\qty{\del x - i \del y}$ and $\delbar z = {1\over 2}\qty{\del x + i \del y}$.
:::

:::{.exercise title="?"}
Let $f(z) = \abs{z}^2$ and check

- $\del \delbar f = \del(z d\bar{z}) = dz \wedge d\zbar = -2i (dx \wedge dy)$.
- $d = \del + \delbar$

Practicing this type of change of variables is important!
:::

:::{.definition title="Levi forms and plurisubharmonicity"}
Let $\phi: X\to \RR$ for $X$ a complex manifold, then the **Levi form** of $\phi$ is
\[
\mcl \phi = \del\delbar \phi = \sum_{i, j} {\del^2 \over \del z_j \delbar \bar{z}_k} dz_j \wedge d\bar{z}_k
,\]
generalizing the Hessian.
The function $\phi$ is **plurisubharmonic** if $\mcl \phi$ is positive semidefinite at every point.
:::

:::{.example title="?"}
Consider $\phi: \CC\to \RR$, then 
\[
\mcl \phi 
&= \del\delbar \phi \\
&= 2\qty{{1\over 2}\qty{\phi_x + i\phi_y}}\\
&= {1\over 2}\qty{ {1\over 2}\qty{\phi_{xx} - \phi_{xy}} + {1\over 2}\qty{ \phi_{yx} - i \phi_{yy}}} \\
&= {1\over 4}\qty{\phi_{xx} + \phi_{yy}}\\
&= {1\over 4}\laplacian \phi
,\]
so plurisubharmonic implies positive Laplacian.
Note that in 1 dimension, $\laplacian f = 0 \implies f'' = 0$, so $(x, f(x))$ is a straight line.
In higher dimensions, $f''>0$ forces convexity, so secant lines are under the straight lines, hence the "sub" in subharmonic.
:::

:::{.proposition title="?"}
If $\phi: X\to \RR$ is plurisubharmonic and $0$ is a regular value, then $(\phi\inv(0), \xi)$ (where $\xi$ is its complex tangencies) forms a contact structure and the sub-level set $\phi\inv(-\infty, 0]$ is a Stein filling.
:::

:::{.example title="A basic example of a plurisubharmonic function"}
The radical function $\phi: \CC^3\to \RR$ where $\phi(z_1,z_2,z_3) = \sum \abs{z_i}^2$ is plurisubharmonic, as is its restriction to any submanifold of $\CC^3$, including any filling of $\Sigma_{p,q,r}$.
Hard theorem: any Stein manifold and any Stein filling essentially comes from this construction.
:::