# Wednesday, March 31

## Polyvector Fields

:::{.remark}
We have a perfect pairing 
\[
\Omega^k \tensor \Omega^{n-k} \to K
,\]
and thus \( \Omega^{n-k} \cong K \tensor (\Omega^{k})\dual \).
So we have 
\[
H^p( \Omega^k )\dual \cong H^{n-p}( (\Omega^{k})\dual \tensor K  ) = H^{n-p}( \Omega^{n-k}) 
,\] 
and thus \( h^{p, k} = h^{n-p, n-k} \), which recovers what we knew about $\hodgestar: \mathcal{H}^{p, q} \to \mathcal{H} ^{n-p, n-q}$.

So we don't get anything new from the Serre duality argument.

What is special when $X\in \CY$ is that 
\[
\Omega^{n-k} \cong ( \Omega^k )\dual = \Wedge^k TX
\]
for $TX$ the tangent bundle.
Note that taking the cotangent bundle gives forms, and instead this gives a bundle of *polyvector fields*.
For $k=1$, we get a holomorphic vector field, which one might think of as an infinitesimal biholomorphism.

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:::

:::{.example title="?"}
$\PP^1$ has a holomorphic vector field in coordinate charts $\CC \cong \ts{ [z: 1] \in \PP^1 }$ which we'll write as $z\dd{}{z}$.
The coordinate chart is $\PP^1 \sm \infty$, so we obtain

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Does this vector field $V$ extend over $\infty$?
The local coordinate at $\infty$ is $w = 1/z$, so $z=1/w$ and we can compute
\[
{1\over w} \dd{}{ {1\over w} } = {1\over w} {\partial \over {-1\over w^2} \partial w} = -w \dd{}{w}
.\]
We have $\ord_0V = 1$ and $\ord_{\infty } V = 1$, and so $\deg T\PP^1 = 2$.
:::

:::{.example title="?"}
For $\Wedge^2 T$, the local sections are of the form $\sum f_I \dd{}{x_I} \wedge \dd{}{x_J}$ instead of e.g. ${d\over d x_I}$.
This yields a **Poisson structure** $H^0(X, \Wedge^2 T)$, which is a generalization of symplectic structure, which would be a section $\omega \in H^0( X, \Wedge^2 T\dual)$ which is nondegenerate.
This would yield an isomorphism $\omega: T\mapsvia{\sim} T\dual$ which is alternating, in which case $\omega\inv: T\dual \mapsvia{\sim} T$ which is also alternating, so $\omega \inv \in H^0(X, \Wedge^2 T)$.
However the Poisson structure need not be nondegenerate.
:::

:::{.remark}
Polyvector fields show up in Hochschild homology!
:::

## Algebraic Surfaces

:::{.definition title="Algebraic Surface"}
An **algebraic surface** is a compact complex 2-fold (so of complex dimension and real dimension 4, admitting local charts to $\CC^2$) which admits a holomorphic embedding into $\CP^N$ for some $N$. 
:::

:::{.remark}
This implies that $S$ is a **projective variety** cut out by homogeneous polynomials in $N+1$ variables in $\CP^N$.
:::

:::{.example title="?"}
A non-example would be $\CC^2 \sm \ts{ (0, 0) } / (x, y) \sim (2x, 2y)$, The *Hopf surface*.
This is a complex manifold of complex dimension 2.
It is compact, but has no projective embedding!
:::

:::{.example title="?"}
Another non-example is $\CC^2 \smz / (x, y) \sim (2x, 2 e^{i\theta} y)$, a *twisted Hopf surface*.
This admits no nontrivial holomorphic line bundles.
:::

:::{.remark}
What makes having a projective embedding special?
If $S \embeds \CP^N$, it admits a line bundle: $\OO_S(1) \da \ro{ \OO_{\CP^N}(1) }{S}$.
:::

:::{.proposition title="Existence of the Fubini-Study form/metric"}
$\CP^N$ is a Kähler manifold, and admits a distinguished 2-form $\omega \da \omega_{\text{FS}}$ the **Fubini-Study form** which induces the Fubini-Study metric $g_{\text{FS}}$.
:::

:::{.remark}
This can be written down as ${i\over 2} \del \delbar \log( \sum_{i=1}^N z_i \bar{z}_i )$, which is well-defined since scaling comes out as a constant.
Being closed follows from $\del \delbar = d\delbar$ since $\delbar^2 = 0$, which implies $d(\del\delbar \cdots) = d^2 \delbar(\cdots) = 0$.
This defines a metric: this follows from checking in local coordinate charts, say $z_0 = 1$, and checking that $g(x ,y) \da \omega(x, Jy)$ yields a metric. 
This involves taking a fussy derivative!
:::

:::{.remark}
Thus given $S\embedsvia{\phi} \CP^N$, we can restrict or take the pullback of $\omega_{\FS}$ to $S$.
Then $\omega \da \phi^* \omega_\FS$ is still Kähler:

1. $\omega$ is closed: this is true for any smooth map at the level of smooth manifolds because of the chain rule.

2. $\omega$ defines a metric: this is true because $S$ is a complex submanifold.
Suppose $v,w \in T_p S$, and we want to check if $g(v, w) \da \omega(v, Jw)$.
This equals $\omega_{\FS}(v, JW)$, viewing $T_p S \subseteq T_p \CP^N$, so this is equal to $g_\FS(v, w)$.

:::

:::{.remark}
Note that a submanifold of a *symplectic* manifold is not necessarily a symplectic submanifold, since there are Lagrangian submanifolds for which the symplectic form restricts to 0 and isn't nondegenerate.
However, Kähler forms do restrict.
:::

:::{.remark}
So we get a Hodge diamond:

\begin{tikzcd}
	&& {h^{2, 2}} \\
	& {h^{2, 1}} && {h^{1, 2}} \\
	{h^{2, 0}} && {h^{1, 1}} && {h^{0, 2}} \\
	& {h^{1, 0}} && {h^{0, 1}} \\
	&& {h^{0, 0}}
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsOSxbMiw0LCJoXnswLCAwfSJdLFsxLDMsImheezEsIDB9Il0sWzMsMywiaF57MCwgMX0iXSxbMCwyLCJoXnsyLCAwfSJdLFsyLDIsImheezAsIDB9Il0sWzQsMiwiaF57MCwgMn0iXSxbMSwxLCJoXnsyLCAxfSJdLFszLDEsImheezEsIDJ9Il0sWzIsMCwiaF57MiwgMn0iXV0=)


Here $h^{2, 0} = h^0( \Omega^2) = h^0(K) = g$ is called the *genus* in analogy with curves.
Similarly, $h^{1, 0} = h^0( \Omega^1)$ is the space of holomorphic 1-forms, sometimes referred to as the *irregularity*.
There is some symmetry:

\begin{tikzcd}
	&& 1 \\
	& q && q \\
	g && {h^{0, 0}} && g \\
	& q && q \\
	&& 1
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsOSxbMiw0LCIxIl0sWzEsMywicSJdLFszLDMsInEiXSxbMCwyLCJnIl0sWzIsMiwiaF57MCwgMH0iXSxbNCwyLCJnIl0sWzEsMSwicSJdLFszLDEsInEiXSxbMiwwLCIxIl1d)

:::

:::{.exercise title="?"}
Solve for $h^{1, 1}$ in terms of $q$ and $g$.
:::