# Monday, March 22

:::{.remark}
Last time: we reviewed Riemann-Roch, Serre duality, sheaves of $p\dash$forms.
Recall a theorem from a few weeks ago:
:::

:::{.theorem title="The Hodge Theorem"}
If $(X,g)$ is a compact oriented Riemannian manifold, then there is a decomposition of the smooth $p\dash$forms on $X$:
\[
\Omega^p(X) = d \Omega^{p-1}(X) \oplus \mch^p(X) + d^\dagger \Omega^{p+1}(X)
.\]
:::

:::{.remark}
Note that \( \mathcal{H}  \)  was the space of harmonic $p\dash$forms, and \( d^\dagger: \da (-1)^? \hodgestar d\hodgestar \) where
\[
\hodgestar: \Omega^{p}(X) &\to \Omega^{n-p}(X) \\
e_{i_1} \wedge \cdots \wedge e_{i_p} 
&\mapsto 
\pm e_{j_1} \wedge \cdots e_{j_{n-p}}
\]
where \( \ts{ e_i}  \) is an orthonormal basis of basis of \( T\dual X \).
Note that this formula is replacing the $e_i$ that do appear with the $e_i$ that don't appear, up to a sign.
The harmonic forms were defined as \( \mch^p(X) = \ker (dd^\dagger + d^\dagger d ) = \ker (d) \intersect \ker(d^\dagger) \).
We proved that assuming this decomposition, there is an isomorphism
\[
\mch^p(X) \cong H^p_{\dR}(X; \RR)
.\]
:::

:::{.example title="The circle $S^1$"}
There's a standard flat metric $g_\std$ on $S^1$ where $g_\std = \dx^2$ with $x$ the coordinate on $\RR$ which is the universal cover of $S^1$.
We can write 
\[ 
\Omega^1(S^1) = \ts{ f(x)\dx \st f \in C^{\infty }(S^1, \RR) } 
,\] 
since every 1-form \( \omega \) looks like this.
Then \( d \omega = 0 \) since this is a 2-form on $S^1$.
On the other hand, what is $d^\dagger$?
We know that $\hodgestar \omega$ is a 0-form, so a function.
The volume form is given by $\sqrt{ \det g_\std} = \sqrt{ [\dx ^2 ] }$, and you can wedge $1\wedge dx = dx$, so $\hodgestar \omega = f(x)$.
Then $d \hodgestar \omega = f'(x) \dx$ and \( d^\dagger x \omega = f'(x) \).
If this is zero, $f'(x) = 0$ and $f$ is a constant function.
So in this metric, \( \mch^1(S^1) = \RR \gens{ \dx }  \cong H^1(S^1; \RR) \).
:::

:::{.remark title="Important"}
The harmonic forms \( \mch^p(X) \) depend on the metric $g$, despite mapping isomorphically to de Rham cohomology.
:::

:::{.remark}
This was just in the case of a real smooth Riemannian manifold.
What extra structure to we have for \( X \in \Mfd(\Hol(\wait, \CC) ) \)?
:::

:::{.definition title="Kähler Forms (Important!)"}
Let \( X\in \Mfd( \Hol(\wait, \CC) ) \) be a complex manifold.
A **Kähler form** \( \omega\in \Omega^2(X_\RR) \) is a closed real (possibly needed: $J\dash$invariant) 2-form on the underlying real manifold of $X$ for which \( \omega(v, Jw) \da g(v, w) \) is a metric on $TX_\RR$ where $J$ is an almost complex structure.
The associated **hermitian metric** is $h\da g + i \omega$, which defines a hermitian form on $TX \in \Vect_\CC$.
:::

:::{.example title="?"}
Take $X \da \CC^n$ and $J(v) \da i\cdot v$.
Note that $X_\RR = \RR^{2n}$, so write its coordinates as $x_k, y_k$ for $k = 1, \cdots, n$ where $z_k = x_k + iy_k$ are the complex coordinates.
Consider $g = g_\std$ on $\RR^{2n}$ -- does this come from a closed 2-form \( g_\std = \sum (\dx_k)^2 + (dy_k)^2 \)? 
Using \( \omega(v, Jw) = g(v, w) \), we have \( \omega(v, J^2 w) = g(v, Jw) \).
The left-hand side is equal to \( - \omega(v, w) \) and the right-hand side is \( \omega(v, w) = -g(v, Jw) \).
What 2-form does this give?
We have 
\[
\omega\qty{ \dd{}{x_k}, \dd{}{x_\ell} } 
&= -g \qty{ \dd{}{x_k}, \dd{}{y_\ell} } = 0 \\
\omega\qty{ \dd{}{y_k}, \dd{}{x_\ell} } 
&= -g \qty{ \dd{}{y_k}, \dd{}{y_\ell} } = 0 \\
\omega\qty{ \dd{}{x_k}, \dd{}{y_\ell} } 
&= -g \qty{ \dd{}{x_k}, \dd{}{y_\ell} } = 0 && \forall k\neq \ell \\
\omega\qty{ \dd{}{x_k}, \dd{}{y_k} } 
&= -g \qty{ \dd{}{x_k}, \dd{}{y_k} } \\
&= (-1)^2 g \qty{ \dd{}{x_k} , \dd{}{x_k} } \\
&= 1 \\
\omega\qty{ \dd{}{y_k}, \dd{}{x_k} } 
&= -1
.\]
So we can write this in block form using blocks
\[
M = 
\begin{bmatrix}
0 & 1 
\\
-1 & 0
\end{bmatrix} &&
\omega =
\begin{bmatrix}
M &  & 
\\
 & M & 
\\
 &  & M
\end{bmatrix}
,\]
which is a closed ($d\omega = 0$) antisymmetric 2-form, i.e. a symplectic form, and 
\[
\omega_\std = dx_1 \wedge dy_1 + dx_2 \wedge dy_2 + \cdots + dx_n \wedge dy_n
,\]
:::

:::{.remark}
So the Kähler geometry is determined by the data \( (\CC^n, g_\std, J, \omega_\std ) \), i.e. a metric, an almost complex structure, and a symplectic form.
Note that the relation \( \omega(x, y) = g(x, Jy) \) can be used to determine the 3rd piece of data from any 2.
This is the fiberwise/local model, i.e. every tangent space at a point looks like this.
:::

:::{.warnings}
But note that a form being closed is not a tensorial property! 
So this local data (looking at a single fiber) is not quite enough to determine the global geometry.
:::

:::{.remark}
Given $g$ and $J$, \( \omega \) is automatically a 2-form.
That it's antisymmetric follows from 
\[
-\omega(w, v)
&= -g(w, Jv) \\
&= -g(Jv, w) \\
&= -g(J^2 v, Jw)\\
&= g(v, Jw)\\
&= \omega(v, w)
.\]

Conversely, we can always define \( g(v, w) \da - \omega(v, Jw) \), but a priori this may not be a metric.
This will be symmetric, but potentially not positive-definite.
:::

:::{.definition title="$\omega\dash$tame almost complex structures"}
An almost complex structure $J$ is **\( \omega\dash \)tame** if $g(v, w) = - \omega(v, Jw)$ is positive definite.
:::

:::{.remark}
Next time: we'll see that if $X$ is Kähler, then 
\[ 
\mch^k(X) = \bigoplus_{p+q=k} \mch^{p, q}(X), 
\] 
so this is compatible with the Hodge decomposition.
This is what people usually call the Hodge decomposition theorem, and gives some invariants of complex manifolds.
By a miracle, this decomposition only depends on $g$ and the complex structure.
:::

:::{.remark}
Note that there is a notion of *hyperkähler* manifolds, which have 3 complex structures $I, J, K$ such that $I^2=J^2=K^2 = IJK = -\one$, yielding 3 "parallel" 2-forms \( \omega_I, \omega_J, \omega_K \) such that the covariant derivative vanishes, i.e. \( \covariant_g \ts{ \omega_I, \omega_J, \omega_K } = 0 \).
With respect to the complex structure $I$, \( \omega_J + \omega_K \) is a holomorphic 2-form.
There is a sphere's worth of almost complex structures, and there is an action \( \SO(4, b_2 - 4) \actson H^*(X) \).
There's no known example where the hyperkähler metric has been explicitly written down.
:::