# Exercises

1. Prove that the canonical almost complex structure, ${\displaystyle J}$, on a complex manifold is independent of the holomorphic coordinates chosen.
2. Show that the following two conditions for an almost complex structure to be a complex structure are equivalent:

${\displaystyle \mathrm {d} \left(\Gamma \left(\Lambda ^{0,1}\right)\right)\subseteq \Gamma \left(\Lambda ^{2,0}\oplus \Lambda ^{1,1}\right)}$
${\displaystyle X,Y\in \Gamma \left(T^{1,0}\right)\implies [X,Y]\in \Gamma \left(T^{1,0}\right)}$

1. Consider an almost complex structure on ${\displaystyle \mathbb {R} ^{4}}$ given by ${\displaystyle \Lambda ^{1,0}=\langle \sigma ^{1},\sigma ^{2}\rangle }$ where ${\displaystyle \sigma ^{1}=\mathrm {d} z^{1}+a\,\mathrm {d} {\bar {z}}^{2}}$, ${\displaystyle \sigma ^{2}=\mathrm {d} z^{2}-a\,\mathrm {d} {\bar {z}}^{1}}$, and ${\displaystyle a}$ is a smooth function of ${\displaystyle z^{1}}$ and ${\displaystyle z^{2}}$.
1. Check ${\displaystyle \Lambda ^{1,0}\cap {\overline {\Lambda ^{1,0}}}={0}}$, so this does define an almost complex structure
2. Show that condition ${\displaystyle (1)}$ in exercise 2 is equivalent to

${\displaystyle {\frac {\partial a}{\partial {\bar {z}}^{1}}}+a{\frac {\partial a}{\partial z^{2}}}=0={\frac {\partial a}{\partial {\bar {z}}^{2}}}-a{\frac {\partial a}{\partial z^{1}}}}$

1. Deduce that ${\displaystyle T^{0,1}=\left\langle {\dfrac {\partial }{\partial {\bar {z}}^{1}}}+a{\dfrac {\partial }{\partial z^{2}}},{\dfrac {\partial }{\partial {\bar {z}}^{2}}}-a{\dfrac {\partial }{\partial z^{1}}}\right\rangle }$
2. The metric associated to the standard inner product on ${\displaystyle \mathbb {R} ^{4}}$ is almost Hermitian, i.e. ${\displaystyle g(JX,JY)=g(X,Y)}$ for all ${\displaystyle X,Y\in T_{m}\mathbb {R} ^{4}}$. Show that this is equivalent to ${\displaystyle \Lambda ^{1,0}}$ being isotropic for the complexification of ${\displaystyle g}$, i.e. ${\displaystyle g(\sigma ^{i},\sigma ^{j})=0}$ for all ${\displaystyle i,j}$.
3. Express ${\displaystyle J}$ as a ${\displaystyle 4\times 4}$ matrix relative to ${\displaystyle {\dfrac {\partial }{\partial x^{1}}},{\dfrac {\partial }{\partial y^{1}}},{\dfrac {\partial }{\partial x^{2}}},{\dfrac {\partial }{\partial y^{2}}}}$
1. Show that, for ${\displaystyle \omega }$ a Hermitian form, ${\displaystyle \omega (X-iJX,Y-iJY)=0}$.
2. Prove the following lemma:

If ${\displaystyle \nabla _{X}\omega \in \Lambda ^{2}}$ with ${\displaystyle \omega \in \Lambda ^{1,1}}$ then ${\displaystyle \nabla _{X}\omega \in \Lambda ^{2,0}\oplus \Lambda ^{0,2}}$, i.e. ${\displaystyle \nabla _{X}\omega (JX,JY)=-\nabla _{X}\omega (X,Y)}$ Starting hint: ${\displaystyle (\nabla _{X}\omega )(Y,Z)=g\left(\left(\nabla _{X}J\right)Y,Z\right)}$

1. Show that projective manifolds are Kähler. The idea is to show that (complex) projective space is Kähler, which requires us to define the Fubini-Study Metric structure on ${\displaystyle \mathbb {P} ^{n}}$.