Exercises

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

\mathrm {d} \left(\Gamma \left(\Lambda ^{0,1}\right)\right)\subseteq \Gamma \left(\Lambda ^{2,0}\oplus \Lambda ^{1,1}\right)

X,Y\in \Gamma \left(T^{1,0}\right)\implies [X,Y]\in \Gamma \left(T^{1,0}\right)

Consider an almost complex structure on $\mathbb {R} ^{4}$ $\mathbb {R} ^{4}$ given by $\Lambda ^{1,0}=\langle \sigma ^{1},\sigma ^{2}\rangle$ $\Lambda ^{1,0}=\langle \sigma ^{1},\sigma ^{2}\rangle$ where $\sigma ^{1}=\mathrm {d} z^{1}+a\,\mathrm {d} {\bar {z}}^{2}$ $\sigma ^{1}=\mathrm {d} z^{1}+a\,\mathrm {d} {\bar {z}}^{2}$ , $\sigma ^{2}=\mathrm {d} z^{2}-a\,\mathrm {d} {\bar {z}}^{1}$ $\sigma ^{2}=\mathrm {d} z^{2}-a\,\mathrm {d} {\bar {z}}^{1}$ , and $a$ $a$ is a smooth function of $z^{1}$ $z^{1}$ and $z^{2}$ $z^{2}$ .

1. Check $\Lambda ^{1,0}\cap {\overline {\Lambda ^{1,0}}}={0}$ $\Lambda ^{1,0}\cap {\overline {\Lambda ^{1,0}}}={0}$ , so this does define an almost complex structure
2. Show that condition $(1)$ $(1)$ in exercise 2 is equivalent to

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

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

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 $\mathbb {P} ^{n}$ $\mathbb {P} ^{n}$ .