L1b. Complex and symplectic str. on a vector space.

We do geometry by looking together at all the tangent spaces of a manifold: the tangent bundle \( T\). We will do generalized geometry by looking at \( T + T^* \). But before generalizing and before doing any geometry, we'd better understand what happens at a single point. Looking at the tangent space at a point is just looking at a vector space. Let us call this vector space \( V \), and let \( n \) be its dimension.

We used almost complex structures as an example when motivating generalized geometry, so let us talk now about (linear) complex structures on a vector space and let us look at them in many possible ways. This will help us to have different points of view when we generalize them.

• A complex structure on \( V \) is given by an endomorphism \( J: V \to V \) such that \( J^2 = - \textrm{Id} \). This implies that the dimension of \( V \) must be even, \(n=2m \), why?
  Well, by looking at the determinant of the endomorphism \( J \), we have \( \textrm{det}(J)^2=\textrm{det}(J^2)=\textrm{det}(-\textrm{Id})=(-1)^n \), so \( n \) must be even, say, \( n=2m \).

The endormophism \( J \) is completely determined by the \( +i \)-eigenspace as an endormophism of \( V_\mathbb{C} \) and the fact that it is a real endormorphism. This suggests another way of describing a complex structure.

• A complex structure on \( V \) is given by a complex subspace \( L \subset V_\mathbb{C} \) such that \( \dim_\mathbb{C} L = m \) and \( L \cap \overline{L} = 0 \).

There is yet another way of describing a complex structure that will be helpful for us. It is based on the definition using the complex subspace \( L \). Take a basis \( \{ \partial_{z_1}, \ldots, \partial_{z_m} \} \) of \( L \). Then \( \{ {\partial}_{\overline{z}_1}, \ldots, {\partial}_{\overline{z}_m} \} \) is a basis of \( \overline{L} \), and we have the dual basis \( \{ d\overline{z}_1, \ldots, d\overline{z}_m \} \) of \( \overline{L^*}  \). The element \( \phi = d\overline{z}_1\wedge \ldots\wedge d\overline{z}_m \) determines \( L \) as follows:
$$ L = \textrm{Ann}(\phi) = \{ X \in V_\mathbb{C} : i_X \phi = 0 \},$$
where \( i_X \) denotes the interior derivative. This inspires the third way. 

• A complex structure on \( V \) is given by an element \( \phi \in \wedge^m V^*_\mathbb{C} \) satisfying... what does it satisfy?

Good! These three will be enough, but also necessary. Make sure you also understand how to pass from one to the others.


Time for symplectic structures. You are probably working on this now. At first glance, we have that \( \omega \), seen as a skew map \( V \to V^* \) and as an element of \( \wedge^2 V^* \), give analogues to \( J\) and \( \phi \). What about an analogue for \( L \)? Check how you recover \( L \) from \( J \) in a complex structure!