L2a. Linear generalized complex structures.

We want to define linear generalized complex structures (linear gcs) as an analogue to linear complex structures, but $V \oplus V^*$ has something that $V$ does not have: a canonical pairing. This pairing must be preserved by the the linear gcs we are defining right now.

Definition. A linear gcs on $V$ is a linear endomorphism
$$\mathcal{J}: V\oplus V^* \to V\oplus V^*$$ such that $\mathcal{J}^2= - \textrm{Id}$ which preserves the pairing:
$$\langle \mathcal{J}v, \mathcal{J}w \rangle = \langle v,w \rangle.$$
We saw that a complex structure on a vector space $V$ of dimension $n=2m$ can alternatively be seen as a complex subspace $L \subset V_{\mathbb{C}}$ such that $\textrm{dim}_\mathbb{C} V = m$ and $L\cap \overline{L} = 0$. Since a linear gcs on $V$ is, in particular, a usual complex structure on $V\oplus V^*$, we get a complex subspace $L$ of $(V\oplus V^*)_\mathbb{C}$ such that $\textrm{dim}_\mathbb{C} V = n$ and $L\cap \overline{L} = 0$. But the linear gcs satisfies an extra condition, $\mathcal{J}$ preserves the pairing. How does this affect $L \subset (V\oplus V^*)_{\mathbb{C}}$?

First, given two elements $l,l' \in L$, we have
$$\langle \mathcal{J}l, \mathcal{J}l' \rangle = \langle il,il' \rangle = - \langle l, l' \rangle,$$ so $\langle l,l' \rangle = 0$ for any $l,l' \in L$. This means that $L$ must be isotropic. Is this the only condition? Conversely, if $L$ is isotropic, we check that the corresponding $\mathcal{J}$ preserves the metric. Notice that $$V = \{ z + \overline{z} \; :\; z\in V_{\mathbb{C}} \}=\{ z + \overline{z} \; :\; z\in L\oplus \overline{L} \}= \{ z + \overline{z} \; :\; z\in L \}.$$ We thus have, for $v=a+\overline{a}, w=b+\overline{b} \in V$,
$$\langle \mathcal{J}(a+\overline{a}), \mathcal{J}(b+\overline{b}) \rangle = \langle ia - i\overline{a}, ib-i\overline{b} \rangle=\langle a,\overline{b}\rangle + \langle \overline{a},b \rangle=\langle a+\overline{a},  b+\overline{b}\rangle.$$

We have just proved a characterization of a linear gcs:

• A linear gcs on $V$ is given by a complex isotropic subspace $L \subset (V\oplus V^*)_\mathbb{C}$ such that $\dim_\mathbb{C} L = n$ and $L \cap \overline{L} = 0$.

Note that when we talk about the pairing or metric on $(V\oplus V^*)_\mathbb{C}$ -we need it to talk about isotropy-, we are extending the pairing linearly: $$\langle a+ib, c+id \rangle = \langle a, b \rangle  - \langle c,d \rangle + i ( \langle a,d\rangle + \langle b, c \rangle).$$

Actually, $n$ is the maximum possible dimension of an isotropic space in a complex metric space of dimension $2n$, so a linear gcs is given by a maximal isotropic subspace $L \subset (V\oplus V^*)_\mathbb{C}$ such that $L \cap \overline{L} = 0$. We will make some comments about this fact later.