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:
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.
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 \( L \cap \overline{L} = 0 \).
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.
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