� 12 Lecture 12 - 17 ( Notes : K . Venkatram ) 12 . 1 Generalized Complex Structures and Topological Obstructions

Definition 19. A generalized complex structure (GCS) on E is an integrable orthogonal complex structure J : E E, i.e. a map s.t. → • �JA, JB� = �A,B� • L = Ker (J − i1) Note. 1. �JA,B� = �JA, JB� = −�A, JB�, and thus �J·, ·� is a symplectic struction on E compatible with �, �. 2. L is maximal isotropic and so is L, and thus E = L ⊕ L = L ⊕ L∗ and we get a Lie bialgebroid. 3. V must be even dimensional: letting x ∈ V ⊕ V ∗ be a null vector then �Jx, x� = 0 and �Jx, Jx� = 0, so we can always enlarge a null set by 2 vectors; thus the maximal null set is even. At the level of structure groups, (T ⊕ T ∗, �, �), J corresponds to O(2n, 2n) → U(n, n) = O(2n, 2n) ∩ GL(2n, C). Problem. Show that O(V ⊕ V ∗) acts transitively by conjugation on a set of GCS