Special Kähler manifolds and generalized geometry
Given a special Kähler manifold ( M , ω , J , ∇ ) we construct a subbundle of the generalized tangent bundle of M endowed with a natural special Kähler structure. Precisely we consider E = T ( M ) ⊕ T ⁎ ( M ) and the subbundle L ω = graph ( ω ) ; we prove that L ω is invariant with respect to the calibrated complex structure J g = ( O − g − 1 g O ) of E defined by the Riemannian metric g = − ω J on M and we define a special connection ∇ ˜ on E by using a natural contravariant connection on T ⁎ ( M ) defined by ω . We prove that ( L ω , ( , ) | L ω , J | L ω g , ∇ ˜ | L ω ) is special Kähler, where ( , ) is the canonical symplectic structure on E . Moreover, by using the identification of T ( M ) ⊕ T ⁎ ( M ) with T ( T ⁎ ( M ) ) defined by the symplectic connection ∇, we describe the corresponding special Kähler subbundle of T ( T ⁎ ( M ) ) . Also we prove that the construction is invariant with respect to the class of connections { ∇ θ } introduced in Alekseevsky et al. [1] .