Special Kähler manifolds and generalized geometry

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