I would like to understand better this point.
In generalized complex geometry the generalized bundle $E$ is defined as a non-trivial fibration of the cotangent bundle $T^*M$ over the tangent bundle $TM$, where the fibration is given by a gerbe.
1) Is this fibration different from the concept of non-trivial fibre bundle? (I don't know anything about gerbes - I just know something about fibre bundles). I mean, can I write locally $E$ as $TM \times T^*M$? Is it because of this fact that sections of $E$ can be written locally as elements in $TM \oplus T^*M$?
2) About sections. By definition, I would expect a section of (the fibre bundle) $E$ to be a map $s: TM \rightarrow E$ satisfying $\pi \circ s = id_{TM}$. In particular, $s(v)$, with $v \in TM$, should be an element of the fibre $T_v^*M$, right? How does this match the answer to the previous question? (In other words, how does a generalized vector locally written as $X = v + \xi$, with $v,\xi$ in $TM,T^*M$ respectively, act on $w \in TM$? )
