Multiple vector bundles and linear generalised complex structures

In this thesis we first study multiple vector bundles, which we define as certain functors from an indexing cube category to the category of smooth manifolds. We describe in detail the cores of n-fold vector bundles and we define an n-pullback of an n-fold vector bundle, as well as n-fold analogues of the core sequences for double vector bundles. We prove the existence of splittings and decompositions of multiple vector bundles, thus showing an equivalent definition in terms of n-fold vector bundle atlases. Furthermore, we define multiply linear sections of an n-fold vector bundle and the category of symmetric n-fold vector bundles as n-fold vector bundles equipped with a certain signed action of the symmetric group $S_n$. Secondly, we study linear generalised complex structures on vector bundles. We show the existence of adapted Dorfman connections, which then give adapted linear splittings. This allows to lift the side morphism on $TM\oplus E^*$ to the generalised complex structure in $TE\oplus T^*E$. We describe under which conditions on the side morphism and the Dorfman connection they induce a linear generalised complex structure, furthermore we show the equivalent description in terms of complex VB-Dirac structures in $T_\C E\oplus T_\C^*E$. Then we study the compatibility of a linear generalised complex structure with an additional Lie algebroid structure and we recover the conditions for morphisms of 2-term representations up to homotopy. We prove that the side and core of the aforementioned complex VB-Dirac structures form complex Lie bialgebroids and we study the induced Drinfeld doubles. In the special case of a complex structure we show that these can be recovered from matched pairs of Courant algebroids. Finally, we translate our results to the abstract setting of VB-Courant algebroids, describing in a splitting the compatibility with the corresponding split Lie 2-algebroid.