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.