Open Access

Luiz Frederic Wagner

• This thesis is split up into two parts: The first one concerns (pseudo)-holomorphic Hamiltonian systems, while the second part is about Kähler structures of complex coadjoint orbits. We begin the first part by investigating basic properties of holomorphic Hamiltonian systems (HHSs) like maximal holomorphic trajectories and holomorphic Hamiltonian foliations. Afterwards, we use these notions to combine HHSs with two structures frequently studied in geometry, namely Lefschetz and almost toric fibrations, leading us to the notion of a holomorphic symplectic Lefschetz fibration. Following this examination, we formulate action functionals for HHSs. As usual in classical mechanics and symplectic geometry, varying these action functionals allows us to recover the trajectories of the given HHS. This is particularly interesting in the case where the trajectories are periodic, as periodic orbits are critical points of the action functional. However, it becomes evident during this investigationThis thesis is split up into two parts: The first one concerns (pseudo)-holomorphic Hamiltonian systems, while the second part is about Kähler structures of complex coadjoint orbits. We begin the first part by investigating basic properties of holomorphic Hamiltonian systems (HHSs) like maximal holomorphic trajectories and holomorphic Hamiltonian foliations. Afterwards, we use these notions to combine HHSs with two structures frequently studied in geometry, namely Lefschetz and almost toric fibrations, leading us to the notion of a holomorphic symplectic Lefschetz fibration. Following this examination, we formulate action functionals for HHSs. As usual in classical mechanics and symplectic geometry, varying these action functionals allows us to recover the trajectories of the given HHS. This is particularly interesting in the case where the trajectories are periodic, as periodic orbits are critical points of the action functional. However, it becomes evident during this investigation that HHSs rarely exhibit periodic orbits. As it turns out, one possible obstruction for a HHS to possess periodic orbits is the integrability of the underlying complex structure. To circumvent this obstruction, we introduce the notion of pseudo-holomorphic Hamiltonian systems (PHHSs) which allow us to describe classical mechanics on almost complex manifolds. We show that PHHSs satisfy almost the same properties as HHSs, for instance, they exhibit pseudo-holomorphic Hamiltonian foliations and their trajectories can be described by action functionals. In the second part, we study coadjoint orbits of complex Lie groups and show that they also exhibit, next to their Hyperkähler structure which was introduced by Kronheimer and Kovalev in the 90s, a holomorphic Kähler structure. A holomorphic Kähler structure can be thought of as a complexification of a usual Kähler structure. We call the fact that a space admits both Hyperkähler and holomorphic Kähler structures ''Kähler duality''. In this thesis, we suspect that the Kähler duality of complex coadjoint orbits can be traced back to double cotangent bundles. Precisely speaking, we conjecture that double cotangent bundles naturally exhibit Kähler duality and that this Kähler duality transfers via reduction to complex coadjoint orbits.…