Pseudo-Holomorphic Hamiltonian Systems and Kähler Duality of Complex Coadjoint Orbits
Luiz Frederic Wagner
TL;DR
The work develops a systematic complexification of Hamiltonian dynamics and Lie-group orbits, introducing pseudo-holomorphic Hamiltonian systems (PHHS) as a robust generalization of holomorphic Hamiltonian systems (HHS) to nonintegrable almost complex structures. It proves that PHHSs retain core features of HHSs, including trajectory foliations and action principles, while clarifying how integrability controls these parallels. A central construction twists standard HHS data by compatible tensors to produce proper PHHSs and shows these deformations are generic, linking to Hyperkähler-type structures. In the Lie-group setting, the thesis establishes holomorphic Kähler structures on complex coadjoint orbits, introduces Kähler duality with Hyperkähler structures, and posits a conceptual framework via double cotangent bundles and reduction, though some details (commutation maps and reduction) remain conjectural or outline-level. Overall, the results illuminate deep connections between complexified Hamiltonian geometry, Lefschetz/almost toric fibrations, and dual Kähler structures on orbit spaces, with potential for further development in holomorphic reductions and explicit PHHS constructions.
Abstract
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. These action functionals are well-suited to find periodic orbits. However, it turns out that HHSs rarely exhibit periodic orbits. 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. 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.