Pseudo-Holomorphic Hamiltonian Systems and K\"ahler Duality of Complex Coadjoint Orbits

TL;DR

This thesis explores (pseudo)-holomorphic Hamiltonian systems and the K"ahler duality of complex coadjoint orbits, introducing new structures and conjecturing a link via double cotangent bundles.

Contribution

It introduces pseudo-holomorphic Hamiltonian systems, holomorphic symplectic Lefschetz fibrations, and proposes a conjecture linking K"ahler duality to double cotangent bundles.

Findings

PHHSs generalize HHSs to almost complex manifolds.

Complex coadjoint orbits admit both Hyperk"ahler and holomorphic K"ahler structures.

Conjecture: Double cotangent bundles exhibit K"ahler duality.

Abstract

This thesis is split up into two parts: The first one concerns (pseudo)-holomorphic Hamiltonian systems, while the second part is about K\"ahler 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 Hyperk\"ahler structure which was introduced by Kronheimer and Kovalev in the 90s, a holomorphic K\"ahler structure. A holomorphic K\"ahler structure can be thought of as a complexification of a usual K\"ahler structure. We call the fact that a space admits both Hyperk\"ahler and holomorphic K\"ahler structures ''K\"ahler duality''. In this thesis, we suspect that the K\"ahler duality of complex coadjoint orbits can be traced back to double cotangent bundles. Precisely speaking, we conjecture that double cotangent bundles naturally exhibit K\"ahler duality and that this K\"ahler duality transfers via reduction to complex coadjoint orbits.