## Departmental PhD Thesis Exam – Lennart Doppenschmitt

Wednesday, August 31, 2022 at 11:00 a.m. (sharp)

PhD Candidate: Lennart Doppenschmitt
Supervisor: Marco Gualtieri
Thesis title: Hamiltonian Geometry of Generalized Kähler Metrics

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Generalized Kähler structures are a natural generalization of Kähler metrics. In this thesis, we pose and investigate the question of finding a generalized Kähler metric with a prescribed volume form in a given generalized Kähler class. This is a natural generalization of the famous Calabi conjecture. We define a generalized Kähler class as a homotopy class of bisections in a holomorphic symplectic Morita equivalence between holomorphic Poisson manifolds. To answer this question we introduce holomorphic families of branes, a novice concept to study variations of generalized complex branes with a complex parameter. We then apply this to families of Lagrangian brane bisections in a symplectic Morita equivalence to analyze variations in generalized Kähler metrics. We construct an almost Kähler metric on the infinite-dimensional space of prequantized generalized Kähler metrics and set up a Hamiltonian group action by gauge transformations. This setup leads to a downward gradient flow of a functional on the space of generalized Kähler metrics towards the metric with prescribed volume form.

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A copy of the thesis can be found here: thesis