Symmetrization of Kähler manifolds

Kähler manifolds, such as smooth projective varieties, are typically intricate, but the presence of symmetries can greatly reduce the complexity, as e.g. seen in the theory of toric varieties.

This project aims to develop methods to symmetrize Kähler manifolds, so that key geometric information more readily can be extracted. More specifically, three different methods of symmetrization will be analysed. The goal is to establish the feasibility of each method and furthermore show that they in fact are equivalent.

Two applications will also be pursued.

The first is to give a new moment map construction of Okounkov bodies, a generalization of moment polytopes from toric geometry, introduced by Okounkov in the 90’s. On this part I will collaborate with Deng Ya, presently at IHES.

The second application is to, using these new tools, prove differentiability properties of the volume of transcendental classes, relating to a fundamental conjecture due to Demailly, postulating a deep connection between two notions of positivity, formulated in terms of duality of cones. The project would run for four years, with me devoting 75% of full time on it.