Departmental PhD Thesis Exam – Mykola Matviichuk

Wednesday, May 20, 2020
1:00 p.m.

PhD Candidate:  Mykola Matviichuk
Supervisor:   Marco Gualtieri
Thesis title:  Quadratic Poisson brackets and co-Higgs fields

This thesis is devoted to studying the geometry of holomorphic Poisson brackets on complex manifolds. We concentrate on the case when the underlying manifold admits a structure of a vector bundle, and the Poisson bracket is invariant under the dilation action of the multiplicative group of the field of complex numbers. We call such a Poisson bracket quadratic, and associate to it a Higgs type tensor, which we call a co-Higgs field. We study the interplay between these two geometric structures. A parallel theory is developed for Poisson brackets on projective bundles. Using the classical tool of the spectral correspondence available for co-Higgs fields, we construct many new examples of Poisson brackets, and provide new classification results in low dimensional cases.

A copy of the thesis can be found here: Quadratic_Poisson_brackets_and_co_Higgs_fields