Symplectic Realizations of Non-Degenerate Poisson-Nijenhuis Manifolds
Open Access
Author:
Broka, Damien
Graduate Program:
Mathematics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
October 23, 2018
Committee Members:
Ping Xu, Dissertation Advisor/Co-Advisor Ping Xu, Committee Chair/Co-Chair Nigel David Higson, Committee Member Adrian Ocneanu, Committee Member Martin Bojowald, Outside Member Mathieu Philippe Stienon, Committee Member
Symplectic realization is a longstanding problem which can be traced back to Sophus Lie. In this dissertation, we present an explicit solution to this problem for an arbitrary holomorphic Poisson manifold. More precisely, for any holomorphic Poisson manifold $(\mathscr{X}, \pi)$, we prove there exists a holomorphic symplectic structure in a neighborhood $Y$ of the zero section of the real cotangent bundle $T^\vee X$ such that the basepoint projection map is a symplectic realization of $(\mathscr{X}, \pi)$. We describe an explicit construction for such a new holomorphic symplectic structure on $Y \subset T^\vee X$.
