## Departmental PhD Thesis Exam – Bradley Hannigan-Daley

Everyone is welcome to attend. Refreshments will be served in the Math Lounge before the exam.

Friday, June 6, 2014
12:10 p.m.
BA6183, 40 St George St.

Supervisor: Joel Kamnitzer
Thesis title: Hypertoric varieties and wall-crossing

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Abstract:

A hypertoric variety is a quaternionic analogue of a toric variety, constructed as an algebraic symplectic quotient of $T \text{*} \mathbb{C}^n$ by the action of a torus $K$, dependent on a choice of character of $K$. The real Lie coalgebra of $K$ contains a hyperplane arrangement called the discriminantal arrangement, with the property that the hypertoric variety corresponding to a given character $\eta$  depends only on which face of the discriminantal arrangement contains $\eta$. We prove two descriptions of the $\eta$-stability condition in terms of a hyperplane arrangement associated to $K$, and using these we give a new proof of  a theorem of Konno that, given two regular characters separated by a single wall of the discriminantal arrangement, the corresponding hypertoric varieties are related by a Mukai flop. We then apply a theorem of Namikawa to construct an equivalence between the bounded derived categories of coherent sheaves of these two hypertoric varieties. We end with a conjecture that these equivalences give rise to a representation of the Deligne groupoid of the complexifi ed discriminantal arrangement.

The thesis can be found in this link:  http://individual.utoronto.ca/hannigandaley/thesis.pdf