Toric Varieties Associated with Moduli Spaces

Any genus g surface, Σg,n with n boundary components may be given a trinion decomposition: a realization of the surface as a union of 2g – 2 + n trinions glued together along 3g – 3 + n of their boundary circles. Together with the flows of Goldman, Jeffrey and Weitsman use the trinion boundary circles in a decomposition of Σg,n to obtain a Hamiltonian action of a compact torus (S1)3 g-3+n' on an open dense subset of the moduli space of certain gauge equivalence classes of flat SU(2)-connections on Σg,n. Jeffrey and Weitsman also provide a complete description of the moment polytopes for these torus actions, and we make use of this description to study the cohomology of associated toric varieties. While we are able to make use of the work of Danilov to obtain the integral (rational) cohomology ring in the smooth (orbifold) case, we show that the aforementioned toric varieties almost always possess singularities worse than those of an orbifold. In these cases we use an algorithm of Bressler and Lunts to recover the intersection cohomology Betti numbers using the combinatorial information provided by the corresponding moment polytopes. The main contribution of this thesis is a computation of the intersection cohomology Betti numbers for the toric varieties associated to trinion decomposed surfaces Σ2,0, Σ 2,1, Σ3,0, Σ3,1, Σ4,0, and Σ4,1.

Department dc:contributor.department

Mathematics

Year dc:date.issued

2012

Author dc:creator

• Uren, James

Advisors dc:contributor.advisor

• Jeffrey, Lisa
• Selick, Paul

• Symplectic Geometry
• Toric Geometry

Statement dc:rights

• Attribution 2.5 Canada

Licence dc:rights.uri

• http://creativecommons.org/licenses/by/2.5/ca/

Language dc:language.iso

en_ca

Handle dc:identifier.uri

http://hdl.handle.net/1807/31960

OAI identifier oai:identifier

oai:utoronto.scholaris.ca:1807/31960