Type
Text
Type
Dissertation
Advisor
Takhtajan, Leon | Starr, Jason | Lyubich, Mikhail | Kra, Irwin.
Date
2013-12-01
Keywords
Mathematics | action functional, Kähler potential, Moduli, Parabolic bundle, Riemann sphere, singular Hermitian metric | action functional, Kähler potential, Moduli, Parabolic bundle, Riemann sphere, singular Hermitian metric
Department
Department of Mathematics.
Language
en_US
Source
This work is sponsored by the Stony Brook University Graduate School in compliance with the requirements for completion of degree.
Identifier
http://hdl.handle.net/11401/76406
Publisher
The Graduate School, Stony Brook University: Stony Brook, NY.
Format
application/pdf
Abstract
We start this work by reinterpreting the Moduli problem of stable parabolic bundles from a complex analytic perspective. This allows us to introduce canonical complex coordinates on the Moduli space, analogous to the Bers' coordinates on Teichmüller spaces. Similarly, a suitable analog of uniformization will play a central role. Finally, the identification of the tangent space at a point with a certain space of automorphic forms leads to the introduction of the parabolic Narasimhan-Atiyah-Bott metric on the Moduli space, which is analogous to the Weil-Petersson metric. Secondly, for each stable parabolic bundle, a regularized WZNW action functional is defined on the space of singular Hermitian metrics with prescribed asymptotics at a finite set of points in the sphere. We prove that these functionals evaluated at their extrema gives rise to a function on the Moduli space that is a Kähler potential for the parabolic Narasimhan-Atiyah-Bott metric over a certain analytic open subset. | We start this work by reinterpreting the Moduli problem of stable parabolic bundles from a complex analytic perspective. This allows us to introduce canonical complex coordinates on the Moduli space, analogous to the Bers' coordinates on Teichmüller spaces. Similarly, a suitable analog of uniformization will play a central role. Finally, the identification of the tangent space at a point with a certain space of automorphic forms leads to the introduction of the parabolic Narasimhan-Atiyah-Bott metric on the Moduli space, which is analogous to the Weil-Petersson metric. Secondly, for each stable parabolic bundle, a regularized WZNW action functional is defined on the space of singular Hermitian metrics with prescribed asymptotics at a finite set of points in the sphere. We prove that these functionals evaluated at their extrema gives rise to a function on the Moduli space that is a Kähler potential for the parabolic Narasimhan-Atiyah-Bott metric over a certain analytic open subset. | 107 pages
Recommended Citation
Meneses-Torres, Claudio, "Kähler potentials on the Moduli space of stable parabolic bundles over the Riemann sphere | Kähler potentials on the Moduli space of stable parabolic bundles over the Riemann sphere" (2013). Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions). 2329.
https://commons.library.stonybrook.edu/stony-brook-theses-and-dissertations-collection/2329