Abstract/Details

The associated map of the nonabelian Gauss-Manin connection


2012 2012

Other formats: Order a copy

Abstract (summary)

In ordinary Hodge theory for a compact kahler manifold, one can look at the Gauss-Manin connection as the complex structure of the manifold varies. The connection satisfies the Griffiths transversality property, and induces a map on the associated graded spaces of the Hodge filtered cohomology spaces of the manifolds. Similarly in nonabelian Hodge theory, where the manifolds are curves and nonabelian cohomology spaces are the moduli spaces of local systems on the curves, the Gauss-Manin connection will be the Isomonodromy deformation, which is a previously known structure on these moduli spaces. One can still define the Hodge filtration and calculate the map induced by the Gauss-Manin connection on the associated graded space. To do this we used deformation theory to express the tangent spaces of the moduli spaces as hypercohomologies of complexes of sheaves over the curves, and write the isomonodromy deformation as a map between such hypercohomology spaces. Under this setting the induced map can be explicitly calculated and is in fact written in an analogous form as the isomonodromy deformation. The induced map turns out to be closely related to another well-known structure called the Hitchin integrable structure, defined on the moduli spaces that correspond to the associated graded spaces. More specifically it is equal up to a factor of 2 to the quadratic Hitchin map.

Indexing (details)


Subject
Applied Mathematics;
Mathematics
Classification
0364: Applied Mathematics
0405: Mathematics
Identifier / keyword
Pure sciences; Applied sciences; Associated maps; Gauss-Manin connection; Manifolds; Nonabelian Hodge theory
Title
The associated map of the nonabelian Gauss-Manin connection
Author
Chen, Ting
Number of pages
21
Publication year
2012
Degree date
2012
School code
0175
Source
DAI-B 73/09(E), Dissertation Abstracts International
Place of publication
Ann Arbor
Country of publication
United States
ISBN
9781267351043
Advisor
Donagi, Ron
Committee member
Block, Jonathan; Donagi, Ron; Pantev, Tony
University/institution
University of Pennsylvania
Department
Mathematics
University location
United States -- Pennsylvania
Degree
Ph.D.
Source type
Dissertations & Theses
Language
English
Document type
Dissertation/Thesis
Dissertation/thesis number
3508983
ProQuest document ID
1019055899
Copyright
Database copyright ProQuest LLC; ProQuest does not claim copyright in the individual underlying works.
Document URL
http://search.proquest.com/docview/1019055899
Access the complete full text

You can get the full text of this document if it is part of your institution's ProQuest subscription.

Try one of the following:

  • Connect to ProQuest through your library network and search for the document from there.
  • Request the document from your library.
  • Go to the ProQuest login page and enter a ProQuest or My Research username / password.