Date of Submission

2-22-1993

Date of Award

2-22-1994

Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name

Mathematics

Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

Supervisor

Ramanan, S.

Abstract (Summary of the Work)

Hitchin (Hi2) realized the importance of studying pairs (E,∅) where E is a vector bundle and ∅ is a homomorphism of E into EOL for a fixed line bundle L. When C is a smooth pro jective algebraic curve this has since been studied quite extensively. One can construct a covering of C in this situation and the given data can be completely recovered by this covering map and a line bundle on the covering curve (see Beauville, Narasimhan and Ramanan (BNR]). When L is the canonical bundle this procedure gives a completely integrable system on the cotangent bundle of the moduli of vector bundles.The aim in the first chapter, which based on a joint work with S. Ra- manan (BR], is firstly to extend some of these results to the case of arbitrary principal bundles, bundles with parabolic structures, etc., but more impor- tantly, to set forth a systematic infinitesimal study of the moduli functor. Some of the results we prove are indeed valid for higher dimensional vari- eties.The computations here are made in the following framework. We start with an algebraic group G and a vector bundle V on a smooth algebraic variety. We consider deformations of pairs (P, Θ) consisting a principal G- bundle P and a section Θ of ad(P) OV satisfying the condition Θ = Θ. We calculate the infinitesimal deformation as the hypercohomology of a complex natually associated to the pair (P, Θ). This result is obtained as a corollary of an identification of infinitesimal deformations of a more general object. We then address the question of smoothness of the related functor and obtain a necessary condition for smoothness.We then take V = K and define a natural 1-form on the moduli space of Higgs bundles on curves; the exterior derivative of this form defines a symplectic structure on the moduli space. If we confine ourselves to the pairs where P is stable, the moduli space can be identified with the cotangent bundle of the moduli space of G-bundles, and the symplectic structure can be identified with the Hamiltonian structure. In the case when G = SL(n), Hitchin considered the global analogue of the map which maps g into Cn-1 given by the coefficients of the characteristic polynomial. We look at the analogue of the Kostant map of g into C for all semisimple groups and show that the fibres are Lagrangian at the smooth points of the fibre and also that the symplectic form vanishes on any smooth variety in the fibre over 0. As in Laumon [L), this leads to the existence of very stable bundles. we then extend these results to pairs with parabolic structures.Now we come to the description of the second chapter.M. Green and R. Lazarsfeld developed a very interesting deformation theory for cohomology of holomorphic vector bundles. Given a family of holomorphic vector bundles Er, on a Kähler manifold X, parametrized by T, the infinitesimal deformation of cohomology is given by a map TI(T) 8 H1(X, E)-H+(X, E). The basic theorem is that the deformation of cohomology is guided by the deformation of the bundle itself. In other words the above map is obtained using the infinitesimal deformation of bun- dles, T(T)H(X, End(E.)), and the cup product, H(X, End(E)) ® H(X,E.)H+1(X, E). In fact, in [GL2), the set up for deformation of cohomology of general elliptic operators is established.

Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28842984

Control Number

ISILib-TH282

Creative Commons License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.

DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

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