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K-Stabilty of Kahler Manifolds

Ossareh, Siavosh | 2022

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 55345 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Bahraini, Alireza
  7. Abstract:
  8. The Yau-Tian-Donaldson conjecture is a challenging problem in complex geometry which currently caught the attention of some birational geometers. This conjecture is about a formulation of the problem of finding the best metric over a Kahler manifold in terms of an asymptotic GIT stability condition. The first stability condition was proposed by Mumford in ICM 1962 which surprisingly just needs the vector bundle to be“more ample” among its subbundles. Another important theorem by Narasimhan and seshadri provides a connection between the slope stability and the existence constant curvature connections for the flat holomorphic vector bundles over compact Rieman surfaces. In 1983, Donaldson proved a generalistion of the theorem to arbitrary holomorphic vector bundles over compact Rieman surfaces. Later on, the theorem was generalised to Donaldson-Uhlenbeck-Yau theorem. The idea behind the proof was imposing the Hilbert-Mumford criterion to the infinite-dimensional action of Hamiltonaian automorphisms to complex structures. In this thesis, we will scrutinize the Donaldson’s generalisation of Narasimhan-Seshadri theorem then we will define the basics of the infinite-dimensional action and finally we will state the YTD conjecture
  9. Keywords:
  10. Kahler Manifold ; Yau-Tian-Donaldson Conjecture ; Mumford-Tate Theorem ; K-Stability ; Mumford Stability ; Narasimhan-Seshadri Theorem

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