Canonical Sasaki metrics and AdS/CFT correspondence

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In this project, we investigated geometric invariant theoretic (GIT) stability and relations between the existence of canonical Kahler metrics and GIT stability of polarized toric manifolds. Then we proved the following three results : (1) The derivative of Hilbert series has the same information with the family of integral invariants, which is an obstruction for asymptotic Chow semistability defined by Futaki. (2) We found the first example of an asymptotically Chow unstable Kahler-Einstein manifold. (3) We gave a combinatorial necessary condition for Chow semistability.