Extremal metrics and K-stability (PhD thesis)

By G'abor Sz'ekelyhidi. 2006

50 citations recorded by the source corpus.

In this thesis we study the relationship between the existence of canonical metrics on a complex manifold and stability in the sense of geometric invariant theory. We introduce a modification of K-stability of a polarised variety which we conjecture to be equivalent to the existence of an extremal metric in the polarisation class. A variant for a complete extremal metric on the complement of a smooth divisor is also given. On toric surfaces we prove a Jordan-Holder type theorem for decomposing semistable surfaces into stable pieces. On a ruled surface we compute the infimum of the Calabi functional for the unstable polarisations, exhibiting a decomposition analogous to the Harder-Narasimhan filtration of an unstable vector bundle.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5b395e96bc4fa870a6299d90e37e2ace1a8c0830

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Stability, birational transformations and the Kahler-Einstein problem (2010, background): “The author is very grateful to Alessio Corti, Paolo Cascini, Frances Kirwan, Miles Reid, Julius Ross, Jacopo Stoppa, Gabor Szekelyhidi and Richard Thomas for helpful discussions.”
Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs (2015, methodology): “Székelyhidi [77, 78] proposed that a uniform notion of K-stability should be used to formulate the Yau–Tian–Donaldson conjecture for general polarizations.”