Extremal metrics and K-stability (PhD thesis)

Details Extremal metrics and K-stability (PhD thesis) Extremal metrics and K-stability (PhD thesis)

2006-10-31

arxiv.org/abs/math/0611002v1

Mathematics

Differential Geometry

85 pages

Scientific paper

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.

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