Extremal metrics and K-stability (PhD thesis)

TL;DR

This thesis explores the link between canonical metrics and stability in complex geometry, proposing a modified K-stability conjecture and analyzing extremal metrics on various complex surfaces.

Contribution

It introduces a new form of K-stability conjecturally equivalent to extremal metric existence and provides results on decomposing semistable surfaces and computing the Calabi functional.

Findings

Proposes a modified K-stability conjecture for extremal metrics

Establishes a Jordan-Holder type theorem for toric surfaces

Calculates the infimum of the Calabi functional on ruled surfaces

Abstract

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.