Extremal metrics and K-stability (PhD thesis)

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.

85 pages

References in corpus (3)

• Interior estimates for solutions of Abreu's equation
• Notes on GIT and symplectic reduction for bundles and varieties
• A study of the Hilbert-Mumford criterion for the stability of projective varieties

Cited by in corpus (13)

• K-stability of constant scalar curvature Kähler manifolds
• A valuative criterion for uniform K-stability of $\mathbb{Q}$-Fano varieties
• Affine techniques on extremal metrics on toric surfaces
• Bergman kernels on punctured Riemann surfaces
• Weighted projective embeddings, stability of orbifolds and constant scalar curvature Kähler metrics
• Extremal Kähler metrics on projective bundles over a curve
• Torus equivariant K-stability
• Extremal metrics on toric surfaces
• Openness results for uniform K-stability
• Twisted cscK metrics and Kähler slope stability
• A decomposition formula for J-stability and its applications
• The Hitchin-cscK system
• A splitting theorem for extremal Kaehler metrics