Extremal metrics and K-stability (PhD thesis)
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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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Cited by 3 Pith papers
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Poincar\'e type J-equation
A two-parameter continuity path characterizes solvability of the Poincaré-type J-equation on Kähler manifolds with divisor singularities, showing subsolutions imply solutions on surfaces and K-energy boundedness under...
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A lower bound on the Calabi functional for a degeneration of polarized varieties
A lower bound on the Calabi functional for degenerations of polarized varieties is proven in terms of CM degree differences, viewed as a discretely valued version of Donaldson's bound.
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Existence of Conical Higher cscK Metrics on a Minimal Ruled Surface
Existence of conical higher cscK metrics is proven in every Kähler class on pseudo-Hirzebruch surfaces via momentum construction, with polyhomogeneous regularity and a conjectural cone-angle relation from the top log ...
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