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Stability conditions and canonical metrics
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abstract
In this thesis we study the principle that extremal objects in differential geometry correspond to stable objects in algebraic geometry. In our introduction we survey the most famous instances of this principle with a view towards the results and background needed in the later chapters. In Part I we discuss the notion of a $Z$-critical metric recently introduced in joint work with Ruadha\'i Dervan and Lars Martin Sektnan. We prove a correspondence for existence with an analogue of Bridgeland stability in the large volume limit, and study important properties of the subsolution condition away from this limit, including identifying the analogues of the Donaldson and Yang-Mills functionals for the equation. In Part II we study the recent theory of optimal symplectic connections on K\"ahler fibrations in the isotrivial case. We prove a correspondence with the existence of Hermite-Einstein metrics on holomorphic principal bundles.
Forward citations
Cited by 2 Pith papers
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The deformed Vortex equations and equivariant stability conditions
On vortex-type SU(2)-equivariant bundles over a product of a curve with P¹, solvability of the deformed Hermitian–Yang–Mills system is equivalent to Z-stability, and Bridgeland stability implies Z-stability.
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Deformed Hermitian-Yang-Mills equation on the manifold of full flags
First irreducible rank-2 dHYM connections are constructed on the full flag manifold F₂, and rank-1 solutions outside the supercritical regime disprove conjectured stability conditions.
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