51 research outputs found
Extremal K\"ahler metrics
This paper is a survey of some recent progress on the study of Calabi's
extremal K\"ahler metrics. We first discuss the Yau-Tian-Donaldson conjecture
relating the existence of extremal metrics to an algebro-geometric stability
notion and we give some example settings where this conjecture has been
established. We then turn to the question of what one expects when no extremal
metric exists.Comment: 17 pages, 4 figures. Contribution to the proceedings of the 2014 IC
Extremal metrics and K-stability
We propose an algebraic geometric stability criterion for a polarised variety
to admit an extremal Kaehler metric. This generalises conjectures by Yau, Tian
and Donaldson which relate to the case of Kaehler-Einstein and constant scalar
curvature metrics. We give a result in geometric invariant theory that
motivates this conjecture, and an example computation that supports it.Comment: 13 pages, v3: fixed typo
Blowing up extremal K\"ahler manifolds II
This is a continuation of the work of Arezzo-Pacard-Singer and the author on
blowups of extremal K\"ahler manifolds. We prove the conjecture stated in [32],
and we relate this result to the K-stability of blown up manifolds. As an
application we prove that if a K\"ahler manifold M of dimension greater than 2
admits a cscK metric, then the blowup of M at a point admits a cscK metric if
and only if it is K-stable, as long as the exceptional divisor is sufficiently
small.Comment: 36 pages, fixed a citatio
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