Uniform Stability of Twisted Constant Scalar Curvature Kähler Metrics
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Abstract
We introduce a norm on the space of test configurations, called the minimum norm. We conjecture that uniform K-stability is equivalent to the existence of a constant scalar curvature Kähler metric. This uniformity is analogous to coercivity of the Mabuchi functional. We show that a test configuration has zero minimum norm if and only if it has zero L2
-norm, if and only if it is almost trivial.
We prove the existence of a twisted constant scalar curvature Kähler metric that implies uniform twisted K-stability with respect to the minimum norm.
We give algebro-geometric proofs of uniform K-stability in the general type and Calabi-Yau cases, as well as Fano case under an alpha invariant condition. Our results hold for nearby line bundles, and in the twisted setting.
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1687-0247