Uniform Stability of Twisted Constant Scalar Curvature Kähler Metrics

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.The author was funded by a studentship associated to an EPSRC Career Acceleration Fellowship (EP/J002062/1).This is the author accepted manuscript. The final version is available from Oxford University Press via http://dx.doi.org/10.1093/imrn/rnv29