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Algebraic and Analytic K-Stability

Sean T. Paul·Gang Tian
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Abstract

In this note we identify the leading terms of the (reduced) K-energy map with a universal linear combination of the principal and subdominant coefficients of the weight of the $mth$ Hilbert point. This shows that the weight $F_{1}(λ;X)$ introduced by Donaldson in [SKD02] is just the weight of the CM-polarisation.The equivalence between the CM-(semi)stability and the K-(semi) stability follows from this. Also, using our previous work, we are able to describe this subdominant coefficient in terms of the weights of some generalised Chow forms, under a multiplicity free hypothesis on the degeneration. This is accomplished by introducing a parameter dependent lift of the CM-polarisation, and letting this parameter tend to infinity. This could be thought of as a ``quantized'' version of the virtual bundle introduced in [Tian94].

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Sean T. Paul, Gang Tian. 2004-05-27. Algebraic and Analytic K-Stability. https://arxiv.org/abs/math/0405530

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