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A study of the Hilbert-Mumford criterion for the stability of projective varieties

J. Ross·R. P. Thomas
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Abstract

We make a systematic study of the Hilbert-Mumford criterion for different notions of stability for polarised algebraic varieties $(X,L)$; in particular for K- and Chow stability. For each type of stability this leads to a concept of slope $μ$ for varieties and their subschemes; if $(X,L)$ is semistable then $μ(Z)\leμ(X)$ for all $Z\subset X$. We give examples such as curves, canonical models and Calabi-Yaus. We prove various foundational technical results towards understanding the converse, leading to partial results; in particular this gives a geometric (rather than combinatorial) proof of the stability of smooth curves.

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J. Ross, R. P. Thomas. 2005-02-02. A study of the Hilbert-Mumford criterion for the stability of projective varieties. https://arxiv.org/abs/math/0412519

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