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

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 $\mu$ for varieties and their subschemes; if $(X,L)$ is semistable then $\mu (Z)\le \mu (X)$ 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.

47 pages, 2 figures. Submitted version; typos corrected

References in corpus (3)

• Minimal models and boundedness of stable varieties
• Algebraic and Analytic K-Stability
• An obstruction to the existence of constant scalar curvature Kähler metrics

Cited by in corpus (5)

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• Notes on GIT and symplectic reduction for bundles and varieties
• An obstruction to the existence of constant scalar curvature Kähler metrics
• Relative Ding Stability and an Obstruction to the Existence of Mabuchi Solitons
• Extremal metrics and K-stability