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

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

2004-12-29

arxiv.org/abs/math/0412519v2

Jour. Alg. Geom. 16 (2007), 201--255.

Mathematics

Algebraic Geometry

47 pages, 2 figures. Submitted version; typos corrected

Scientific paper

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.

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