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Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



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

Authors: Julius Ross and Richard Thomas
Journal: J. Algebraic Geom. 16 (2007), 201-255
Published electronically: November 28, 2006
MathSciNet review: 2274514
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Abstract | References | Additional Information

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.

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Additional Information

Julius Ross
Affiliation: Department of Mathematics, Columbia University, New York, New York 10027

Richard Thomas
Affiliation: Department of Mathematics, Imperial College, London SW7 2AZ, United Kingdom
MR Author ID: 636321

Received by editor(s): April 1, 2005
Received by editor(s) in revised form: May 1, 2006
Published electronically: November 28, 2006
Additional Notes: The first author was supported by an EPSRC Ph.D. studentship. The second author was partially supported by the Royal Society and the Leverhulme Trust.