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
DOI: https://doi.org/10.1090/S1056-3911-06-00461-9
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
Email: jaross@math.columbia.edu

Richard Thomas
Affiliation: Department of Mathematics, Imperial College, London SW7 2AZ, United Kingdom
Email: richard.thomas@imperial.ac.uk

DOI: https://doi.org/10.1090/S1056-3911-06-00461-9
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.