Journal of Algebraic Geometry

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

Author(s): Julius Ross; Richard Thomas
Journal: J. Algebraic Geom. 16 (2007), 201-255.
Posted: November 28, 2006
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Abstract: We make a systematic study of the Hilbert-Mumford criterion for different notions of stability for polarised algebraic varieties

; 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

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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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

PII: S 1056-3911(06)00461-9
Received by editor(s): April 1, 2005
Received by editor(s) in revised form: May 1, 2006
Posted: 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.

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