Inventiones mathematicae

, Volume 96, Issue 2, pp 349–383

# Stability and equivariant maps

• Zinovy Reichstein
Article

## Summary

Consider a linearized action of a reductive algebraic group on a projective algebraic varietyX over an algebraically closed field. In this situation Mumford [1] defined the concept of stability for points ofX. Given an equivariant morphismYX we introduce a suitable linearization of the action onY and relate stability inY to stability inX. In particular, we prove a relative Hilbert-Mumford theorem which says that stability inX andY can be tested simultaneously by 1-parameter subgroups. It is hoped that this result will have applications to moduli problems.

In the caseYX is a blowing up of a sheaf of ideals inX we give a simple explicit description of the properly stable and semi-stable loci inY. Our relative Hilbert-Mumford theorem is used here in an essential way.

We also consider the following resolution problem introduced by Kirwan [2]. The equivariant mapYX is called a stable resolution if it is an isomorphism over the properly stable locus inX and every point ofY is either unstable or properly stable. Kirwan [2] gave a canonical procedure for constructing a stable resolution over the complex numbers. We show that a stable resolution can be obtained by resolving the singularities of a naturally defined subvariety ofX. This gives an alternative (non-canonical) procedure for constructing stable resolutions.

## Keywords

Complex Number Algebraic Group Stable Locus Explicit Description Resolution Problem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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