Towards non-reductive geometric invariant theory

By Brent Doran, Frances Kirwan. 2007

80 citations recorded by the source corpus.

We study linear actions of algebraic groups on smooth projective varieties X. A guiding goal for us is to understand the cohomology of "quotients" under such actions, by generalizing (from reductive to non-reductive group actions) existing methods involving Mumford's geometric invariant theory (GIT). We concentrate on actions of unipotent groups H, and define sets of stable points X^s and semistable points X^{ss}, often explicitly computable via the methods of reductive GIT, which reduce to the standard definitions due to Mumford in the case of reductive actions. We compare these with definitions in the literature. Results include (1) a geometric criterion determining whether or not a ring of invariants is finitely generated, (2) the existence of a geometric quotient of X^s, and (3) the existence of a canonical "enveloping quotient" variety of X^{ss}, denoted X//H, which (4) has a projective completion given by a reductive GIT quotient and (5) is itself projective and isomorphic to Proj(k[X]^H) when k[X]^H is finitely generated.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9c6f25b669a6a8b622c5761af07e7e823f112661

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Richard P. Thomas
2. J. Ross
3. G'abor Sz'ekelyhidi
4. Gergely Bérczi
5. András Szenes
6. Y. Odaka
7. Yanir A. Rubinstein
8. David Witt Nyström

In Mathematical Physics

1. Richard P. Thomas
2. J. Ross
3. Gergely Bérczi
4. András Szenes
5. G'abor Sz'ekelyhidi
6. Y. Odaka
7. Yanir A. Rubinstein
8. David Witt Nyström

Citations in the corpus

• Thom polynomials of Morin singularities (2006, result): “ the action of the nonreductive group Diffd(1). The problem of finding such an analog of the Geometric Invariant Theory of Mumford [36] is addressed in the recent work by Brent Doran and Frances Kirwan [10]; the comparison of our constructions with their results should provide us with new insights. Th”
• Weighted Projective Embeddings, Stability of Orbifolds, and Constant Scarla Curvature Kär Metrics (2009, background): “ sight this seems dicult because Aut(P) is not reductive. Classical GIT works only for reductive groups (though a remarkable amount of the theory has now been pushed through in the nonreductive case [DK]). As a trivial example consider P(1;2) embedded by the identity map in itself. The automorphisms”