3 hours of lectures per week.
Lecturer
Marcel Bökstedt
Content
Geometric invariant theory is about how to construct quotients of algebraic varieties under the action of algebraic groups. These quotients are just the usual quotient spaces, but provided with an essential algebraic structure.
The course is a topology course, but it is all-important in this subject to consider the algebraic aspects at all times. The standard text [1] is steeped in notions from algebraic geometry. It is impossible to avoid these notions entirely, but you can actually do a lot to soften their impact. I will use topological ideas and proofs in preference to algebraic whenever this seems possible.
One effect of this choice is that most of the time we will only consider actions on varieties defined over the complex number. The theory is actually far more general than this.
Applications of the theory are for instance the construction of various moduli spaces.
Some subjects will be:
Reductive linear groups.
Variants of reductive groups.
Geometrical description of stable points, according to Kempf, Ness and Kirwan.
Example:
The conjugation action of PGLn on matrices.
The Hilbert-Mumford criterion.
Example:
The action of PGLn on sets of subspaces.
Quotients by reductive groups: The affine case. The projective case.
Prerequisites
Commutative rings on the level of Algebra 2. Some background in geometry/topology.
Literature
[1] Mumford, Fogarty, Kirwan : Geometric Invariant Theory, third edition, Springer-Verlag 1994.
[2] Notes, articles.
Evaluation
Students who do not intend to take a degree in Mathematics or Statistics from the University of Aarhus, but wish to earn credits for a 2.dels course from the Department of Mathematics, should indicate at the beginning of the course that they wish to be examined.
The form of examination for these students will be active participation together with oral or written contributions.
Credits
10 ECTS
Semester
Spring 2003