Seminar Algebra and Geometry: Hanspeter Kraft (Basel)

Small G-varieties

Let G be a semisimple algebraic group acting on an affine variety X. An orbit O⊂X is called minimal if it is G-isomorphic to the orbit of highest weight vectors in an irreducible representation of G. These orbits have many interesting properties. E.g. the closure of a minimal orbit in any affine G-variety X is of the form $\bar O = O \cup \{x_0\}$ where x₀ ∈ X is a fixed point, and they are even characterised by this property.
An affine G-variety X is called small if all non-trivial orbits in X are minimal. It turns out that these varieties have many remarkable properties. The most interesting one is that the coordinate ring is a  graded G-algebra. This allows a classification. In fact, there is an equivalence of categories of small G-varieties with so-called fix-pointed k^*-varieties, a class of well-understood objects which have been studied very carefully in different contexts.
A striking consequence is the following result.
Theorem. Let n > 4. Then a smooth $\SL_n$-variety of dimension d < 2n-2 is an $\SL_n$-vector bundle over a smooth variety of dimension d-n. There are also interesting applications to actions of the affine group $\Aff_n$. This was the starting point of this joint work with with Andriy Regeta and Susanna Zimmermann.

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