Abstract

Constructive and algorithmic aspects of Hesselink-type stratifications

Let X ⊂ P(V ) be a projective manifold acted upon by a connected complex reductive Lie group G ⊂ GL(V ). There is a G-equivariant stratification of X, known as Hesselink, or Kirwan-Ness, stratification, whose strata are indexed by certain pairs (β,C ), where β is a one-parameter subgroup of G and C ⊂ X is a connected component of the set of fixed points X^β . The conditions for a pair (β,C ) to occur in this paramatrization are nontrivial, and will be in the focus of this series of talks.

In the case X = P(V ), there is an algorithm due to Popov, using rooted trees, allowing to determine all relevant pairs by a combinatorial procedure. I will describe a generalization of Popov’s method, also applicable in the case where G is real reductive and X ⊂ P(V ) is an analitic locally closed submanifold. The stratification theorem in the latter case is due to Heinzner, Schwarz and Stötzel. There will be an emphasis on the case of homogeneous (under a larger reductive group) varieties, for which many of the constructions can be carried out explicitely and further general properties of the strata can be deduced.

The first lecture will be devoted to the general definitions of the stratification and simple examples. The second - to Popov’s algorithm. The third - to the real case and further examples.