 
Summary: Note for a conference
Actions of Linear Algebraic Groups on Projective manifolds
Marco Andreatta
MSC numb.: Prim.:14L30, 14E30 Sec.:14L35, 14J40
Set up. In this talk X will be a smooth projective variety of dimension n on which a
linear connected algebraic group G acts algebraically and non trivially.
We will actually assume that G is semisimple and thus we may also assume that G is
simply connected (via the universal cover) (note that once we assume the existence of a
closed orbit of dimension > 0 then G is semisimple).
General Problem Fixed G, classify such X.
Special case: Homogeneous manifolds.
These are completely classied and they correspond to the parabolic subgroups which in
turn correspond to a choice of a subset of nodes of the Dynkin diagram.
Note that an homogeneous manifold is a rational Fano manifold ( KX is ample and TX
is generated by g.s.) and that they are fundamental examples in various aspects of Alg.
Geom. (for instance Severi varieties, i.e. variety in P m of dimension 2=3(m 2)) which
are embedded into P m 1 by the projection of a general point).
The approach at the general problem we choose is to start with projective varieties of small
dimension with respect to G, i.e. with too many symmetries; to state this properly we
introduce a denition.
