Date:	Tue, November 9, 2010
Time:	14:00
Place:	Research I Seminar Room

Abstract: Let M be a smooth complex variety. It is called Stein if there is an embedding of M into an n-dimensional complex space as a closed subvariety. Let G be a group acting freely on a variety M. It is good to know whether or not quotient M/G is Stein.

To begin with I assume that G_+=(C, +) and provide examples of free G-actions such that quotients M/G_+ are(and are not) Stein. There is a more abstract definition of Stein varieties due to H. Cartan: M is a Stein variety if and only if cohomologies H^i(M, F) vanishes for any i>0 and any holomorphic sheaf F on M. A corollary of this abstract definition is that if M/G_+ is Stein then M is Stein too and moreover M=M/G_+ x G_+.

I will explain next theorem due to Y. Matsushima: Let G be an affine complex(=Stein) reductive group (say GL_n(C)) and H be a subgroup. Then the quotient G/H is Stein if and only if H is a reductive subgroup (good example is GL_{n-1}(C)).