I'm talking about today about the joint work with Nadimia Schoon.

And I would like to mention that an essential part of this work has been done at the conference

in Moscow for the 80th anniversary of Wienberg in 2017.

And also for that reason we dedicated this paper which we wrote to Wienberg who died

to this day 11 months ago of COVID-19.

Okay now let me start with some very old results.

Let's say first K is any ground field.

D is connected reductive group over K and X is the G-variety.

And now assume first we have the algebraic closed situation that K is equal to K-var.

And then we have a parenter group in G and we call X is called spherical if it contains

an open B orbit.

And then there was a theorem about the other orbits.

I mean an open B orbit there are very easy examples for example the translation group

acting on pn where you have an open orbit but actually everything in the boundary is

a fixed point.

So there might very well be infinitely many orbits but this does not happen.

This is the theorem of Brion and coincidentally it's at the same time proved by Wienberg in

1986 which says that if X is spherical then X has only finitely many orbits.

Actually Wienberg's result was a little bit more general in that it applied not only to

spherical varieties but it had a non-trivial meaning also for arbitrary varieties.

For that I have to define the complexity which is in the title of this talk.

So if H say acts on some variety Y and H is another not necessarily reductive group and

then we define the complexity as the dimension of the generic orbits, I mean the space of

generic orbits.

That means you take the field of rational functions on Y, take its H invariance and

then take its transcendence degree over K and this I call C of Y mod H.

This is completely formal, I do not assume that a quotient exists in any sense.

So this is called the complexity of the action of H and Y, Wienberg called it modularity

but it's the same.

And now the theorem of Wienberg of which this year the first theorem is a corollary of says

that if Y in X is B stable sub-variety then the complexity of Y with respect to B is at

most the complexity of the generic orbits, generic case.

So the most orbits occur always generically.

So the proof was done by a technique which was quite new at that time which it was as

Pfizer remember pioneered by Boho and Kraft and it involves the deformation argument to

so-called horospherical variety.

Don't yet really know what that is and both proofs, Viprion's and Wienberg's used the

same technique.

So a couple of years later Matsuki gave a talk at the ICM in 1990 and he gave a much

shorter proof.

So he gave a simplified proof by reducing it to the rank one, to groups of rank one

by reduction.

Okay and in this very short note, very same note he asked also what about other fields.

Well he was mostly interested in complex numbers and real numbers so he for example asked what

about if K is equal to R.

And he already noted that well what you should do for other fields of course is no Borel subgroup.

The replacement for a Borel subgroup is now a minimal parabolic P. So let's say P and

G is minimal K parabolic and what is definitely not true is that if now everything is defined

over R and if P has a dense open orbit in X then this does not imply that the number

of P orbits in X is finite.