Of course, it is a question about isometries of a Kahler or a hyperkahler manifold, but you already guessed it I suppose.

That's reasonable, thanks. I was thinking of applying Howe duality, but it is too much. I misprinted the question, by the way, it shoud be $A\in GL(2n, {\Bbb R})$ - sorry.

You don't need the base change theorem, just notice that the higher direct images vanish, because the projection is locally equivalent to a product with Stein fibers, and use it to compute the higher direct image explicitly

PU(2) is just as good. There are compact K3 obtained as instanton spaces: one needs to choose a stable U(2)-bundle which cannot be deformed to reducible semistable because its c_1 is not divisible by 2. The corresponding principal PU(2)-bundle has the same deformation space, which is naturally hyperkahler and compact.

Dear Oliver, it was not you who made this claim! However, some people said that it implies this. Of course, if we could obtain the space of Yang-Mills connection on K3 via hyperkahler reduction, this gives a compact hyperkahler K3. Indeed, there are compact components, isomorphic to K3, in the moduli of PGL(2)-instantons.

For the application I have in mind, non-algebraic tori are more than sufficient