Seminar Algebra and Geometry: Matthias Leuenberger (Universität Bern)

Holomorphic Automorphisms of the Koras-Russell Cubic

The hypersurface X given by x²y+x+z²+t³ = 0 is called Koras-Russell cubic threefold and it played an important role in affine algebraic geometry. It is an example of a threefold that is diffeomorphic to R⁶ but not isomorphic to C³ as an algebraic variety. The latter statement follows from the fact that the Makar-Limanov invariant is non-trivial, which means that there is a lack of algebraic automorphisms compared with C³. However, in the holomorphic setting the situation is completely different: We will see that there are significantly more holomorphic autmorphisms on the Koras-Russel cubic X than algebraic ones. In fact the holomorphic automorphisms act transitively on the cubic. Actually X even has the density property, which means that the group of holomorphic automorphisms is in some sense huge. The question if X is biholomorphic to C³ is still open.

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