G. Roos (St. Petersburg)
"Symplectic duality for symmetric spaces"

Abstract. It is well-known that the complex unit disc and the complex space
are not biholomorphic, by Liouville's theorem. However, they are
diffeomorphic; moreover, there exists a diffeomorphism which maps the flat
metic of the unit disc onto the Fubini-Study metric restricted to the
complex plane. The same diffeomorphism maps the hyperbolic metric of the
unit disc to the flat metric of the complex plane. (I have found no trace of
this elementary result in the classical literature on functions of one
complex variable).

A. Di Scala (Torino) and A. Loi (Cagliari) have shown (2005) that this
situation extends to all classical bounded symmetric domains and their
ambient vector space; the diffeomorphism they use is a map introduced in
1995 by G. Roos to show that the volume of a bounded symmetric domain,
suitably but canonically normalized, is equal to the algebraic degree of
some projective embedding of its compact dual. It has been proved by G. Roos
in 2006 that the result of Di Scala and Loi holds for all bounded symmetric
domains; the proof is independent of the classification of these domains.

The result is a new indication for the importance of considering symplectic
Kaehlerian structures in complex multivariate analysis.