Moscow-Petersburg seminar on low-dimensional mathematics
January 30, 2004

Guy ROOS
Compactification and volume of bounded symmetric domains

We describe, with the help of Hermitian positive Jordan triple systems
(HPJTS), a canonical projective realization of the compactification of a
bounded symmetric domain. For a natural normalization, the Euclidean volume
of a bounded circled homogeneous complex domain is an integer which is equal
to the projective degree of this compactification.

We will first give some elementary examples: the unit disc and the Riemann
sphere, the Hermitian unit ball and the complex projective space of the same
complex dimension; in these cases, there are natural normalizations of the
flat metric and the Fubini-Study metric, such that all the above spaces have
a volume equal to 1.

A typical example for the general result is the comparison of the Euclidean
volume of the generalized unit ball of rectangular matrices, with the degree
of the Pluecker embedding of the complex Grassmannian.

After this example, we will give the generalization to all irreducible bounded
complex symmetric domains.

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