Holomorphic invariants, normal forms, and the moduli space of convex domains.(English)Zbl 0658.32015

Consider $$(C^{\infty})$$ smoothly bounded strongly convex domains containing the origin of $${\mathbb{C}}^ n$$ (n$$\geq 2)$$; call two such domains equivalent if there is an origin preserving biholomorphic mapping between them whose differential at the origin is the identity. The space $${\mathcal X}$$ of such domains carries a natural topology, hence so does the moduli space $${\mathcal X}/\sim$$ of equivalence classes. With each $$D\in {\mathcal X}$$ the author associates a triple (I,H,Q) of invariants; I is the Kobayashi indicatrix $$(=the$$ Carathéodory indicatrix) of D at the origin; the smooth hermitian form H and the smooth quadratic form Q are defined on the rank n-1 vector bundle of (1,0) vectors tangent to the boundary of I. Two domains are equivalent if and only if they have the same triple of invariants; it follows that this association defines a homeomorphism of $${\mathcal X}/\sim$$ onto a contractible subset of a Fréchet bundle whose base space $${\mathcal C}=\{I\in {\mathcal X}:$$ I is circular$$\}$$ is an open convex cone in a Frèchet space. When $$n=2$$ the forms H and Q can be replaced with a single quadratic form on the same (line) bundle; the resulting association defines a homeomorphism of $${\mathcal X}/\sim$$ onto an open neighborhood of the zero section in a somewhat simpler Fréchet bundle $${\mathcal E}$$ over $${\mathcal C}$$; this gives $${\mathcal X}/\sim$$ the structure of a smooth Fréchet manifold. To verify that the image of $${\mathcal X}/\sim$$ is open, the author constructs mutually inequivalent 2- dimensional marked Stein manifolds associated with the elements of $${\mathcal E}$$; these manifolds may be regarded as normal forms for the domains in $${\mathcal X}$$ with respect to the given equivalence relation.
The author announced these results in his 45 minute invited address at the Berkeley International Congress; his published report [Proc. Int. Congr. Math. Berkeley/Calif. 1986, Vol. 1, 759-765 (1987)] describes the geometric context of this work and sketches the proof for the case $$n=2$$.
Reviewer: T.J.Barth

MSC:

 32G13 Complex-analytic moduli problems 32F45 Invariant metrics and pseudodistances in several complex variables

Keywords:

Kobayashi indicatrix
Full Text: