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


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