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By Chern S., Osserman R. (eds.)

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Then there exists a Newton diagram D with support K ⊃ K such that D |K = D and such that #(D ) ≤ #(D) and such that K contains all points ( j, k) with j + k ≤ m. Proof. We change the 0-points at level k into alternating N and P-points, one connected group of 0-points of level k at a time. There must be at least one P or N-point at the k-level. There are two cases that we should consider. In the first case, the group of 0-points has N or P-points on both sides. In this case no new nodes can possibly be created.

Let U(N, 1) be the set of matrices M such that M ∗V M = V . U(N, 1) is called the generalized unitary group. It is left as an easy exercise that up to scaling U(N, 1) represent linear fractional transformations preserving the sphere. The scaling comes about because we are working in homogeneous coordinates. Therefore, the set of linear fractional transformations that preserve the sphere is the Lie group SU(N, 1) (the generalized special unitary group), that is, matrices in U(N, 1) with determinant 1.

It is not necessarily true that this must extend to a unitary operator on 2 (for example it could happen that codimension of H2 is finite, while codimension of H1 is infinite). But we are allowed to direct sum with a zero component, producing f˜ and g, ˜ ensuring that the codimensions are infinite. Since all infinite dimensional separable Hilbert spaces are isometric, we can extend U to a unitary operator on the resulting space. The result follows. Notice that the theorem implies that any germ of a subvariety extends to a subvariety of ∆.

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