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Extra resources for Hermitian forms meet several complex variables: Minicourse on CR geometry

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1. 1: Moduli space of degree 2 proper maps of B2 to BN in the s,t plane. The minimal target dimension N is marked for each vertex, edge, and the interior. 44 CHAPTER 3. PROPER MAPS OF BALLS Similarly, the moduli space for n dimensions is an n + 1 dimensional simplex with one vertex for the identity, and the other vertices for maps obtained by tensoring with the identity on 1, 2, . . , n dimensional subspace. We will prove that this is the moduli space for all rational degree 2 proper maps of balls.

1 (Rudin ’84). Suppose that F : Bn → BN be proper and a homogeneous polynomial of degree d, then F is spherically equivalent (actually unitarily equivalent) to Hn,d . There is an elegant proof of this theorem by D’Angelo. Proof. 13) Now take any w ∈ Cn . Write w = tz, where t ≥ 0 and z = 1. 14) Hence F(w) = Hn,d (w) for all w ∈ Cn . 15) If F already had the same number of components as Hn,d then the zeros are not necessary. We can also only tensor on a subspace. For example, let z = (z , zn ). Then we can create the so-called Whitney map z ⊕ (zn ⊗ z) = (z1 , z2 , .

Notice the quantitative aspects of the theorem. 4 (D’Angelo). Let M ⊂ Cn be an real-analytic hypersurface defined by r(z, z¯) = 0 near p and let ∆ be a polydisc centered at p such that the series for r converges in a neighborhood of ∆ × ∆∗ . 49) r(z, z¯) = 2 Re h(z) + f (z) 2 − g(z) 2 , be the holomorphic decomposition of r where h, f , and g are holomorphic in ∆ and vanish at p, and f and g are 2 valued. Then M contains a germ of complex variety at p if and only if there exists a unitary U such that V (U, p) is nontrivial.

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