3. Mappings
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Problem 3.1.
[D’Angelo] Given proper holomorphic maps f,g: \mathbb{B}^n\to \mathbb{B}^N, how does one show that f and g are not homotopic? Find homotopy invariants for proper holomorphic maps between balls.
\bullet Special case (Lebl): Is the Faran map from \mathbb{B}^2 to \mathbb{B}^4 given by (z,w) \to (z^3,\sqrt{3}zw, w^3,0)homotopic to the map (z,w)\to (z,w, 0, 0)? -
Problem 3.2.
[D’Angelo] Let R(z,\overline{z}) be a bihomogeneous polynomial. Find necessary and sufficient conditions such that there exists an integer N with (R(z,\overline{z}))^N=\sum_{j=1}^{m}|p_j(z)|^2.Here p_j(z)’s are linearly independent holomorphic polynomials. See [MR1682713] and [MR2770459]. -
Problem 3.3.
[Ebenfelt] Let R(z,\overline{z}) be a Hermitian polynomial that is a sum of squares (SOS) R(z,\overline{z})=\sum_{j=1}^m|p_j(z)|^2,where p_j(z)’s are linearly independent holomorphic polynomials. Suppose that R(z,\overline{z}) is divisible by ||z||^2 (i.e. R(z,\overline{z})=||z||^2A(z,\overline{z})). What are the possible values of m?
\bullet Huang’s lemma [MR1703603] implies that m is either 0 or at least n.
\bullet See [MR2869101] for a generalization of the Huang’s lemma when ||z||^2 is replaced by ||z||^{2d}. Consider the same question in this case. -
Problem 3.4.
[Zaitsev and Mok] Let D_1 and D_2 be two bounded symmetric domains such that neither is a ball. Let F:D_1\to D_2 be a proper holomorphic map. Is F a trivial map, that is F(z)=(z,g(z)), in suitable coordinates, where g(z) is a vector valued holomorphic function? By a result of Tsai, the answer is yes when the rank (as a bounded symmetric domain) of D_2 is not greater than the rank of D_1. -
Problem 3.5.
[Epstein] It was proven by Eliashberg that any embeddable CR-structure on \mathbb{S}^3 bounds a Stein manifold X\simeq \mathbb{B}^4 with a strictly plurisubharmonic exhaustion function with a single critical point. Can X always be embedded into \mathbb{C}^2?
Cite this as: AimPL: Cauchy-Riemann equations in several variables, available at http://aimpl.org/crscv.