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3. Mappings

    1. 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)?
          It is known that the Faran map is homotopic to this embedding when the target dimension is 5, and it is not homotopic when the target dimension is 3.
        • 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.